YES Problem 1: (VAR v_NonEmpty:S M:S N:S V1:S V2:S X:S X1:S X2:S X3:S) (RULES U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ) (STRATEGY INNERMOST) Problem 1: Dependency Pairs Processor: -> Pairs: U11#(active(X1:S),X2:S,X3:S) -> U11#(X1:S,X2:S,X3:S) U11#(mark(X1:S),X2:S,X3:S) -> U11#(X1:S,X2:S,X3:S) U11#(X1:S,active(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S) U11#(X1:S,mark(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S) U11#(X1:S,X2:S,active(X3:S)) -> U11#(X1:S,X2:S,X3:S) U11#(X1:S,X2:S,mark(X3:S)) -> U11#(X1:S,X2:S,X3:S) U12#(active(X1:S),X2:S,X3:S) -> U12#(X1:S,X2:S,X3:S) U12#(mark(X1:S),X2:S,X3:S) -> U12#(X1:S,X2:S,X3:S) U12#(X1:S,active(X2:S),X3:S) -> U12#(X1:S,X2:S,X3:S) U12#(X1:S,mark(X2:S),X3:S) -> U12#(X1:S,X2:S,X3:S) U12#(X1:S,X2:S,active(X3:S)) -> U12#(X1:S,X2:S,X3:S) U12#(X1:S,X2:S,mark(X3:S)) -> U12#(X1:S,X2:S,X3:S) U13#(active(X1:S),X2:S,X3:S) -> U13#(X1:S,X2:S,X3:S) U13#(mark(X1:S),X2:S,X3:S) -> U13#(X1:S,X2:S,X3:S) U13#(X1:S,active(X2:S),X3:S) -> U13#(X1:S,X2:S,X3:S) U13#(X1:S,mark(X2:S),X3:S) -> U13#(X1:S,X2:S,X3:S) U13#(X1:S,X2:S,active(X3:S)) -> U13#(X1:S,X2:S,X3:S) U13#(X1:S,X2:S,mark(X3:S)) -> U13#(X1:S,X2:S,X3:S) U14#(active(X1:S),X2:S,X3:S) -> U14#(X1:S,X2:S,X3:S) U14#(mark(X1:S),X2:S,X3:S) -> U14#(X1:S,X2:S,X3:S) U14#(X1:S,active(X2:S),X3:S) -> U14#(X1:S,X2:S,X3:S) U14#(X1:S,mark(X2:S),X3:S) -> U14#(X1:S,X2:S,X3:S) U14#(X1:S,X2:S,active(X3:S)) -> U14#(X1:S,X2:S,X3:S) U14#(X1:S,X2:S,mark(X3:S)) -> U14#(X1:S,X2:S,X3:S) U15#(active(X1:S),X2:S) -> U15#(X1:S,X2:S) U15#(mark(X1:S),X2:S) -> U15#(X1:S,X2:S) U15#(X1:S,active(X2:S)) -> U15#(X1:S,X2:S) U15#(X1:S,mark(X2:S)) -> U15#(X1:S,X2:S) U16#(active(X:S)) -> U16#(X:S) U16#(mark(X:S)) -> U16#(X:S) U21#(active(X1:S),X2:S) -> U21#(X1:S,X2:S) U21#(mark(X1:S),X2:S) -> U21#(X1:S,X2:S) U21#(X1:S,active(X2:S)) -> U21#(X1:S,X2:S) U21#(X1:S,mark(X2:S)) -> U21#(X1:S,X2:S) U22#(active(X1:S),X2:S) -> U22#(X1:S,X2:S) U22#(mark(X1:S),X2:S) -> U22#(X1:S,X2:S) U22#(X1:S,active(X2:S)) -> U22#(X1:S,X2:S) U22#(X1:S,mark(X2:S)) -> U22#(X1:S,X2:S) U23#(active(X:S)) -> U23#(X:S) U23#(mark(X:S)) -> U23#(X:S) U31#(active(X1:S),X2:S) -> U31#(X1:S,X2:S) U31#(mark(X1:S),X2:S) -> U31#(X1:S,X2:S) U31#(X1:S,active(X2:S)) -> U31#(X1:S,X2:S) U31#(X1:S,mark(X2:S)) -> U31#(X1:S,X2:S) U32#(active(X:S)) -> U32#(X:S) U32#(mark(X:S)) -> U32#(X:S) U41#(active(X:S)) -> U41#(X:S) U41#(mark(X:S)) -> U41#(X:S) U51#(active(X1:S),X2:S) -> U51#(X1:S,X2:S) U51#(mark(X1:S),X2:S) -> U51#(X1:S,X2:S) U51#(X1:S,active(X2:S)) -> U51#(X1:S,X2:S) U51#(X1:S,mark(X2:S)) -> U51#(X1:S,X2:S) U52#(active(X1:S),X2:S) -> U52#(X1:S,X2:S) U52#(mark(X1:S),X2:S) -> U52#(X1:S,X2:S) U52#(X1:S,active(X2:S)) -> U52#(X1:S,X2:S) U52#(X1:S,mark(X2:S)) -> U52#(X1:S,X2:S) U61#(active(X1:S),X2:S,X3:S) -> U61#(X1:S,X2:S,X3:S) U61#(mark(X1:S),X2:S,X3:S) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,active(X2:S),X3:S) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,mark(X2:S),X3:S) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,X2:S,active(X3:S)) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,X2:S,mark(X3:S)) -> U61#(X1:S,X2:S,X3:S) U62#(active(X1:S),X2:S,X3:S) -> U62#(X1:S,X2:S,X3:S) U62#(mark(X1:S),X2:S,X3:S) -> U62#(X1:S,X2:S,X3:S) U62#(X1:S,active(X2:S),X3:S) -> U62#(X1:S,X2:S,X3:S) U62#(X1:S,mark(X2:S),X3:S) -> U62#(X1:S,X2:S,X3:S) U62#(X1:S,X2:S,active(X3:S)) -> U62#(X1:S,X2:S,X3:S) U62#(X1:S,X2:S,mark(X3:S)) -> U62#(X1:S,X2:S,X3:S) U63#(active(X1:S),X2:S,X3:S) -> U63#(X1:S,X2:S,X3:S) U63#(mark(X1:S),X2:S,X3:S) -> U63#(X1:S,X2:S,X3:S) U63#(X1:S,active(X2:S),X3:S) -> U63#(X1:S,X2:S,X3:S) U63#(X1:S,mark(X2:S),X3:S) -> U63#(X1:S,X2:S,X3:S) U63#(X1:S,X2:S,active(X3:S)) -> U63#(X1:S,X2:S,X3:S) U63#(X1:S,X2:S,mark(X3:S)) -> U63#(X1:S,X2:S,X3:S) U64#(active(X1:S),X2:S,X3:S) -> U64#(X1:S,X2:S,X3:S) U64#(mark(X1:S),X2:S,X3:S) -> U64#(X1:S,X2:S,X3:S) U64#(X1:S,active(X2:S),X3:S) -> U64#(X1:S,X2:S,X3:S) U64#(X1:S,mark(X2:S),X3:S) -> U64#(X1:S,X2:S,X3:S) U64#(X1:S,X2:S,active(X3:S)) -> U64#(X1:S,X2:S,X3:S) U64#(X1:S,X2:S,mark(X3:S)) -> U64#(X1:S,X2:S,X3:S) ACTIVE(U11(tt,V1:S,V2:S)) -> U12#(isNatKind(V1:S),V1:S,V2:S) ACTIVE(U11(tt,V1:S,V2:S)) -> ISNATKIND(V1:S) ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> U13#(isNatKind(V2:S),V1:S,V2:S) ACTIVE(U12(tt,V1:S,V2:S)) -> ISNATKIND(V2:S) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> U14#(isNatKind(V2:S),V1:S,V2:S) ACTIVE(U13(tt,V1:S,V2:S)) -> ISNATKIND(V2:S) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> U15#(isNat(V1:S),V2:S) ACTIVE(U14(tt,V1:S,V2:S)) -> ISNAT(V1:S) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> U16#(isNat(V2:S)) ACTIVE(U15(tt,V2:S)) -> ISNAT(V2:S) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U16(tt)) -> MARK(tt) ACTIVE(U21(tt,V1:S)) -> U22#(isNatKind(V1:S),V1:S) ACTIVE(U21(tt,V1:S)) -> ISNATKIND(V1:S) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> U23#(isNat(V1:S)) ACTIVE(U22(tt,V1:S)) -> ISNAT(V1:S) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U23(tt)) -> MARK(tt) ACTIVE(U31(tt,V2:S)) -> U32#(isNatKind(V2:S)) ACTIVE(U31(tt,V2:S)) -> ISNATKIND(V2:S) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(U32(tt)) -> MARK(tt) ACTIVE(U41(tt)) -> MARK(tt) ACTIVE(U51(tt,N:S)) -> U52#(isNatKind(N:S),N:S) ACTIVE(U51(tt,N:S)) -> ISNATKIND(N:S) ACTIVE(U51(tt,N:S)) -> MARK(U52(isNatKind(N:S),N:S)) ACTIVE(U52(tt,N:S)) -> MARK(N:S) ACTIVE(U61(tt,M:S,N:S)) -> U62#(isNatKind(M:S),M:S,N:S) ACTIVE(U61(tt,M:S,N:S)) -> ISNATKIND(M:S) ACTIVE(U61(tt,M:S,N:S)) -> MARK(U62(isNatKind(M:S),M:S,N:S)) ACTIVE(U62(tt,M:S,N:S)) -> U63#(isNat(N:S),M:S,N:S) ACTIVE(U62(tt,M:S,N:S)) -> ISNAT(N:S) ACTIVE(U62(tt,M:S,N:S)) -> MARK(U63(isNat(N:S),M:S,N:S)) ACTIVE(U63(tt,M:S,N:S)) -> U64#(isNatKind(N:S),M:S,N:S) ACTIVE(U63(tt,M:S,N:S)) -> ISNATKIND(N:S) ACTIVE(U63(tt,M:S,N:S)) -> MARK(U64(isNatKind(N:S),M:S,N:S)) ACTIVE(U64(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S))) ACTIVE(U64(tt,M:S,N:S)) -> PLUS(N:S,M:S) ACTIVE(U64(tt,M:S,N:S)) -> S(plus(N:S,M:S)) ACTIVE(isNat(plus(V1:S,V2:S))) -> U11#(isNatKind(V1:S),V1:S,V2:S) ACTIVE(isNat(plus(V1:S,V2:S))) -> ISNATKIND(V1:S) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> U21#(isNatKind(V1:S),V1:S) ACTIVE(isNat(s(V1:S))) -> ISNATKIND(V1:S) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNat(0)) -> MARK(tt) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> U31#(isNatKind(V1:S),V2:S) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> ISNATKIND(V1:S) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> U41#(isNatKind(V1:S)) ACTIVE(isNatKind(s(V1:S))) -> ISNATKIND(V1:S) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) ACTIVE(isNatKind(0)) -> MARK(tt) ACTIVE(plus(N:S,s(M:S))) -> U61#(isNat(M:S),M:S,N:S) ACTIVE(plus(N:S,s(M:S))) -> ISNAT(M:S) ACTIVE(plus(N:S,s(M:S))) -> MARK(U61(isNat(M:S),M:S,N:S)) ACTIVE(plus(N:S,0)) -> U51#(isNat(N:S),N:S) ACTIVE(plus(N:S,0)) -> ISNAT(N:S) ACTIVE(plus(N:S,0)) -> MARK(U51(isNat(N:S),N:S)) ISNAT(active(X:S)) -> ISNAT(X:S) ISNAT(mark(X:S)) -> ISNAT(X:S) ISNATKIND(active(X:S)) -> ISNATKIND(X:S) ISNATKIND(mark(X:S)) -> ISNATKIND(X:S) MARK(U11(X1:S,X2:S,X3:S)) -> U11#(mark(X1:S),X2:S,X3:S) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> U12#(mark(X1:S),X2:S,X3:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> U13#(mark(X1:S),X2:S,X3:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> U14#(mark(X1:S),X2:S,X3:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> U15#(mark(X1:S),X2:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> U16#(mark(X:S)) MARK(U16(X:S)) -> ACTIVE(U16(mark(X:S))) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> U21#(mark(X1:S),X2:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> U22#(mark(X1:S),X2:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> U23#(mark(X:S)) MARK(U23(X:S)) -> ACTIVE(U23(mark(X:S))) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> U31#(mark(X1:S),X2:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> U32#(mark(X:S)) MARK(U32(X:S)) -> ACTIVE(U32(mark(X:S))) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> U41#(mark(X:S)) MARK(U41(X:S)) -> ACTIVE(U41(mark(X:S))) MARK(U41(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> U51#(mark(X1:S),X2:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X1:S,X2:S)) -> U52#(mark(X1:S),X2:S) MARK(U52(X1:S,X2:S)) -> ACTIVE(U52(mark(X1:S),X2:S)) MARK(U52(X1:S,X2:S)) -> MARK(X1:S) MARK(U61(X1:S,X2:S,X3:S)) -> U61#(mark(X1:S),X2:S,X3:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> U62#(mark(X1:S),X2:S,X3:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> U63#(mark(X1:S),X2:S,X3:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> U64#(mark(X1:S),X2:S,X3:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(plus(X1:S,X2:S)) -> PLUS(mark(X1:S),mark(X2:S)) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(s(X:S)) -> S(mark(X:S)) PLUS(active(X1:S),X2:S) -> PLUS(X1:S,X2:S) PLUS(mark(X1:S),X2:S) -> PLUS(X1:S,X2:S) PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S) PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S) S(active(X:S)) -> S(X:S) S(mark(X:S)) -> S(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1: SCC Processor: -> Pairs: U11#(active(X1:S),X2:S,X3:S) -> U11#(X1:S,X2:S,X3:S) U11#(mark(X1:S),X2:S,X3:S) -> U11#(X1:S,X2:S,X3:S) U11#(X1:S,active(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S) U11#(X1:S,mark(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S) U11#(X1:S,X2:S,active(X3:S)) -> U11#(X1:S,X2:S,X3:S) U11#(X1:S,X2:S,mark(X3:S)) -> U11#(X1:S,X2:S,X3:S) U12#(active(X1:S),X2:S,X3:S) -> U12#(X1:S,X2:S,X3:S) U12#(mark(X1:S),X2:S,X3:S) -> U12#(X1:S,X2:S,X3:S) U12#(X1:S,active(X2:S),X3:S) -> U12#(X1:S,X2:S,X3:S) U12#(X1:S,mark(X2:S),X3:S) -> U12#(X1:S,X2:S,X3:S) U12#(X1:S,X2:S,active(X3:S)) -> U12#(X1:S,X2:S,X3:S) U12#(X1:S,X2:S,mark(X3:S)) -> U12#(X1:S,X2:S,X3:S) U13#(active(X1:S),X2:S,X3:S) -> U13#(X1:S,X2:S,X3:S) U13#(mark(X1:S),X2:S,X3:S) -> U13#(X1:S,X2:S,X3:S) U13#(X1:S,active(X2:S),X3:S) -> U13#(X1:S,X2:S,X3:S) U13#(X1:S,mark(X2:S),X3:S) -> U13#(X1:S,X2:S,X3:S) U13#(X1:S,X2:S,active(X3:S)) -> U13#(X1:S,X2:S,X3:S) U13#(X1:S,X2:S,mark(X3:S)) -> U13#(X1:S,X2:S,X3:S) U14#(active(X1:S),X2:S,X3:S) -> U14#(X1:S,X2:S,X3:S) U14#(mark(X1:S),X2:S,X3:S) -> U14#(X1:S,X2:S,X3:S) U14#(X1:S,active(X2:S),X3:S) -> U14#(X1:S,X2:S,X3:S) U14#(X1:S,mark(X2:S),X3:S) -> U14#(X1:S,X2:S,X3:S) U14#(X1:S,X2:S,active(X3:S)) -> U14#(X1:S,X2:S,X3:S) U14#(X1:S,X2:S,mark(X3:S)) -> U14#(X1:S,X2:S,X3:S) U15#(active(X1:S),X2:S) -> U15#(X1:S,X2:S) U15#(mark(X1:S),X2:S) -> U15#(X1:S,X2:S) U15#(X1:S,active(X2:S)) -> U15#(X1:S,X2:S) U15#(X1:S,mark(X2:S)) -> U15#(X1:S,X2:S) U16#(active(X:S)) -> U16#(X:S) U16#(mark(X:S)) -> U16#(X:S) U21#(active(X1:S),X2:S) -> U21#(X1:S,X2:S) U21#(mark(X1:S),X2:S) -> U21#(X1:S,X2:S) U21#(X1:S,active(X2:S)) -> U21#(X1:S,X2:S) U21#(X1:S,mark(X2:S)) -> U21#(X1:S,X2:S) U22#(active(X1:S),X2:S) -> U22#(X1:S,X2:S) U22#(mark(X1:S),X2:S) -> U22#(X1:S,X2:S) U22#(X1:S,active(X2:S)) -> U22#(X1:S,X2:S) U22#(X1:S,mark(X2:S)) -> U22#(X1:S,X2:S) U23#(active(X:S)) -> U23#(X:S) U23#(mark(X:S)) -> U23#(X:S) U31#(active(X1:S),X2:S) -> U31#(X1:S,X2:S) U31#(mark(X1:S),X2:S) -> U31#(X1:S,X2:S) U31#(X1:S,active(X2:S)) -> U31#(X1:S,X2:S) U31#(X1:S,mark(X2:S)) -> U31#(X1:S,X2:S) U32#(active(X:S)) -> U32#(X:S) U32#(mark(X:S)) -> U32#(X:S) U41#(active(X:S)) -> U41#(X:S) U41#(mark(X:S)) -> U41#(X:S) U51#(active(X1:S),X2:S) -> U51#(X1:S,X2:S) U51#(mark(X1:S),X2:S) -> U51#(X1:S,X2:S) U51#(X1:S,active(X2:S)) -> U51#(X1:S,X2:S) U51#(X1:S,mark(X2:S)) -> U51#(X1:S,X2:S) U52#(active(X1:S),X2:S) -> U52#(X1:S,X2:S) U52#(mark(X1:S),X2:S) -> U52#(X1:S,X2:S) U52#(X1:S,active(X2:S)) -> U52#(X1:S,X2:S) U52#(X1:S,mark(X2:S)) -> U52#(X1:S,X2:S) U61#(active(X1:S),X2:S,X3:S) -> U61#(X1:S,X2:S,X3:S) U61#(mark(X1:S),X2:S,X3:S) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,active(X2:S),X3:S) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,mark(X2:S),X3:S) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,X2:S,active(X3:S)) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,X2:S,mark(X3:S)) -> U61#(X1:S,X2:S,X3:S) U62#(active(X1:S),X2:S,X3:S) -> U62#(X1:S,X2:S,X3:S) U62#(mark(X1:S),X2:S,X3:S) -> U62#(X1:S,X2:S,X3:S) U62#(X1:S,active(X2:S),X3:S) -> U62#(X1:S,X2:S,X3:S) U62#(X1:S,mark(X2:S),X3:S) -> U62#(X1:S,X2:S,X3:S) U62#(X1:S,X2:S,active(X3:S)) -> U62#(X1:S,X2:S,X3:S) U62#(X1:S,X2:S,mark(X3:S)) -> U62#(X1:S,X2:S,X3:S) U63#(active(X1:S),X2:S,X3:S) -> U63#(X1:S,X2:S,X3:S) U63#(mark(X1:S),X2:S,X3:S) -> U63#(X1:S,X2:S,X3:S) U63#(X1:S,active(X2:S),X3:S) -> U63#(X1:S,X2:S,X3:S) U63#(X1:S,mark(X2:S),X3:S) -> U63#(X1:S,X2:S,X3:S) U63#(X1:S,X2:S,active(X3:S)) -> U63#(X1:S,X2:S,X3:S) U63#(X1:S,X2:S,mark(X3:S)) -> U63#(X1:S,X2:S,X3:S) U64#(active(X1:S),X2:S,X3:S) -> U64#(X1:S,X2:S,X3:S) U64#(mark(X1:S),X2:S,X3:S) -> U64#(X1:S,X2:S,X3:S) U64#(X1:S,active(X2:S),X3:S) -> U64#(X1:S,X2:S,X3:S) U64#(X1:S,mark(X2:S),X3:S) -> U64#(X1:S,X2:S,X3:S) U64#(X1:S,X2:S,active(X3:S)) -> U64#(X1:S,X2:S,X3:S) U64#(X1:S,X2:S,mark(X3:S)) -> U64#(X1:S,X2:S,X3:S) ACTIVE(U11(tt,V1:S,V2:S)) -> U12#(isNatKind(V1:S),V1:S,V2:S) ACTIVE(U11(tt,V1:S,V2:S)) -> ISNATKIND(V1:S) ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> U13#(isNatKind(V2:S),V1:S,V2:S) ACTIVE(U12(tt,V1:S,V2:S)) -> ISNATKIND(V2:S) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> U14#(isNatKind(V2:S),V1:S,V2:S) ACTIVE(U13(tt,V1:S,V2:S)) -> ISNATKIND(V2:S) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> U15#(isNat(V1:S),V2:S) ACTIVE(U14(tt,V1:S,V2:S)) -> ISNAT(V1:S) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> U16#(isNat(V2:S)) ACTIVE(U15(tt,V2:S)) -> ISNAT(V2:S) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U16(tt)) -> MARK(tt) ACTIVE(U21(tt,V1:S)) -> U22#(isNatKind(V1:S),V1:S) ACTIVE(U21(tt,V1:S)) -> ISNATKIND(V1:S) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> U23#(isNat(V1:S)) ACTIVE(U22(tt,V1:S)) -> ISNAT(V1:S) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U23(tt)) -> MARK(tt) ACTIVE(U31(tt,V2:S)) -> U32#(isNatKind(V2:S)) ACTIVE(U31(tt,V2:S)) -> ISNATKIND(V2:S) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(U32(tt)) -> MARK(tt) ACTIVE(U41(tt)) -> MARK(tt) ACTIVE(U51(tt,N:S)) -> U52#(isNatKind(N:S),N:S) ACTIVE(U51(tt,N:S)) -> ISNATKIND(N:S) ACTIVE(U51(tt,N:S)) -> MARK(U52(isNatKind(N:S),N:S)) ACTIVE(U52(tt,N:S)) -> MARK(N:S) ACTIVE(U61(tt,M:S,N:S)) -> U62#(isNatKind(M:S),M:S,N:S) ACTIVE(U61(tt,M:S,N:S)) -> ISNATKIND(M:S) ACTIVE(U61(tt,M:S,N:S)) -> MARK(U62(isNatKind(M:S),M:S,N:S)) ACTIVE(U62(tt,M:S,N:S)) -> U63#(isNat(N:S),M:S,N:S) ACTIVE(U62(tt,M:S,N:S)) -> ISNAT(N:S) ACTIVE(U62(tt,M:S,N:S)) -> MARK(U63(isNat(N:S),M:S,N:S)) ACTIVE(U63(tt,M:S,N:S)) -> U64#(isNatKind(N:S),M:S,N:S) ACTIVE(U63(tt,M:S,N:S)) -> ISNATKIND(N:S) ACTIVE(U63(tt,M:S,N:S)) -> MARK(U64(isNatKind(N:S),M:S,N:S)) ACTIVE(U64(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S))) ACTIVE(U64(tt,M:S,N:S)) -> PLUS(N:S,M:S) ACTIVE(U64(tt,M:S,N:S)) -> S(plus(N:S,M:S)) ACTIVE(isNat(plus(V1:S,V2:S))) -> U11#(isNatKind(V1:S),V1:S,V2:S) ACTIVE(isNat(plus(V1:S,V2:S))) -> ISNATKIND(V1:S) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> U21#(isNatKind(V1:S),V1:S) ACTIVE(isNat(s(V1:S))) -> ISNATKIND(V1:S) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNat(0)) -> MARK(tt) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> U31#(isNatKind(V1:S),V2:S) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> ISNATKIND(V1:S) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> U41#(isNatKind(V1:S)) ACTIVE(isNatKind(s(V1:S))) -> ISNATKIND(V1:S) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) ACTIVE(isNatKind(0)) -> MARK(tt) ACTIVE(plus(N:S,s(M:S))) -> U61#(isNat(M:S),M:S,N:S) ACTIVE(plus(N:S,s(M:S))) -> ISNAT(M:S) ACTIVE(plus(N:S,s(M:S))) -> MARK(U61(isNat(M:S),M:S,N:S)) ACTIVE(plus(N:S,0)) -> U51#(isNat(N:S),N:S) ACTIVE(plus(N:S,0)) -> ISNAT(N:S) ACTIVE(plus(N:S,0)) -> MARK(U51(isNat(N:S),N:S)) ISNAT(active(X:S)) -> ISNAT(X:S) ISNAT(mark(X:S)) -> ISNAT(X:S) ISNATKIND(active(X:S)) -> ISNATKIND(X:S) ISNATKIND(mark(X:S)) -> ISNATKIND(X:S) MARK(U11(X1:S,X2:S,X3:S)) -> U11#(mark(X1:S),X2:S,X3:S) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> U12#(mark(X1:S),X2:S,X3:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> U13#(mark(X1:S),X2:S,X3:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> U14#(mark(X1:S),X2:S,X3:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> U15#(mark(X1:S),X2:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> U16#(mark(X:S)) MARK(U16(X:S)) -> ACTIVE(U16(mark(X:S))) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> U21#(mark(X1:S),X2:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> U22#(mark(X1:S),X2:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> U23#(mark(X:S)) MARK(U23(X:S)) -> ACTIVE(U23(mark(X:S))) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> U31#(mark(X1:S),X2:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> U32#(mark(X:S)) MARK(U32(X:S)) -> ACTIVE(U32(mark(X:S))) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> U41#(mark(X:S)) MARK(U41(X:S)) -> ACTIVE(U41(mark(X:S))) MARK(U41(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> U51#(mark(X1:S),X2:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X1:S,X2:S)) -> U52#(mark(X1:S),X2:S) MARK(U52(X1:S,X2:S)) -> ACTIVE(U52(mark(X1:S),X2:S)) MARK(U52(X1:S,X2:S)) -> MARK(X1:S) MARK(U61(X1:S,X2:S,X3:S)) -> U61#(mark(X1:S),X2:S,X3:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> U62#(mark(X1:S),X2:S,X3:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> U63#(mark(X1:S),X2:S,X3:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> U64#(mark(X1:S),X2:S,X3:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(plus(X1:S,X2:S)) -> PLUS(mark(X1:S),mark(X2:S)) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(s(X:S)) -> S(mark(X:S)) PLUS(active(X1:S),X2:S) -> PLUS(X1:S,X2:S) PLUS(mark(X1:S),X2:S) -> PLUS(X1:S,X2:S) PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S) PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S) S(active(X:S)) -> S(X:S) S(mark(X:S)) -> S(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: S(active(X:S)) -> S(X:S) S(mark(X:S)) -> S(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: PLUS(active(X1:S),X2:S) -> PLUS(X1:S,X2:S) PLUS(mark(X1:S),X2:S) -> PLUS(X1:S,X2:S) PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S) PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: ISNATKIND(active(X:S)) -> ISNATKIND(X:S) ISNATKIND(mark(X:S)) -> ISNATKIND(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: ISNAT(active(X:S)) -> ISNAT(X:S) ISNAT(mark(X:S)) -> ISNAT(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: U64#(active(X1:S),X2:S,X3:S) -> U64#(X1:S,X2:S,X3:S) U64#(mark(X1:S),X2:S,X3:S) -> U64#(X1:S,X2:S,X3:S) U64#(X1:S,active(X2:S),X3:S) -> U64#(X1:S,X2:S,X3:S) U64#(X1:S,mark(X2:S),X3:S) -> U64#(X1:S,X2:S,X3:S) U64#(X1:S,X2:S,active(X3:S)) -> U64#(X1:S,X2:S,X3:S) U64#(X1:S,X2:S,mark(X3:S)) -> U64#(X1:S,X2:S,X3:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: U63#(active(X1:S),X2:S,X3:S) -> U63#(X1:S,X2:S,X3:S) U63#(mark(X1:S),X2:S,X3:S) -> U63#(X1:S,X2:S,X3:S) U63#(X1:S,active(X2:S),X3:S) -> U63#(X1:S,X2:S,X3:S) U63#(X1:S,mark(X2:S),X3:S) -> U63#(X1:S,X2:S,X3:S) U63#(X1:S,X2:S,active(X3:S)) -> U63#(X1:S,X2:S,X3:S) U63#(X1:S,X2:S,mark(X3:S)) -> U63#(X1:S,X2:S,X3:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: U62#(active(X1:S),X2:S,X3:S) -> U62#(X1:S,X2:S,X3:S) U62#(mark(X1:S),X2:S,X3:S) -> U62#(X1:S,X2:S,X3:S) U62#(X1:S,active(X2:S),X3:S) -> U62#(X1:S,X2:S,X3:S) U62#(X1:S,mark(X2:S),X3:S) -> U62#(X1:S,X2:S,X3:S) U62#(X1:S,X2:S,active(X3:S)) -> U62#(X1:S,X2:S,X3:S) U62#(X1:S,X2:S,mark(X3:S)) -> U62#(X1:S,X2:S,X3:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: U61#(active(X1:S),X2:S,X3:S) -> U61#(X1:S,X2:S,X3:S) U61#(mark(X1:S),X2:S,X3:S) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,active(X2:S),X3:S) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,mark(X2:S),X3:S) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,X2:S,active(X3:S)) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,X2:S,mark(X3:S)) -> U61#(X1:S,X2:S,X3:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: U52#(active(X1:S),X2:S) -> U52#(X1:S,X2:S) U52#(mark(X1:S),X2:S) -> U52#(X1:S,X2:S) U52#(X1:S,active(X2:S)) -> U52#(X1:S,X2:S) U52#(X1:S,mark(X2:S)) -> U52#(X1:S,X2:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: U51#(active(X1:S),X2:S) -> U51#(X1:S,X2:S) U51#(mark(X1:S),X2:S) -> U51#(X1:S,X2:S) U51#(X1:S,active(X2:S)) -> U51#(X1:S,X2:S) U51#(X1:S,mark(X2:S)) -> U51#(X1:S,X2:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: U41#(active(X:S)) -> U41#(X:S) U41#(mark(X:S)) -> U41#(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: U32#(active(X:S)) -> U32#(X:S) U32#(mark(X:S)) -> U32#(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: U31#(active(X1:S),X2:S) -> U31#(X1:S,X2:S) U31#(mark(X1:S),X2:S) -> U31#(X1:S,X2:S) U31#(X1:S,active(X2:S)) -> U31#(X1:S,X2:S) U31#(X1:S,mark(X2:S)) -> U31#(X1:S,X2:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: U23#(active(X:S)) -> U23#(X:S) U23#(mark(X:S)) -> U23#(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: U22#(active(X1:S),X2:S) -> U22#(X1:S,X2:S) U22#(mark(X1:S),X2:S) -> U22#(X1:S,X2:S) U22#(X1:S,active(X2:S)) -> U22#(X1:S,X2:S) U22#(X1:S,mark(X2:S)) -> U22#(X1:S,X2:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: U21#(active(X1:S),X2:S) -> U21#(X1:S,X2:S) U21#(mark(X1:S),X2:S) -> U21#(X1:S,X2:S) U21#(X1:S,active(X2:S)) -> U21#(X1:S,X2:S) U21#(X1:S,mark(X2:S)) -> U21#(X1:S,X2:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: U16#(active(X:S)) -> U16#(X:S) U16#(mark(X:S)) -> U16#(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: U15#(active(X1:S),X2:S) -> U15#(X1:S,X2:S) U15#(mark(X1:S),X2:S) -> U15#(X1:S,X2:S) U15#(X1:S,active(X2:S)) -> U15#(X1:S,X2:S) U15#(X1:S,mark(X2:S)) -> U15#(X1:S,X2:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: U14#(active(X1:S),X2:S,X3:S) -> U14#(X1:S,X2:S,X3:S) U14#(mark(X1:S),X2:S,X3:S) -> U14#(X1:S,X2:S,X3:S) U14#(X1:S,active(X2:S),X3:S) -> U14#(X1:S,X2:S,X3:S) U14#(X1:S,mark(X2:S),X3:S) -> U14#(X1:S,X2:S,X3:S) U14#(X1:S,X2:S,active(X3:S)) -> U14#(X1:S,X2:S,X3:S) U14#(X1:S,X2:S,mark(X3:S)) -> U14#(X1:S,X2:S,X3:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: U13#(active(X1:S),X2:S,X3:S) -> U13#(X1:S,X2:S,X3:S) U13#(mark(X1:S),X2:S,X3:S) -> U13#(X1:S,X2:S,X3:S) U13#(X1:S,active(X2:S),X3:S) -> U13#(X1:S,X2:S,X3:S) U13#(X1:S,mark(X2:S),X3:S) -> U13#(X1:S,X2:S,X3:S) U13#(X1:S,X2:S,active(X3:S)) -> U13#(X1:S,X2:S,X3:S) U13#(X1:S,X2:S,mark(X3:S)) -> U13#(X1:S,X2:S,X3:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: U12#(active(X1:S),X2:S,X3:S) -> U12#(X1:S,X2:S,X3:S) U12#(mark(X1:S),X2:S,X3:S) -> U12#(X1:S,X2:S,X3:S) U12#(X1:S,active(X2:S),X3:S) -> U12#(X1:S,X2:S,X3:S) U12#(X1:S,mark(X2:S),X3:S) -> U12#(X1:S,X2:S,X3:S) U12#(X1:S,X2:S,active(X3:S)) -> U12#(X1:S,X2:S,X3:S) U12#(X1:S,X2:S,mark(X3:S)) -> U12#(X1:S,X2:S,X3:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: U11#(active(X1:S),X2:S,X3:S) -> U11#(X1:S,X2:S,X3:S) U11#(mark(X1:S),X2:S,X3:S) -> U11#(X1:S,X2:S,X3:S) U11#(X1:S,active(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S) U11#(X1:S,mark(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S) U11#(X1:S,X2:S,active(X3:S)) -> U11#(X1:S,X2:S,X3:S) U11#(X1:S,X2:S,mark(X3:S)) -> U11#(X1:S,X2:S,X3:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(U51(tt,N:S)) -> MARK(U52(isNatKind(N:S),N:S)) ACTIVE(U52(tt,N:S)) -> MARK(N:S) ACTIVE(U61(tt,M:S,N:S)) -> MARK(U62(isNatKind(M:S),M:S,N:S)) ACTIVE(U62(tt,M:S,N:S)) -> MARK(U63(isNat(N:S),M:S,N:S)) ACTIVE(U63(tt,M:S,N:S)) -> MARK(U64(isNatKind(N:S),M:S,N:S)) ACTIVE(U64(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) ACTIVE(plus(N:S,s(M:S))) -> MARK(U61(isNat(M:S),M:S,N:S)) ACTIVE(plus(N:S,0)) -> MARK(U51(isNat(N:S),N:S)) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> ACTIVE(U16(mark(X:S))) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> ACTIVE(U23(mark(X:S))) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> ACTIVE(U32(mark(X:S))) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> ACTIVE(U41(mark(X:S))) MARK(U41(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X1:S,X2:S)) -> ACTIVE(U52(mark(X1:S),X2:S)) MARK(U52(X1:S,X2:S)) -> MARK(X1:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) The problem is decomposed in 23 subproblems. Problem 1.1: Subterm Processor: -> Pairs: S(active(X:S)) -> S(X:S) S(mark(X:S)) -> S(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(S) = 1 Problem 1.1: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Subterm Processor: -> Pairs: PLUS(active(X1:S),X2:S) -> PLUS(X1:S,X2:S) PLUS(mark(X1:S),X2:S) -> PLUS(X1:S,X2:S) PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S) PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(PLUS) = 1 Problem 1.2: SCC Processor: -> Pairs: PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S) PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S) PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.2: Subterm Processor: -> Pairs: PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S) PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(PLUS) = 2 Problem 1.2: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.3: Subterm Processor: -> Pairs: ISNATKIND(active(X:S)) -> ISNATKIND(X:S) ISNATKIND(mark(X:S)) -> ISNATKIND(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(ISNATKIND) = 1 Problem 1.3: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.4: Subterm Processor: -> Pairs: ISNAT(active(X:S)) -> ISNAT(X:S) ISNAT(mark(X:S)) -> ISNAT(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(ISNAT) = 1 Problem 1.4: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.5: Subterm Processor: -> Pairs: U64#(active(X1:S),X2:S,X3:S) -> U64#(X1:S,X2:S,X3:S) U64#(mark(X1:S),X2:S,X3:S) -> U64#(X1:S,X2:S,X3:S) U64#(X1:S,active(X2:S),X3:S) -> U64#(X1:S,X2:S,X3:S) U64#(X1:S,mark(X2:S),X3:S) -> U64#(X1:S,X2:S,X3:S) U64#(X1:S,X2:S,active(X3:S)) -> U64#(X1:S,X2:S,X3:S) U64#(X1:S,X2:S,mark(X3:S)) -> U64#(X1:S,X2:S,X3:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U64#) = 1 Problem 1.5: SCC Processor: -> Pairs: U64#(X1:S,active(X2:S),X3:S) -> U64#(X1:S,X2:S,X3:S) U64#(X1:S,mark(X2:S),X3:S) -> U64#(X1:S,X2:S,X3:S) U64#(X1:S,X2:S,active(X3:S)) -> U64#(X1:S,X2:S,X3:S) U64#(X1:S,X2:S,mark(X3:S)) -> U64#(X1:S,X2:S,X3:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U64#(X1:S,active(X2:S),X3:S) -> U64#(X1:S,X2:S,X3:S) U64#(X1:S,mark(X2:S),X3:S) -> U64#(X1:S,X2:S,X3:S) U64#(X1:S,X2:S,active(X3:S)) -> U64#(X1:S,X2:S,X3:S) U64#(X1:S,X2:S,mark(X3:S)) -> U64#(X1:S,X2:S,X3:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.5: Subterm Processor: -> Pairs: U64#(X1:S,active(X2:S),X3:S) -> U64#(X1:S,X2:S,X3:S) U64#(X1:S,mark(X2:S),X3:S) -> U64#(X1:S,X2:S,X3:S) U64#(X1:S,X2:S,active(X3:S)) -> U64#(X1:S,X2:S,X3:S) U64#(X1:S,X2:S,mark(X3:S)) -> U64#(X1:S,X2:S,X3:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U64#) = 2 Problem 1.5: SCC Processor: -> Pairs: U64#(X1:S,X2:S,active(X3:S)) -> U64#(X1:S,X2:S,X3:S) U64#(X1:S,X2:S,mark(X3:S)) -> U64#(X1:S,X2:S,X3:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U64#(X1:S,X2:S,active(X3:S)) -> U64#(X1:S,X2:S,X3:S) U64#(X1:S,X2:S,mark(X3:S)) -> U64#(X1:S,X2:S,X3:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.5: Subterm Processor: -> Pairs: U64#(X1:S,X2:S,active(X3:S)) -> U64#(X1:S,X2:S,X3:S) U64#(X1:S,X2:S,mark(X3:S)) -> U64#(X1:S,X2:S,X3:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U64#) = 3 Problem 1.5: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.6: Subterm Processor: -> Pairs: U63#(active(X1:S),X2:S,X3:S) -> U63#(X1:S,X2:S,X3:S) U63#(mark(X1:S),X2:S,X3:S) -> U63#(X1:S,X2:S,X3:S) U63#(X1:S,active(X2:S),X3:S) -> U63#(X1:S,X2:S,X3:S) U63#(X1:S,mark(X2:S),X3:S) -> U63#(X1:S,X2:S,X3:S) U63#(X1:S,X2:S,active(X3:S)) -> U63#(X1:S,X2:S,X3:S) U63#(X1:S,X2:S,mark(X3:S)) -> U63#(X1:S,X2:S,X3:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U63#) = 1 Problem 1.6: SCC Processor: -> Pairs: U63#(X1:S,active(X2:S),X3:S) -> U63#(X1:S,X2:S,X3:S) U63#(X1:S,mark(X2:S),X3:S) -> U63#(X1:S,X2:S,X3:S) U63#(X1:S,X2:S,active(X3:S)) -> U63#(X1:S,X2:S,X3:S) U63#(X1:S,X2:S,mark(X3:S)) -> U63#(X1:S,X2:S,X3:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U63#(X1:S,active(X2:S),X3:S) -> U63#(X1:S,X2:S,X3:S) U63#(X1:S,mark(X2:S),X3:S) -> U63#(X1:S,X2:S,X3:S) U63#(X1:S,X2:S,active(X3:S)) -> U63#(X1:S,X2:S,X3:S) U63#(X1:S,X2:S,mark(X3:S)) -> U63#(X1:S,X2:S,X3:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.6: Subterm Processor: -> Pairs: U63#(X1:S,active(X2:S),X3:S) -> U63#(X1:S,X2:S,X3:S) U63#(X1:S,mark(X2:S),X3:S) -> U63#(X1:S,X2:S,X3:S) U63#(X1:S,X2:S,active(X3:S)) -> U63#(X1:S,X2:S,X3:S) U63#(X1:S,X2:S,mark(X3:S)) -> U63#(X1:S,X2:S,X3:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U63#) = 2 Problem 1.6: SCC Processor: -> Pairs: U63#(X1:S,X2:S,active(X3:S)) -> U63#(X1:S,X2:S,X3:S) U63#(X1:S,X2:S,mark(X3:S)) -> U63#(X1:S,X2:S,X3:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U63#(X1:S,X2:S,active(X3:S)) -> U63#(X1:S,X2:S,X3:S) U63#(X1:S,X2:S,mark(X3:S)) -> U63#(X1:S,X2:S,X3:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.6: Subterm Processor: -> Pairs: U63#(X1:S,X2:S,active(X3:S)) -> U63#(X1:S,X2:S,X3:S) U63#(X1:S,X2:S,mark(X3:S)) -> U63#(X1:S,X2:S,X3:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U63#) = 3 Problem 1.6: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.7: Subterm Processor: -> Pairs: U62#(active(X1:S),X2:S,X3:S) -> U62#(X1:S,X2:S,X3:S) U62#(mark(X1:S),X2:S,X3:S) -> U62#(X1:S,X2:S,X3:S) U62#(X1:S,active(X2:S),X3:S) -> U62#(X1:S,X2:S,X3:S) U62#(X1:S,mark(X2:S),X3:S) -> U62#(X1:S,X2:S,X3:S) U62#(X1:S,X2:S,active(X3:S)) -> U62#(X1:S,X2:S,X3:S) U62#(X1:S,X2:S,mark(X3:S)) -> U62#(X1:S,X2:S,X3:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U62#) = 1 Problem 1.7: SCC Processor: -> Pairs: U62#(X1:S,active(X2:S),X3:S) -> U62#(X1:S,X2:S,X3:S) U62#(X1:S,mark(X2:S),X3:S) -> U62#(X1:S,X2:S,X3:S) U62#(X1:S,X2:S,active(X3:S)) -> U62#(X1:S,X2:S,X3:S) U62#(X1:S,X2:S,mark(X3:S)) -> U62#(X1:S,X2:S,X3:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U62#(X1:S,active(X2:S),X3:S) -> U62#(X1:S,X2:S,X3:S) U62#(X1:S,mark(X2:S),X3:S) -> U62#(X1:S,X2:S,X3:S) U62#(X1:S,X2:S,active(X3:S)) -> U62#(X1:S,X2:S,X3:S) U62#(X1:S,X2:S,mark(X3:S)) -> U62#(X1:S,X2:S,X3:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.7: Subterm Processor: -> Pairs: U62#(X1:S,active(X2:S),X3:S) -> U62#(X1:S,X2:S,X3:S) U62#(X1:S,mark(X2:S),X3:S) -> U62#(X1:S,X2:S,X3:S) U62#(X1:S,X2:S,active(X3:S)) -> U62#(X1:S,X2:S,X3:S) U62#(X1:S,X2:S,mark(X3:S)) -> U62#(X1:S,X2:S,X3:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U62#) = 2 Problem 1.7: SCC Processor: -> Pairs: U62#(X1:S,X2:S,active(X3:S)) -> U62#(X1:S,X2:S,X3:S) U62#(X1:S,X2:S,mark(X3:S)) -> U62#(X1:S,X2:S,X3:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U62#(X1:S,X2:S,active(X3:S)) -> U62#(X1:S,X2:S,X3:S) U62#(X1:S,X2:S,mark(X3:S)) -> U62#(X1:S,X2:S,X3:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.7: Subterm Processor: -> Pairs: U62#(X1:S,X2:S,active(X3:S)) -> U62#(X1:S,X2:S,X3:S) U62#(X1:S,X2:S,mark(X3:S)) -> U62#(X1:S,X2:S,X3:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U62#) = 3 Problem 1.7: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.8: Subterm Processor: -> Pairs: U61#(active(X1:S),X2:S,X3:S) -> U61#(X1:S,X2:S,X3:S) U61#(mark(X1:S),X2:S,X3:S) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,active(X2:S),X3:S) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,mark(X2:S),X3:S) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,X2:S,active(X3:S)) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,X2:S,mark(X3:S)) -> U61#(X1:S,X2:S,X3:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U61#) = 1 Problem 1.8: SCC Processor: -> Pairs: U61#(X1:S,active(X2:S),X3:S) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,mark(X2:S),X3:S) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,X2:S,active(X3:S)) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,X2:S,mark(X3:S)) -> U61#(X1:S,X2:S,X3:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U61#(X1:S,active(X2:S),X3:S) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,mark(X2:S),X3:S) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,X2:S,active(X3:S)) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,X2:S,mark(X3:S)) -> U61#(X1:S,X2:S,X3:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.8: Subterm Processor: -> Pairs: U61#(X1:S,active(X2:S),X3:S) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,mark(X2:S),X3:S) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,X2:S,active(X3:S)) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,X2:S,mark(X3:S)) -> U61#(X1:S,X2:S,X3:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U61#) = 2 Problem 1.8: SCC Processor: -> Pairs: U61#(X1:S,X2:S,active(X3:S)) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,X2:S,mark(X3:S)) -> U61#(X1:S,X2:S,X3:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U61#(X1:S,X2:S,active(X3:S)) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,X2:S,mark(X3:S)) -> U61#(X1:S,X2:S,X3:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.8: Subterm Processor: -> Pairs: U61#(X1:S,X2:S,active(X3:S)) -> U61#(X1:S,X2:S,X3:S) U61#(X1:S,X2:S,mark(X3:S)) -> U61#(X1:S,X2:S,X3:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U61#) = 3 Problem 1.8: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.9: Subterm Processor: -> Pairs: U52#(active(X1:S),X2:S) -> U52#(X1:S,X2:S) U52#(mark(X1:S),X2:S) -> U52#(X1:S,X2:S) U52#(X1:S,active(X2:S)) -> U52#(X1:S,X2:S) U52#(X1:S,mark(X2:S)) -> U52#(X1:S,X2:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U52#) = 1 Problem 1.9: SCC Processor: -> Pairs: U52#(X1:S,active(X2:S)) -> U52#(X1:S,X2:S) U52#(X1:S,mark(X2:S)) -> U52#(X1:S,X2:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U52#(X1:S,active(X2:S)) -> U52#(X1:S,X2:S) U52#(X1:S,mark(X2:S)) -> U52#(X1:S,X2:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.9: Subterm Processor: -> Pairs: U52#(X1:S,active(X2:S)) -> U52#(X1:S,X2:S) U52#(X1:S,mark(X2:S)) -> U52#(X1:S,X2:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U52#) = 2 Problem 1.9: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.10: Subterm Processor: -> Pairs: U51#(active(X1:S),X2:S) -> U51#(X1:S,X2:S) U51#(mark(X1:S),X2:S) -> U51#(X1:S,X2:S) U51#(X1:S,active(X2:S)) -> U51#(X1:S,X2:S) U51#(X1:S,mark(X2:S)) -> U51#(X1:S,X2:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U51#) = 1 Problem 1.10: SCC Processor: -> Pairs: U51#(X1:S,active(X2:S)) -> U51#(X1:S,X2:S) U51#(X1:S,mark(X2:S)) -> U51#(X1:S,X2:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U51#(X1:S,active(X2:S)) -> U51#(X1:S,X2:S) U51#(X1:S,mark(X2:S)) -> U51#(X1:S,X2:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.10: Subterm Processor: -> Pairs: U51#(X1:S,active(X2:S)) -> U51#(X1:S,X2:S) U51#(X1:S,mark(X2:S)) -> U51#(X1:S,X2:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U51#) = 2 Problem 1.10: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.11: Subterm Processor: -> Pairs: U41#(active(X:S)) -> U41#(X:S) U41#(mark(X:S)) -> U41#(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U41#) = 1 Problem 1.11: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.12: Subterm Processor: -> Pairs: U32#(active(X:S)) -> U32#(X:S) U32#(mark(X:S)) -> U32#(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U32#) = 1 Problem 1.12: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.13: Subterm Processor: -> Pairs: U31#(active(X1:S),X2:S) -> U31#(X1:S,X2:S) U31#(mark(X1:S),X2:S) -> U31#(X1:S,X2:S) U31#(X1:S,active(X2:S)) -> U31#(X1:S,X2:S) U31#(X1:S,mark(X2:S)) -> U31#(X1:S,X2:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U31#) = 1 Problem 1.13: SCC Processor: -> Pairs: U31#(X1:S,active(X2:S)) -> U31#(X1:S,X2:S) U31#(X1:S,mark(X2:S)) -> U31#(X1:S,X2:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U31#(X1:S,active(X2:S)) -> U31#(X1:S,X2:S) U31#(X1:S,mark(X2:S)) -> U31#(X1:S,X2:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.13: Subterm Processor: -> Pairs: U31#(X1:S,active(X2:S)) -> U31#(X1:S,X2:S) U31#(X1:S,mark(X2:S)) -> U31#(X1:S,X2:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U31#) = 2 Problem 1.13: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.14: Subterm Processor: -> Pairs: U23#(active(X:S)) -> U23#(X:S) U23#(mark(X:S)) -> U23#(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U23#) = 1 Problem 1.14: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.15: Subterm Processor: -> Pairs: U22#(active(X1:S),X2:S) -> U22#(X1:S,X2:S) U22#(mark(X1:S),X2:S) -> U22#(X1:S,X2:S) U22#(X1:S,active(X2:S)) -> U22#(X1:S,X2:S) U22#(X1:S,mark(X2:S)) -> U22#(X1:S,X2:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U22#) = 1 Problem 1.15: SCC Processor: -> Pairs: U22#(X1:S,active(X2:S)) -> U22#(X1:S,X2:S) U22#(X1:S,mark(X2:S)) -> U22#(X1:S,X2:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U22#(X1:S,active(X2:S)) -> U22#(X1:S,X2:S) U22#(X1:S,mark(X2:S)) -> U22#(X1:S,X2:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.15: Subterm Processor: -> Pairs: U22#(X1:S,active(X2:S)) -> U22#(X1:S,X2:S) U22#(X1:S,mark(X2:S)) -> U22#(X1:S,X2:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U22#) = 2 Problem 1.15: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.16: Subterm Processor: -> Pairs: U21#(active(X1:S),X2:S) -> U21#(X1:S,X2:S) U21#(mark(X1:S),X2:S) -> U21#(X1:S,X2:S) U21#(X1:S,active(X2:S)) -> U21#(X1:S,X2:S) U21#(X1:S,mark(X2:S)) -> U21#(X1:S,X2:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U21#) = 1 Problem 1.16: SCC Processor: -> Pairs: U21#(X1:S,active(X2:S)) -> U21#(X1:S,X2:S) U21#(X1:S,mark(X2:S)) -> U21#(X1:S,X2:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U21#(X1:S,active(X2:S)) -> U21#(X1:S,X2:S) U21#(X1:S,mark(X2:S)) -> U21#(X1:S,X2:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.16: Subterm Processor: -> Pairs: U21#(X1:S,active(X2:S)) -> U21#(X1:S,X2:S) U21#(X1:S,mark(X2:S)) -> U21#(X1:S,X2:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U21#) = 2 Problem 1.16: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.17: Subterm Processor: -> Pairs: U16#(active(X:S)) -> U16#(X:S) U16#(mark(X:S)) -> U16#(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U16#) = 1 Problem 1.17: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.18: Subterm Processor: -> Pairs: U15#(active(X1:S),X2:S) -> U15#(X1:S,X2:S) U15#(mark(X1:S),X2:S) -> U15#(X1:S,X2:S) U15#(X1:S,active(X2:S)) -> U15#(X1:S,X2:S) U15#(X1:S,mark(X2:S)) -> U15#(X1:S,X2:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U15#) = 1 Problem 1.18: SCC Processor: -> Pairs: U15#(X1:S,active(X2:S)) -> U15#(X1:S,X2:S) U15#(X1:S,mark(X2:S)) -> U15#(X1:S,X2:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U15#(X1:S,active(X2:S)) -> U15#(X1:S,X2:S) U15#(X1:S,mark(X2:S)) -> U15#(X1:S,X2:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.18: Subterm Processor: -> Pairs: U15#(X1:S,active(X2:S)) -> U15#(X1:S,X2:S) U15#(X1:S,mark(X2:S)) -> U15#(X1:S,X2:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U15#) = 2 Problem 1.18: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.19: Subterm Processor: -> Pairs: U14#(active(X1:S),X2:S,X3:S) -> U14#(X1:S,X2:S,X3:S) U14#(mark(X1:S),X2:S,X3:S) -> U14#(X1:S,X2:S,X3:S) U14#(X1:S,active(X2:S),X3:S) -> U14#(X1:S,X2:S,X3:S) U14#(X1:S,mark(X2:S),X3:S) -> U14#(X1:S,X2:S,X3:S) U14#(X1:S,X2:S,active(X3:S)) -> U14#(X1:S,X2:S,X3:S) U14#(X1:S,X2:S,mark(X3:S)) -> U14#(X1:S,X2:S,X3:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U14#) = 1 Problem 1.19: SCC Processor: -> Pairs: U14#(X1:S,active(X2:S),X3:S) -> U14#(X1:S,X2:S,X3:S) U14#(X1:S,mark(X2:S),X3:S) -> U14#(X1:S,X2:S,X3:S) U14#(X1:S,X2:S,active(X3:S)) -> U14#(X1:S,X2:S,X3:S) U14#(X1:S,X2:S,mark(X3:S)) -> U14#(X1:S,X2:S,X3:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U14#(X1:S,active(X2:S),X3:S) -> U14#(X1:S,X2:S,X3:S) U14#(X1:S,mark(X2:S),X3:S) -> U14#(X1:S,X2:S,X3:S) U14#(X1:S,X2:S,active(X3:S)) -> U14#(X1:S,X2:S,X3:S) U14#(X1:S,X2:S,mark(X3:S)) -> U14#(X1:S,X2:S,X3:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.19: Subterm Processor: -> Pairs: U14#(X1:S,active(X2:S),X3:S) -> U14#(X1:S,X2:S,X3:S) U14#(X1:S,mark(X2:S),X3:S) -> U14#(X1:S,X2:S,X3:S) U14#(X1:S,X2:S,active(X3:S)) -> U14#(X1:S,X2:S,X3:S) U14#(X1:S,X2:S,mark(X3:S)) -> U14#(X1:S,X2:S,X3:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U14#) = 2 Problem 1.19: SCC Processor: -> Pairs: U14#(X1:S,X2:S,active(X3:S)) -> U14#(X1:S,X2:S,X3:S) U14#(X1:S,X2:S,mark(X3:S)) -> U14#(X1:S,X2:S,X3:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U14#(X1:S,X2:S,active(X3:S)) -> U14#(X1:S,X2:S,X3:S) U14#(X1:S,X2:S,mark(X3:S)) -> U14#(X1:S,X2:S,X3:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.19: Subterm Processor: -> Pairs: U14#(X1:S,X2:S,active(X3:S)) -> U14#(X1:S,X2:S,X3:S) U14#(X1:S,X2:S,mark(X3:S)) -> U14#(X1:S,X2:S,X3:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U14#) = 3 Problem 1.19: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.20: Subterm Processor: -> Pairs: U13#(active(X1:S),X2:S,X3:S) -> U13#(X1:S,X2:S,X3:S) U13#(mark(X1:S),X2:S,X3:S) -> U13#(X1:S,X2:S,X3:S) U13#(X1:S,active(X2:S),X3:S) -> U13#(X1:S,X2:S,X3:S) U13#(X1:S,mark(X2:S),X3:S) -> U13#(X1:S,X2:S,X3:S) U13#(X1:S,X2:S,active(X3:S)) -> U13#(X1:S,X2:S,X3:S) U13#(X1:S,X2:S,mark(X3:S)) -> U13#(X1:S,X2:S,X3:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U13#) = 1 Problem 1.20: SCC Processor: -> Pairs: U13#(X1:S,active(X2:S),X3:S) -> U13#(X1:S,X2:S,X3:S) U13#(X1:S,mark(X2:S),X3:S) -> U13#(X1:S,X2:S,X3:S) U13#(X1:S,X2:S,active(X3:S)) -> U13#(X1:S,X2:S,X3:S) U13#(X1:S,X2:S,mark(X3:S)) -> U13#(X1:S,X2:S,X3:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U13#(X1:S,active(X2:S),X3:S) -> U13#(X1:S,X2:S,X3:S) U13#(X1:S,mark(X2:S),X3:S) -> U13#(X1:S,X2:S,X3:S) U13#(X1:S,X2:S,active(X3:S)) -> U13#(X1:S,X2:S,X3:S) U13#(X1:S,X2:S,mark(X3:S)) -> U13#(X1:S,X2:S,X3:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.20: Subterm Processor: -> Pairs: U13#(X1:S,active(X2:S),X3:S) -> U13#(X1:S,X2:S,X3:S) U13#(X1:S,mark(X2:S),X3:S) -> U13#(X1:S,X2:S,X3:S) U13#(X1:S,X2:S,active(X3:S)) -> U13#(X1:S,X2:S,X3:S) U13#(X1:S,X2:S,mark(X3:S)) -> U13#(X1:S,X2:S,X3:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U13#) = 2 Problem 1.20: SCC Processor: -> Pairs: U13#(X1:S,X2:S,active(X3:S)) -> U13#(X1:S,X2:S,X3:S) U13#(X1:S,X2:S,mark(X3:S)) -> U13#(X1:S,X2:S,X3:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U13#(X1:S,X2:S,active(X3:S)) -> U13#(X1:S,X2:S,X3:S) U13#(X1:S,X2:S,mark(X3:S)) -> U13#(X1:S,X2:S,X3:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.20: Subterm Processor: -> Pairs: U13#(X1:S,X2:S,active(X3:S)) -> U13#(X1:S,X2:S,X3:S) U13#(X1:S,X2:S,mark(X3:S)) -> U13#(X1:S,X2:S,X3:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U13#) = 3 Problem 1.20: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.21: Subterm Processor: -> Pairs: U12#(active(X1:S),X2:S,X3:S) -> U12#(X1:S,X2:S,X3:S) U12#(mark(X1:S),X2:S,X3:S) -> U12#(X1:S,X2:S,X3:S) U12#(X1:S,active(X2:S),X3:S) -> U12#(X1:S,X2:S,X3:S) U12#(X1:S,mark(X2:S),X3:S) -> U12#(X1:S,X2:S,X3:S) U12#(X1:S,X2:S,active(X3:S)) -> U12#(X1:S,X2:S,X3:S) U12#(X1:S,X2:S,mark(X3:S)) -> U12#(X1:S,X2:S,X3:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U12#) = 1 Problem 1.21: SCC Processor: -> Pairs: U12#(X1:S,active(X2:S),X3:S) -> U12#(X1:S,X2:S,X3:S) U12#(X1:S,mark(X2:S),X3:S) -> U12#(X1:S,X2:S,X3:S) U12#(X1:S,X2:S,active(X3:S)) -> U12#(X1:S,X2:S,X3:S) U12#(X1:S,X2:S,mark(X3:S)) -> U12#(X1:S,X2:S,X3:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U12#(X1:S,active(X2:S),X3:S) -> U12#(X1:S,X2:S,X3:S) U12#(X1:S,mark(X2:S),X3:S) -> U12#(X1:S,X2:S,X3:S) U12#(X1:S,X2:S,active(X3:S)) -> U12#(X1:S,X2:S,X3:S) U12#(X1:S,X2:S,mark(X3:S)) -> U12#(X1:S,X2:S,X3:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.21: Subterm Processor: -> Pairs: U12#(X1:S,active(X2:S),X3:S) -> U12#(X1:S,X2:S,X3:S) U12#(X1:S,mark(X2:S),X3:S) -> U12#(X1:S,X2:S,X3:S) U12#(X1:S,X2:S,active(X3:S)) -> U12#(X1:S,X2:S,X3:S) U12#(X1:S,X2:S,mark(X3:S)) -> U12#(X1:S,X2:S,X3:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U12#) = 2 Problem 1.21: SCC Processor: -> Pairs: U12#(X1:S,X2:S,active(X3:S)) -> U12#(X1:S,X2:S,X3:S) U12#(X1:S,X2:S,mark(X3:S)) -> U12#(X1:S,X2:S,X3:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U12#(X1:S,X2:S,active(X3:S)) -> U12#(X1:S,X2:S,X3:S) U12#(X1:S,X2:S,mark(X3:S)) -> U12#(X1:S,X2:S,X3:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.21: Subterm Processor: -> Pairs: U12#(X1:S,X2:S,active(X3:S)) -> U12#(X1:S,X2:S,X3:S) U12#(X1:S,X2:S,mark(X3:S)) -> U12#(X1:S,X2:S,X3:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U12#) = 3 Problem 1.21: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.22: Subterm Processor: -> Pairs: U11#(active(X1:S),X2:S,X3:S) -> U11#(X1:S,X2:S,X3:S) U11#(mark(X1:S),X2:S,X3:S) -> U11#(X1:S,X2:S,X3:S) U11#(X1:S,active(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S) U11#(X1:S,mark(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S) U11#(X1:S,X2:S,active(X3:S)) -> U11#(X1:S,X2:S,X3:S) U11#(X1:S,X2:S,mark(X3:S)) -> U11#(X1:S,X2:S,X3:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U11#) = 1 Problem 1.22: SCC Processor: -> Pairs: U11#(X1:S,active(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S) U11#(X1:S,mark(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S) U11#(X1:S,X2:S,active(X3:S)) -> U11#(X1:S,X2:S,X3:S) U11#(X1:S,X2:S,mark(X3:S)) -> U11#(X1:S,X2:S,X3:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U11#(X1:S,active(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S) U11#(X1:S,mark(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S) U11#(X1:S,X2:S,active(X3:S)) -> U11#(X1:S,X2:S,X3:S) U11#(X1:S,X2:S,mark(X3:S)) -> U11#(X1:S,X2:S,X3:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.22: Subterm Processor: -> Pairs: U11#(X1:S,active(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S) U11#(X1:S,mark(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S) U11#(X1:S,X2:S,active(X3:S)) -> U11#(X1:S,X2:S,X3:S) U11#(X1:S,X2:S,mark(X3:S)) -> U11#(X1:S,X2:S,X3:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U11#) = 2 Problem 1.22: SCC Processor: -> Pairs: U11#(X1:S,X2:S,active(X3:S)) -> U11#(X1:S,X2:S,X3:S) U11#(X1:S,X2:S,mark(X3:S)) -> U11#(X1:S,X2:S,X3:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U11#(X1:S,X2:S,active(X3:S)) -> U11#(X1:S,X2:S,X3:S) U11#(X1:S,X2:S,mark(X3:S)) -> U11#(X1:S,X2:S,X3:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.22: Subterm Processor: -> Pairs: U11#(X1:S,X2:S,active(X3:S)) -> U11#(X1:S,X2:S,X3:S) U11#(X1:S,X2:S,mark(X3:S)) -> U11#(X1:S,X2:S,X3:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(U11#) = 3 Problem 1.22: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(U51(tt,N:S)) -> MARK(U52(isNatKind(N:S),N:S)) ACTIVE(U52(tt,N:S)) -> MARK(N:S) ACTIVE(U61(tt,M:S,N:S)) -> MARK(U62(isNatKind(M:S),M:S,N:S)) ACTIVE(U62(tt,M:S,N:S)) -> MARK(U63(isNat(N:S),M:S,N:S)) ACTIVE(U63(tt,M:S,N:S)) -> MARK(U64(isNatKind(N:S),M:S,N:S)) ACTIVE(U64(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) ACTIVE(plus(N:S,s(M:S))) -> MARK(U61(isNat(M:S),M:S,N:S)) ACTIVE(plus(N:S,0)) -> MARK(U51(isNat(N:S),N:S)) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> ACTIVE(U16(mark(X:S))) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> ACTIVE(U23(mark(X:S))) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> ACTIVE(U32(mark(X:S))) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> ACTIVE(U41(mark(X:S))) MARK(U41(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X1:S,X2:S)) -> ACTIVE(U52(mark(X1:S),X2:S)) MARK(U52(X1:S,X2:S)) -> MARK(X1:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = X1 [U12](X1,X2,X3) = X1 [U13](X1,X2,X3) = X1 [U14](X1,X2,X3) = X1 [U15](X1,X2) = X1 [U16](X) = X [U21](X1,X2) = X1 [U22](X1,X2) = X1 [U23](X) = X [U31](X1,X2) = X1 [U32](X) = X [U41](X) = X [U51](X1,X2) = 2.X1 + 2.X2 + 1 [U52](X1,X2) = X1 + X2 + 2 [U61](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U62](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U63](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U64](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [active](X) = X [isNat](X) = 2 [isNatKind](X) = 2 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X + 2 [0] = 2 [fSNonEmpty] = 0 [tt] = 2 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = 2.X + 2 [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = 2.X + 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(U52(tt,N:S)) -> MARK(N:S) ACTIVE(U61(tt,M:S,N:S)) -> MARK(U62(isNatKind(M:S),M:S,N:S)) ACTIVE(U62(tt,M:S,N:S)) -> MARK(U63(isNat(N:S),M:S,N:S)) ACTIVE(U63(tt,M:S,N:S)) -> MARK(U64(isNatKind(N:S),M:S,N:S)) ACTIVE(U64(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) ACTIVE(plus(N:S,s(M:S))) -> MARK(U61(isNat(M:S),M:S,N:S)) ACTIVE(plus(N:S,0)) -> MARK(U51(isNat(N:S),N:S)) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> ACTIVE(U16(mark(X:S))) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> ACTIVE(U23(mark(X:S))) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> ACTIVE(U32(mark(X:S))) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> ACTIVE(U41(mark(X:S))) MARK(U41(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X1:S,X2:S)) -> ACTIVE(U52(mark(X1:S),X2:S)) MARK(U52(X1:S,X2:S)) -> MARK(X1:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(U52(tt,N:S)) -> MARK(N:S) ACTIVE(U61(tt,M:S,N:S)) -> MARK(U62(isNatKind(M:S),M:S,N:S)) ACTIVE(U62(tt,M:S,N:S)) -> MARK(U63(isNat(N:S),M:S,N:S)) ACTIVE(U63(tt,M:S,N:S)) -> MARK(U64(isNatKind(N:S),M:S,N:S)) ACTIVE(U64(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) ACTIVE(plus(N:S,s(M:S))) -> MARK(U61(isNat(M:S),M:S,N:S)) ACTIVE(plus(N:S,0)) -> MARK(U51(isNat(N:S),N:S)) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> ACTIVE(U16(mark(X:S))) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> ACTIVE(U23(mark(X:S))) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> ACTIVE(U32(mark(X:S))) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> ACTIVE(U41(mark(X:S))) MARK(U41(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X1:S,X2:S)) -> ACTIVE(U52(mark(X1:S),X2:S)) MARK(U52(X1:S,X2:S)) -> MARK(X1:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(U52(tt,N:S)) -> MARK(N:S) ACTIVE(U61(tt,M:S,N:S)) -> MARK(U62(isNatKind(M:S),M:S,N:S)) ACTIVE(U62(tt,M:S,N:S)) -> MARK(U63(isNat(N:S),M:S,N:S)) ACTIVE(U63(tt,M:S,N:S)) -> MARK(U64(isNatKind(N:S),M:S,N:S)) ACTIVE(U64(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) ACTIVE(plus(N:S,s(M:S))) -> MARK(U61(isNat(M:S),M:S,N:S)) ACTIVE(plus(N:S,0)) -> MARK(U51(isNat(N:S),N:S)) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> ACTIVE(U16(mark(X:S))) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> ACTIVE(U23(mark(X:S))) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> ACTIVE(U32(mark(X:S))) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> ACTIVE(U41(mark(X:S))) MARK(U41(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X1:S,X2:S)) -> ACTIVE(U52(mark(X1:S),X2:S)) MARK(U52(X1:S,X2:S)) -> MARK(X1:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = X1 [U12](X1,X2,X3) = X1 [U13](X1,X2,X3) = X1 [U14](X1,X2,X3) = X1 [U15](X1,X2) = X1 [U16](X) = X [U21](X1,X2) = X1 [U22](X1,X2) = X1 [U23](X) = X [U31](X1,X2) = X1 [U32](X) = X [U41](X) = X [U51](X1,X2) = 2.X1 + 2.X2 + 2 [U52](X1,X2) = 2.X1 + X2 + 1 [U61](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U62](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U63](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U64](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [active](X) = X [isNat](X) = 2 [isNatKind](X) = 2 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X + 2 [0] = 2 [fSNonEmpty] = 0 [tt] = 2 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = 2.X + 2 [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = 2.X + 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(U61(tt,M:S,N:S)) -> MARK(U62(isNatKind(M:S),M:S,N:S)) ACTIVE(U62(tt,M:S,N:S)) -> MARK(U63(isNat(N:S),M:S,N:S)) ACTIVE(U63(tt,M:S,N:S)) -> MARK(U64(isNatKind(N:S),M:S,N:S)) ACTIVE(U64(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) ACTIVE(plus(N:S,s(M:S))) -> MARK(U61(isNat(M:S),M:S,N:S)) ACTIVE(plus(N:S,0)) -> MARK(U51(isNat(N:S),N:S)) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> ACTIVE(U16(mark(X:S))) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> ACTIVE(U23(mark(X:S))) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> ACTIVE(U32(mark(X:S))) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> ACTIVE(U41(mark(X:S))) MARK(U41(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X1:S,X2:S)) -> ACTIVE(U52(mark(X1:S),X2:S)) MARK(U52(X1:S,X2:S)) -> MARK(X1:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(U61(tt,M:S,N:S)) -> MARK(U62(isNatKind(M:S),M:S,N:S)) ACTIVE(U62(tt,M:S,N:S)) -> MARK(U63(isNat(N:S),M:S,N:S)) ACTIVE(U63(tt,M:S,N:S)) -> MARK(U64(isNatKind(N:S),M:S,N:S)) ACTIVE(U64(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) ACTIVE(plus(N:S,s(M:S))) -> MARK(U61(isNat(M:S),M:S,N:S)) ACTIVE(plus(N:S,0)) -> MARK(U51(isNat(N:S),N:S)) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> ACTIVE(U16(mark(X:S))) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> ACTIVE(U23(mark(X:S))) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> ACTIVE(U32(mark(X:S))) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> ACTIVE(U41(mark(X:S))) MARK(U41(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X1:S,X2:S)) -> ACTIVE(U52(mark(X1:S),X2:S)) MARK(U52(X1:S,X2:S)) -> MARK(X1:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(U61(tt,M:S,N:S)) -> MARK(U62(isNatKind(M:S),M:S,N:S)) ACTIVE(U62(tt,M:S,N:S)) -> MARK(U63(isNat(N:S),M:S,N:S)) ACTIVE(U63(tt,M:S,N:S)) -> MARK(U64(isNatKind(N:S),M:S,N:S)) ACTIVE(U64(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) ACTIVE(plus(N:S,s(M:S))) -> MARK(U61(isNat(M:S),M:S,N:S)) ACTIVE(plus(N:S,0)) -> MARK(U51(isNat(N:S),N:S)) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> ACTIVE(U16(mark(X:S))) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> ACTIVE(U23(mark(X:S))) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> ACTIVE(U32(mark(X:S))) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> ACTIVE(U41(mark(X:S))) MARK(U41(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X1:S,X2:S)) -> ACTIVE(U52(mark(X1:S),X2:S)) MARK(U52(X1:S,X2:S)) -> MARK(X1:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = X1 [U12](X1,X2,X3) = X1 [U13](X1,X2,X3) = X1 [U14](X1,X2,X3) = X1 [U15](X1,X2) = X1 [U16](X) = X [U21](X1,X2) = X1 [U22](X1,X2) = X1 [U23](X) = X [U31](X1,X2) = X1 [U32](X) = X [U41](X) = X [U51](X1,X2) = 2.X1 + 2.X2 + 2 [U52](X1,X2) = X1 + X2 + 2 [U61](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U62](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U63](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U64](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [active](X) = X [isNat](X) = 2 [isNatKind](X) = 2 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X + 2 [0] = 2 [fSNonEmpty] = 0 [tt] = 2 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = X + 2 [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = X + 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(U62(tt,M:S,N:S)) -> MARK(U63(isNat(N:S),M:S,N:S)) ACTIVE(U63(tt,M:S,N:S)) -> MARK(U64(isNatKind(N:S),M:S,N:S)) ACTIVE(U64(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) ACTIVE(plus(N:S,s(M:S))) -> MARK(U61(isNat(M:S),M:S,N:S)) ACTIVE(plus(N:S,0)) -> MARK(U51(isNat(N:S),N:S)) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> ACTIVE(U16(mark(X:S))) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> ACTIVE(U23(mark(X:S))) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> ACTIVE(U32(mark(X:S))) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> ACTIVE(U41(mark(X:S))) MARK(U41(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X1:S,X2:S)) -> ACTIVE(U52(mark(X1:S),X2:S)) MARK(U52(X1:S,X2:S)) -> MARK(X1:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(U62(tt,M:S,N:S)) -> MARK(U63(isNat(N:S),M:S,N:S)) ACTIVE(U63(tt,M:S,N:S)) -> MARK(U64(isNatKind(N:S),M:S,N:S)) ACTIVE(U64(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) ACTIVE(plus(N:S,s(M:S))) -> MARK(U61(isNat(M:S),M:S,N:S)) ACTIVE(plus(N:S,0)) -> MARK(U51(isNat(N:S),N:S)) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> ACTIVE(U16(mark(X:S))) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> ACTIVE(U23(mark(X:S))) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> ACTIVE(U32(mark(X:S))) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> ACTIVE(U41(mark(X:S))) MARK(U41(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X1:S,X2:S)) -> ACTIVE(U52(mark(X1:S),X2:S)) MARK(U52(X1:S,X2:S)) -> MARK(X1:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(U62(tt,M:S,N:S)) -> MARK(U63(isNat(N:S),M:S,N:S)) ACTIVE(U63(tt,M:S,N:S)) -> MARK(U64(isNatKind(N:S),M:S,N:S)) ACTIVE(U64(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) ACTIVE(plus(N:S,s(M:S))) -> MARK(U61(isNat(M:S),M:S,N:S)) ACTIVE(plus(N:S,0)) -> MARK(U51(isNat(N:S),N:S)) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> ACTIVE(U16(mark(X:S))) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> ACTIVE(U23(mark(X:S))) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> ACTIVE(U32(mark(X:S))) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> ACTIVE(U41(mark(X:S))) MARK(U41(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X1:S,X2:S)) -> ACTIVE(U52(mark(X1:S),X2:S)) MARK(U52(X1:S,X2:S)) -> MARK(X1:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = X1 [U12](X1,X2,X3) = X1 [U13](X1,X2,X3) = X1 [U14](X1,X2,X3) = X1 [U15](X1,X2) = X1 [U16](X) = X [U21](X1,X2) = X1 [U22](X1,X2) = X1 [U23](X) = X [U31](X1,X2) = X1 [U32](X) = X [U41](X) = X [U51](X1,X2) = 2.X1 + 2.X2 + 2 [U52](X1,X2) = 2.X1 + 2.X2 + 2 [U61](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U62](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U63](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 1 [U64](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 [active](X) = X [isNat](X) = 2 [isNatKind](X) = 2 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X + 2 [0] = 2 [fSNonEmpty] = 0 [tt] = 2 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = 2.X + 2 [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = 2.X + 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(U63(tt,M:S,N:S)) -> MARK(U64(isNatKind(N:S),M:S,N:S)) ACTIVE(U64(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) ACTIVE(plus(N:S,s(M:S))) -> MARK(U61(isNat(M:S),M:S,N:S)) ACTIVE(plus(N:S,0)) -> MARK(U51(isNat(N:S),N:S)) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> ACTIVE(U16(mark(X:S))) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> ACTIVE(U23(mark(X:S))) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> ACTIVE(U32(mark(X:S))) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> ACTIVE(U41(mark(X:S))) MARK(U41(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X1:S,X2:S)) -> ACTIVE(U52(mark(X1:S),X2:S)) MARK(U52(X1:S,X2:S)) -> MARK(X1:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(U63(tt,M:S,N:S)) -> MARK(U64(isNatKind(N:S),M:S,N:S)) ACTIVE(U64(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) ACTIVE(plus(N:S,s(M:S))) -> MARK(U61(isNat(M:S),M:S,N:S)) ACTIVE(plus(N:S,0)) -> MARK(U51(isNat(N:S),N:S)) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> ACTIVE(U16(mark(X:S))) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> ACTIVE(U23(mark(X:S))) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> ACTIVE(U32(mark(X:S))) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> ACTIVE(U41(mark(X:S))) MARK(U41(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X1:S,X2:S)) -> ACTIVE(U52(mark(X1:S),X2:S)) MARK(U52(X1:S,X2:S)) -> MARK(X1:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(U63(tt,M:S,N:S)) -> MARK(U64(isNatKind(N:S),M:S,N:S)) ACTIVE(U64(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) ACTIVE(plus(N:S,s(M:S))) -> MARK(U61(isNat(M:S),M:S,N:S)) ACTIVE(plus(N:S,0)) -> MARK(U51(isNat(N:S),N:S)) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> ACTIVE(U16(mark(X:S))) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> ACTIVE(U23(mark(X:S))) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> ACTIVE(U32(mark(X:S))) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> ACTIVE(U41(mark(X:S))) MARK(U41(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X1:S,X2:S)) -> ACTIVE(U52(mark(X1:S),X2:S)) MARK(U52(X1:S,X2:S)) -> MARK(X1:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = X1 [U12](X1,X2,X3) = X1 [U13](X1,X2,X3) = X1 [U14](X1,X2,X3) = X1 [U15](X1,X2) = X1 [U16](X) = X [U21](X1,X2) = X1 [U22](X1,X2) = X1 [U23](X) = X [U31](X1,X2) = X1 [U32](X) = X [U41](X) = X [U51](X1,X2) = X1 + 2.X2 + 2 [U52](X1,X2) = X1 + 2.X2 + 2 [U61](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U62](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U63](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U64](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 1 [active](X) = X [isNat](X) = 2 [isNatKind](X) = 2 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X + 2 [0] = 2 [fSNonEmpty] = 0 [tt] = 2 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = 2.X + 2 [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = 2.X + 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(U64(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) ACTIVE(plus(N:S,s(M:S))) -> MARK(U61(isNat(M:S),M:S,N:S)) ACTIVE(plus(N:S,0)) -> MARK(U51(isNat(N:S),N:S)) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> ACTIVE(U16(mark(X:S))) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> ACTIVE(U23(mark(X:S))) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> ACTIVE(U32(mark(X:S))) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> ACTIVE(U41(mark(X:S))) MARK(U41(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X1:S,X2:S)) -> ACTIVE(U52(mark(X1:S),X2:S)) MARK(U52(X1:S,X2:S)) -> MARK(X1:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(U64(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) ACTIVE(plus(N:S,s(M:S))) -> MARK(U61(isNat(M:S),M:S,N:S)) ACTIVE(plus(N:S,0)) -> MARK(U51(isNat(N:S),N:S)) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> ACTIVE(U16(mark(X:S))) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> ACTIVE(U23(mark(X:S))) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> ACTIVE(U32(mark(X:S))) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> ACTIVE(U41(mark(X:S))) MARK(U41(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X1:S,X2:S)) -> ACTIVE(U52(mark(X1:S),X2:S)) MARK(U52(X1:S,X2:S)) -> MARK(X1:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(U64(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) ACTIVE(plus(N:S,s(M:S))) -> MARK(U61(isNat(M:S),M:S,N:S)) ACTIVE(plus(N:S,0)) -> MARK(U51(isNat(N:S),N:S)) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> ACTIVE(U16(mark(X:S))) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> ACTIVE(U23(mark(X:S))) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> ACTIVE(U32(mark(X:S))) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> ACTIVE(U41(mark(X:S))) MARK(U41(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X1:S,X2:S)) -> ACTIVE(U52(mark(X1:S),X2:S)) MARK(U52(X1:S,X2:S)) -> MARK(X1:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = X1 [U12](X1,X2,X3) = X1 [U13](X1,X2,X3) = X1 [U14](X1,X2,X3) = X1 [U15](X1,X2) = X1 [U16](X) = X [U21](X1,X2) = X1 [U22](X1,X2) = X1 [U23](X) = X [U31](X1,X2) = X1 [U32](X) = X [U41](X) = X [U51](X1,X2) = 2.X1 + 2.X2 + 1 [U52](X1,X2) = X1 + 2.X2 + 2 [U61](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U62](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U63](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U64](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [active](X) = X [isNat](X) = 2 [isNatKind](X) = 2 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X + 2 [0] = 2 [fSNonEmpty] = 0 [tt] = 2 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = 2.X + 2 [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = 2.X + 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) ACTIVE(plus(N:S,s(M:S))) -> MARK(U61(isNat(M:S),M:S,N:S)) ACTIVE(plus(N:S,0)) -> MARK(U51(isNat(N:S),N:S)) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> ACTIVE(U16(mark(X:S))) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> ACTIVE(U23(mark(X:S))) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> ACTIVE(U32(mark(X:S))) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> ACTIVE(U41(mark(X:S))) MARK(U41(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X1:S,X2:S)) -> ACTIVE(U52(mark(X1:S),X2:S)) MARK(U52(X1:S,X2:S)) -> MARK(X1:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) ACTIVE(plus(N:S,s(M:S))) -> MARK(U61(isNat(M:S),M:S,N:S)) ACTIVE(plus(N:S,0)) -> MARK(U51(isNat(N:S),N:S)) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> ACTIVE(U16(mark(X:S))) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> ACTIVE(U23(mark(X:S))) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> ACTIVE(U32(mark(X:S))) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> ACTIVE(U41(mark(X:S))) MARK(U41(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X1:S,X2:S)) -> ACTIVE(U52(mark(X1:S),X2:S)) MARK(U52(X1:S,X2:S)) -> MARK(X1:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) ACTIVE(plus(N:S,s(M:S))) -> MARK(U61(isNat(M:S),M:S,N:S)) ACTIVE(plus(N:S,0)) -> MARK(U51(isNat(N:S),N:S)) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> ACTIVE(U16(mark(X:S))) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> ACTIVE(U23(mark(X:S))) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> ACTIVE(U32(mark(X:S))) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> ACTIVE(U41(mark(X:S))) MARK(U41(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X1:S,X2:S)) -> ACTIVE(U52(mark(X1:S),X2:S)) MARK(U52(X1:S,X2:S)) -> MARK(X1:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = X1 [U12](X1,X2,X3) = X1 [U13](X1,X2,X3) = X1 [U14](X1,X2,X3) = X1 [U15](X1,X2) = X1 [U16](X) = X [U21](X1,X2) = X1 [U22](X1,X2) = X1 [U23](X) = X [U31](X1,X2) = X1 [U32](X) = X [U41](X) = X [U51](X1,X2) = 2.X1 + 2.X2 + 2 [U52](X1,X2) = X1 + 2.X2 + 1 [U61](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 1 [U62](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 1 [U63](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U64](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [active](X) = X [isNat](X) = 2 [isNatKind](X) = 2 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X + 2 [0] = 2 [fSNonEmpty] = 0 [tt] = 2 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = 2.X + 2 [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = 2.X + 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) ACTIVE(plus(N:S,0)) -> MARK(U51(isNat(N:S),N:S)) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> ACTIVE(U16(mark(X:S))) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> ACTIVE(U23(mark(X:S))) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> ACTIVE(U32(mark(X:S))) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> ACTIVE(U41(mark(X:S))) MARK(U41(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X1:S,X2:S)) -> ACTIVE(U52(mark(X1:S),X2:S)) MARK(U52(X1:S,X2:S)) -> MARK(X1:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) ACTIVE(plus(N:S,0)) -> MARK(U51(isNat(N:S),N:S)) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> ACTIVE(U16(mark(X:S))) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> ACTIVE(U23(mark(X:S))) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> ACTIVE(U32(mark(X:S))) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> ACTIVE(U41(mark(X:S))) MARK(U41(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X1:S,X2:S)) -> ACTIVE(U52(mark(X1:S),X2:S)) MARK(U52(X1:S,X2:S)) -> MARK(X1:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) ACTIVE(plus(N:S,0)) -> MARK(U51(isNat(N:S),N:S)) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> ACTIVE(U16(mark(X:S))) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> ACTIVE(U23(mark(X:S))) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> ACTIVE(U32(mark(X:S))) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> ACTIVE(U41(mark(X:S))) MARK(U41(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X1:S,X2:S)) -> ACTIVE(U52(mark(X1:S),X2:S)) MARK(U52(X1:S,X2:S)) -> MARK(X1:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = X1 [U12](X1,X2,X3) = X1 [U13](X1,X2,X3) = X1 [U14](X1,X2,X3) = X1 [U15](X1,X2) = X1 [U16](X) = X [U21](X1,X2) = X1 [U22](X1,X2) = X1 [U23](X) = X [U31](X1,X2) = X1 [U32](X) = X [U41](X) = X [U51](X1,X2) = X1 + 2.X2 + 1 [U52](X1,X2) = X1 + 2.X2 [U61](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U62](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U63](X1,X2,X3) = X1 + 2.X2 + 2.X3 [U64](X1,X2,X3) = X1 + 2.X2 + 2.X3 [active](X) = X [isNat](X) = 2 [isNatKind](X) = 2 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 [s](X) = X + 2 [0] = 2 [fSNonEmpty] = 0 [tt] = 2 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = 2.X + 2 [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = 2.X + 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> ACTIVE(U16(mark(X:S))) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> ACTIVE(U23(mark(X:S))) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> ACTIVE(U32(mark(X:S))) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> ACTIVE(U41(mark(X:S))) MARK(U41(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X1:S,X2:S)) -> ACTIVE(U52(mark(X1:S),X2:S)) MARK(U52(X1:S,X2:S)) -> MARK(X1:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> ACTIVE(U16(mark(X:S))) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> ACTIVE(U23(mark(X:S))) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> ACTIVE(U32(mark(X:S))) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> ACTIVE(U41(mark(X:S))) MARK(U41(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X1:S,X2:S)) -> ACTIVE(U52(mark(X1:S),X2:S)) MARK(U52(X1:S,X2:S)) -> MARK(X1:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> ACTIVE(U16(mark(X:S))) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> ACTIVE(U23(mark(X:S))) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> ACTIVE(U32(mark(X:S))) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> ACTIVE(U41(mark(X:S))) MARK(U41(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X1:S,X2:S)) -> ACTIVE(U52(mark(X1:S),X2:S)) MARK(U52(X1:S,X2:S)) -> MARK(X1:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 1 [U12](X1,X2,X3) = 1 [U13](X1,X2,X3) = 1 [U14](X1,X2,X3) = 1 [U15](X1,X2) = 1 [U16](X) = 0 [U21](X1,X2) = 1 [U22](X1,X2) = 1 [U23](X) = 1 [U31](X1,X2) = 1 [U32](X) = 1 [U41](X) = 0 [U51](X1,X2) = 1 [U52](X1,X2) = 1 [U61](X1,X2,X3) = 1 [U62](X1,X2,X3) = 1 [U63](X1,X2,X3) = 1 [U64](X1,X2,X3) = 1 [active](X) = 2 [isNat](X) = 1 [isNatKind](X) = 1 [mark](X) = 2 [plus](X1,X2) = 1 [s](X) = 1 [0] = 2 [fSNonEmpty] = 0 [tt] = 2 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = 2.X [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> ACTIVE(U23(mark(X:S))) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> ACTIVE(U32(mark(X:S))) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> ACTIVE(U41(mark(X:S))) MARK(U41(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X1:S,X2:S)) -> ACTIVE(U52(mark(X1:S),X2:S)) MARK(U52(X1:S,X2:S)) -> MARK(X1:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> ACTIVE(U23(mark(X:S))) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> ACTIVE(U32(mark(X:S))) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> ACTIVE(U41(mark(X:S))) MARK(U41(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X1:S,X2:S)) -> ACTIVE(U52(mark(X1:S),X2:S)) MARK(U52(X1:S,X2:S)) -> MARK(X1:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> ACTIVE(U23(mark(X:S))) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> ACTIVE(U32(mark(X:S))) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> ACTIVE(U41(mark(X:S))) MARK(U41(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X1:S,X2:S)) -> ACTIVE(U52(mark(X1:S),X2:S)) MARK(U52(X1:S,X2:S)) -> MARK(X1:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 1 [U12](X1,X2,X3) = 1 [U13](X1,X2,X3) = 1 [U14](X1,X2,X3) = 1 [U15](X1,X2) = 1 [U16](X) = 0 [U21](X1,X2) = 1 [U22](X1,X2) = 1 [U23](X) = 0 [U31](X1,X2) = 1 [U32](X) = 0 [U41](X) = 0 [U51](X1,X2) = 1 [U52](X1,X2) = 1 [U61](X1,X2,X3) = 0 [U62](X1,X2,X3) = 0 [U63](X1,X2,X3) = 1 [U64](X1,X2,X3) = 0 [active](X) = 2 [isNat](X) = 1 [isNatKind](X) = 1 [mark](X) = 2 [plus](X1,X2) = 1 [s](X) = 1 [0] = 1 [fSNonEmpty] = 0 [tt] = 0 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = 2.X [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> ACTIVE(U32(mark(X:S))) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> ACTIVE(U41(mark(X:S))) MARK(U41(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X1:S,X2:S)) -> ACTIVE(U52(mark(X1:S),X2:S)) MARK(U52(X1:S,X2:S)) -> MARK(X1:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> ACTIVE(U32(mark(X:S))) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> ACTIVE(U41(mark(X:S))) MARK(U41(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X1:S,X2:S)) -> ACTIVE(U52(mark(X1:S),X2:S)) MARK(U52(X1:S,X2:S)) -> MARK(X1:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> ACTIVE(U32(mark(X:S))) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> ACTIVE(U41(mark(X:S))) MARK(U41(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X1:S,X2:S)) -> ACTIVE(U52(mark(X1:S),X2:S)) MARK(U52(X1:S,X2:S)) -> MARK(X1:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 1 [U12](X1,X2,X3) = 1 [U13](X1,X2,X3) = 1 [U14](X1,X2,X3) = 1 [U15](X1,X2) = 1 [U16](X) = 2 [U21](X1,X2) = 1 [U22](X1,X2) = 1 [U23](X) = 2 [U31](X1,X2) = 1 [U32](X) = 0 [U41](X) = 0 [U51](X1,X2) = 1 [U52](X1,X2) = 1 [U61](X1,X2,X3) = 1 [U62](X1,X2,X3) = 1 [U63](X1,X2,X3) = 1 [U64](X1,X2,X3) = 0 [active](X) = 2 [isNat](X) = 1 [isNatKind](X) = 1 [mark](X) = 2 [plus](X1,X2) = 0 [s](X) = 0 [0] = 2 [fSNonEmpty] = 0 [tt] = 1 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = 2.X [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> ACTIVE(U41(mark(X:S))) MARK(U41(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X1:S,X2:S)) -> ACTIVE(U52(mark(X1:S),X2:S)) MARK(U52(X1:S,X2:S)) -> MARK(X1:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> ACTIVE(U41(mark(X:S))) MARK(U41(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X1:S,X2:S)) -> ACTIVE(U52(mark(X1:S),X2:S)) MARK(U52(X1:S,X2:S)) -> MARK(X1:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> ACTIVE(U41(mark(X:S))) MARK(U41(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X1:S,X2:S)) -> ACTIVE(U52(mark(X1:S),X2:S)) MARK(U52(X1:S,X2:S)) -> MARK(X1:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 2 [U12](X1,X2,X3) = 2 [U13](X1,X2,X3) = 2 [U14](X1,X2,X3) = 2 [U15](X1,X2) = 2 [U16](X) = 0 [U21](X1,X2) = 2 [U22](X1,X2) = 2 [U23](X) = 0 [U31](X1,X2) = 2 [U32](X) = 0 [U41](X) = 1 [U51](X1,X2) = 1 [U52](X1,X2) = 0 [U61](X1,X2,X3) = 2 [U62](X1,X2,X3) = 0 [U63](X1,X2,X3) = 2 [U64](X1,X2,X3) = 0 [active](X) = 2 [isNat](X) = 2 [isNatKind](X) = 2 [mark](X) = 2 [plus](X1,X2) = 2 [s](X) = 1 [0] = 2 [fSNonEmpty] = 0 [tt] = 0 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = X [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X1:S,X2:S)) -> ACTIVE(U52(mark(X1:S),X2:S)) MARK(U52(X1:S,X2:S)) -> MARK(X1:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X1:S,X2:S)) -> ACTIVE(U52(mark(X1:S),X2:S)) MARK(U52(X1:S,X2:S)) -> MARK(X1:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> ACTIVE(U51(mark(X1:S),X2:S)) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X1:S,X2:S)) -> ACTIVE(U52(mark(X1:S),X2:S)) MARK(U52(X1:S,X2:S)) -> MARK(X1:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 1 [U12](X1,X2,X3) = 1 [U13](X1,X2,X3) = 1 [U14](X1,X2,X3) = 1 [U15](X1,X2) = 1 [U16](X) = 0 [U21](X1,X2) = 1 [U22](X1,X2) = 1 [U23](X) = 0 [U31](X1,X2) = 1 [U32](X) = 2 [U41](X) = 0 [U51](X1,X2) = 0 [U52](X1,X2) = 1 [U61](X1,X2,X3) = 1 [U62](X1,X2,X3) = 1 [U63](X1,X2,X3) = 1 [U64](X1,X2,X3) = 0 [active](X) = 2 [isNat](X) = 1 [isNatKind](X) = 1 [mark](X) = 2 [plus](X1,X2) = 1 [s](X) = 1 [0] = 1 [fSNonEmpty] = 0 [tt] = 2 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = X + 1 [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X1:S,X2:S)) -> ACTIVE(U52(mark(X1:S),X2:S)) MARK(U52(X1:S,X2:S)) -> MARK(X1:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X1:S,X2:S)) -> ACTIVE(U52(mark(X1:S),X2:S)) MARK(U52(X1:S,X2:S)) -> MARK(X1:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U51(X1:S,X2:S)) -> MARK(X1:S) MARK(U52(X1:S,X2:S)) -> ACTIVE(U52(mark(X1:S),X2:S)) MARK(U52(X1:S,X2:S)) -> MARK(X1:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = X1 [U12](X1,X2,X3) = X1 [U13](X1,X2,X3) = X1 [U14](X1,X2,X3) = X1 [U15](X1,X2) = X1 [U16](X) = X [U21](X1,X2) = X1 [U22](X1,X2) = X1 [U23](X) = X [U31](X1,X2) = X1 [U32](X) = X [U41](X) = X [U51](X1,X2) = X1 + 2.X2 + 2 [U52](X1,X2) = 2.X1 + X2 [U61](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U62](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U63](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U64](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [active](X) = X [isNat](X) = 2 [isNatKind](X) = 2 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X + 2 [0] = 1 [fSNonEmpty] = 0 [tt] = 2 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = X + 2 [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = X + 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U52(X1:S,X2:S)) -> ACTIVE(U52(mark(X1:S),X2:S)) MARK(U52(X1:S,X2:S)) -> MARK(X1:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U52(X1:S,X2:S)) -> ACTIVE(U52(mark(X1:S),X2:S)) MARK(U52(X1:S,X2:S)) -> MARK(X1:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U52(X1:S,X2:S)) -> ACTIVE(U52(mark(X1:S),X2:S)) MARK(U52(X1:S,X2:S)) -> MARK(X1:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 2.X1 [U12](X1,X2,X3) = 2.X1 [U13](X1,X2,X3) = 2.X1 [U14](X1,X2,X3) = 2.X1 [U15](X1,X2) = 2.X1 [U16](X) = 2.X [U21](X1,X2) = 2.X1 [U22](X1,X2) = 2.X1 [U23](X) = 2.X [U31](X1,X2) = 2.X1 [U32](X) = 2.X [U41](X) = 2.X [U51](X1,X2) = 2.X1 + 2.X2 + 2 [U52](X1,X2) = 2.X1 + 2.X2 + 2 [U61](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U62](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U63](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U64](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [active](X) = X [isNat](X) = 0 [isNatKind](X) = 0 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 [s](X) = X + 2 [0] = 2 [fSNonEmpty] = 0 [tt] = 0 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = 2 [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = 2.X + 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U52(X1:S,X2:S)) -> MARK(X1:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U52(X1:S,X2:S)) -> MARK(X1:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U52(X1:S,X2:S)) -> MARK(X1:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = X1 [U12](X1,X2,X3) = X1 [U13](X1,X2,X3) = X1 [U14](X1,X2,X3) = X1 [U15](X1,X2) = X1 [U16](X) = X [U21](X1,X2) = X1 [U22](X1,X2) = X1 [U23](X) = X [U31](X1,X2) = X1 [U32](X) = X [U41](X) = X [U51](X1,X2) = 2.X1 + 2.X2 + 2 [U52](X1,X2) = X1 + 2.X2 + 2 [U61](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U62](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U63](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U64](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [active](X) = X [isNat](X) = 2 [isNatKind](X) = 2 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X + 2 [0] = 2 [fSNonEmpty] = 0 [tt] = 2 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = 2.X + 2 [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = 2.X + 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> ACTIVE(U61(mark(X1:S),X2:S,X3:S)) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 2.X1 [U12](X1,X2,X3) = 2.X1 [U13](X1,X2,X3) = 2.X1 [U14](X1,X2,X3) = 2.X1 [U15](X1,X2) = 2.X1 [U16](X) = 2.X [U21](X1,X2) = 2.X1 [U22](X1,X2) = 2.X1 [U23](X) = 2.X [U31](X1,X2) = 2.X1 [U32](X) = 2.X [U41](X) = 2.X [U51](X1,X2) = 2.X1 + 2.X2 + 2 [U52](X1,X2) = 2.X1 + 2.X2 + 2 [U61](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U62](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U63](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U64](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [active](X) = X [isNat](X) = 0 [isNatKind](X) = 0 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X [0] = 2 [fSNonEmpty] = 0 [tt] = 0 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = 2 [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = 2.X + 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = X1 [U12](X1,X2,X3) = X1 [U13](X1,X2,X3) = X1 [U14](X1,X2,X3) = X1 [U15](X1,X2) = X1 [U16](X) = X [U21](X1,X2) = X1 [U22](X1,X2) = X1 [U23](X) = X [U31](X1,X2) = X1 [U32](X) = X [U41](X) = X [U51](X1,X2) = 2.X2 + 2 [U52](X1,X2) = X1 + 2.X2 [U61](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 1 [U62](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U63](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 [U64](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 1 [active](X) = X [isNat](X) = 2 [isNatKind](X) = 2 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 1 [s](X) = X + 2 [0] = 2 [fSNonEmpty] = 0 [tt] = 2 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = 2.X + 2 [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = 2.X + 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U62(X1:S,X2:S,X3:S)) -> ACTIVE(U62(mark(X1:S),X2:S,X3:S)) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 2.X1 [U12](X1,X2,X3) = 2.X1 [U13](X1,X2,X3) = 2.X1 [U14](X1,X2,X3) = 2.X1 [U15](X1,X2) = 2.X1 [U16](X) = 2.X [U21](X1,X2) = 2.X1 [U22](X1,X2) = 2.X1 [U23](X) = 2.X [U31](X1,X2) = 2.X1 [U32](X) = 2.X [U41](X) = 2.X [U51](X1,X2) = 2.X1 + 2.X2 + 2 [U52](X1,X2) = 2.X1 + 2.X2 + 2 [U61](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U62](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U63](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U64](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [active](X) = X [isNat](X) = 0 [isNatKind](X) = 0 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X [0] = 0 [fSNonEmpty] = 0 [tt] = 0 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = X + 2 [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = 2.X + 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 2.X1 [U12](X1,X2,X3) = 2.X1 [U13](X1,X2,X3) = 2.X1 [U14](X1,X2,X3) = 2.X1 [U15](X1,X2) = 2.X1 [U16](X) = 2.X [U21](X1,X2) = 2.X1 [U22](X1,X2) = 2.X1 [U23](X) = 2.X [U31](X1,X2) = 2.X1 [U32](X) = X [U41](X) = 2.X [U51](X1,X2) = 2.X1 + 2.X2 + 2 [U52](X1,X2) = X2 + 2 [U61](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U62](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U63](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U64](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [active](X) = X [isNat](X) = 0 [isNatKind](X) = 0 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X [0] = 1 [fSNonEmpty] = 0 [tt] = 0 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = 2 [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = 2.X + 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U63(X1:S,X2:S,X3:S)) -> ACTIVE(U63(mark(X1:S),X2:S,X3:S)) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = X1 [U12](X1,X2,X3) = 2.X1 [U13](X1,X2,X3) = 2.X1 [U14](X1,X2,X3) = 2.X1 [U15](X1,X2) = 2.X1 [U16](X) = 2.X [U21](X1,X2) = 2.X1 [U22](X1,X2) = 2.X1 [U23](X) = 2.X [U31](X1,X2) = X1 [U32](X) = 2.X [U41](X) = 2.X [U51](X1,X2) = 2.X1 + 2.X2 + 1 [U52](X1,X2) = 2.X1 + 2.X2 + 1 [U61](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U62](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U63](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U64](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [active](X) = X [isNat](X) = 0 [isNatKind](X) = 0 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X [0] = 0 [fSNonEmpty] = 0 [tt] = 0 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = X + 2 [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = 2.X + 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = X1 [U12](X1,X2,X3) = X1 [U13](X1,X2,X3) = X1 [U14](X1,X2,X3) = X1 [U15](X1,X2) = X1 [U16](X) = X [U21](X1,X2) = X1 [U22](X1,X2) = X1 [U23](X) = X [U31](X1,X2) = X1 [U32](X) = X [U41](X) = X [U51](X1,X2) = 2.X1 + 2.X2 + 2 [U52](X1,X2) = 2.X1 + 2.X2 + 1 [U61](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U62](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U63](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U64](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [active](X) = X [isNat](X) = 2 [isNatKind](X) = 2 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X + 2 [0] = 2 [fSNonEmpty] = 0 [tt] = 2 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = 2.X + 2 [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = 2.X + 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U12](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 1 [U13](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U14](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U15](X1,X2) = X1 + 2.X2 + 2 [U16](X) = X + 2 [U21](X1,X2) = X1 + 2.X2 [U22](X1,X2) = X1 + 2.X2 [U23](X) = X [U31](X1,X2) = X1 [U32](X) = X [U41](X) = X [U51](X1,X2) = 2.X2 + 2 [U52](X1,X2) = X2 + 2 [U61](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U62](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U63](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U64](X1,X2,X3) = X1 + 2.X2 + 2.X3 [active](X) = X [isNat](X) = 2.X + 2 [isNatKind](X) = 2 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X [0] = 2 [fSNonEmpty] = 0 [tt] = 2 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = 2.X + 1 [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = 2.X + 1 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(U12(tt,V1:S,V2:S)) -> MARK(U13(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U12](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U13](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U14](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 [U15](X1,X2) = X1 + 2.X2 + 2 [U16](X) = X + 2 [U21](X1,X2) = X1 + 2.X2 [U22](X1,X2) = X1 + 2.X2 [U23](X) = X [U31](X1,X2) = X1 [U32](X) = X [U41](X) = X [U51](X1,X2) = 2.X2 + 2 [U52](X1,X2) = 2.X2 + 2 [U61](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U62](X1,X2,X3) = X1 + 2.X2 + 2.X3 [U63](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U64](X1,X2,X3) = X1 + 2.X2 + 2.X3 [active](X) = X [isNat](X) = 2.X + 2 [isNatKind](X) = 2 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X [0] = 1 [fSNonEmpty] = 0 [tt] = 2 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = 2.X [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = 2.X [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(U13(tt,V1:S,V2:S)) -> MARK(U14(isNatKind(V2:S),V1:S,V2:S)) ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U12](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U13](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U14](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 1 [U15](X1,X2) = X1 + 2.X2 + 1 [U16](X) = X + 1 [U21](X1,X2) = X1 + 2.X2 [U22](X1,X2) = X1 + 2.X2 [U23](X) = X [U31](X1,X2) = X1 [U32](X) = X [U41](X) = X [U51](X1,X2) = 2.X2 + 2 [U52](X1,X2) = 2.X2 + 2 [U61](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U62](X1,X2,X3) = X1 + 2.X2 + 2.X3 [U63](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U64](X1,X2,X3) = X1 + 2.X2 + 2.X3 [active](X) = X [isNat](X) = 2.X + 2 [isNatKind](X) = 2 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X [0] = 0 [fSNonEmpty] = 0 [tt] = 2 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = 2.X + 2 [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = 2.X + 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(U14(tt,V1:S,V2:S)) -> MARK(U15(isNat(V1:S),V2:S)) ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U12](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U13](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U14](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U15](X1,X2) = X1 + 2.X2 + 2 [U16](X) = X + 2 [U21](X1,X2) = X1 + 2.X2 [U22](X1,X2) = X1 + 2.X2 [U23](X) = X [U31](X1,X2) = X1 [U32](X) = X [U41](X) = X [U51](X1,X2) = 2.X2 + 2 [U52](X1,X2) = 2.X2 + 2 [U61](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U62](X1,X2,X3) = X1 + 2.X2 + 2.X3 [U63](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U64](X1,X2,X3) = X1 + 2.X2 + 2.X3 [active](X) = X [isNat](X) = 2.X + 2 [isNatKind](X) = 2 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X [0] = 2 [fSNonEmpty] = 0 [tt] = 2 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = 2.X + 2 [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = 2.X + 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(U15(tt,V2:S)) -> MARK(U16(isNat(V2:S))) ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U12](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 [U13](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U14](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U15](X1,X2) = X1 + 2.X2 + 2 [U16](X) = X + 1 [U21](X1,X2) = X1 + 2.X2 [U22](X1,X2) = X1 + 2.X2 [U23](X) = X [U31](X1,X2) = X1 [U32](X) = X [U41](X) = X [U51](X1,X2) = X2 + 2 [U52](X1,X2) = X1 + X2 [U61](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U62](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U63](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U64](X1,X2,X3) = X1 + 2.X2 + 2.X3 [active](X) = X [isNat](X) = 2.X + 2 [isNatKind](X) = 2 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X [0] = 1 [fSNonEmpty] = 0 [tt] = 2 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = 2.X + 2 [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = 2.X + 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNatKind(V1:S),V1:S)) ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = X1 + 2.X2 + 2.X3 [U12](X1,X2,X3) = X1 + 2.X2 + 2.X3 [U13](X1,X2,X3) = X1 + 2.X2 + 2.X3 [U14](X1,X2,X3) = X1 + 2.X2 + 2.X3 [U15](X1,X2) = X1 + 2.X2 [U16](X) = X [U21](X1,X2) = 2.X1 + 2.X2 + 1 [U22](X1,X2) = X1 + 2.X2 + 2 [U23](X) = X + 2 [U31](X1,X2) = X1 [U32](X) = X [U41](X) = X [U51](X1,X2) = 2.X2 + 2 [U52](X1,X2) = X1 + 2.X2 [U61](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U62](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U63](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U64](X1,X2,X3) = X1 + 2.X2 + 2.X3 [active](X) = X [isNat](X) = 2.X + 2 [isNatKind](X) = 2 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 [s](X) = X + 2 [0] = 2 [fSNonEmpty] = 0 [tt] = 2 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = 2.X + 2 [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = 2.X + 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(U22(tt,V1:S)) -> MARK(U23(isNat(V1:S))) ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 [U12](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U13](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U14](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U15](X1,X2) = X1 + 2.X2 + 2 [U16](X) = X + 2 [U21](X1,X2) = X1 + 2.X2 + 2 [U22](X1,X2) = 2.X1 + 2.X2 [U23](X) = X + 1 [U31](X1,X2) = X1 [U32](X) = X [U41](X) = X [U51](X1,X2) = 2.X2 + 2 [U52](X1,X2) = X1 + 2.X2 [U61](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U62](X1,X2,X3) = X1 + 2.X2 + 2.X3 [U63](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U64](X1,X2,X3) = X1 + 2.X2 + 2.X3 [active](X) = X [isNat](X) = 2.X + 2 [isNatKind](X) = 2 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 1 [s](X) = X + 1 [0] = 2 [fSNonEmpty] = 0 [tt] = 2 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = 2.X + 2 [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = 2.X + 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(isNatKind(V1:S),V1:S,V2:S)) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U12](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 [U13](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 [U14](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U15](X1,X2) = X1 + 2.X2 + 2 [U16](X) = X [U21](X1,X2) = X1 + 2.X2 [U22](X1,X2) = X1 + 2.X2 [U23](X) = X [U31](X1,X2) = X1 [U32](X) = X [U41](X) = X [U51](X1,X2) = 2.X2 + 2 [U52](X1,X2) = X1 + 2.X2 [U61](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U62](X1,X2,X3) = X1 + 2.X2 + 2.X3 [U63](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U64](X1,X2,X3) = X1 + 2.X2 + 2.X3 [active](X) = X [isNat](X) = 2.X + 2 [isNatKind](X) = 2 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X [0] = 2 [fSNonEmpty] = 0 [tt] = 2 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = 2.X + 2 [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = 2.X + 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S)) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 [U12](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 [U13](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 [U14](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U15](X1,X2) = X1 + 2.X2 + 1 [U16](X) = X [U21](X1,X2) = X1 + 2.X2 + 1 [U22](X1,X2) = X1 + 2.X2 + 1 [U23](X) = X [U31](X1,X2) = X1 [U32](X) = X [U41](X) = X [U51](X1,X2) = 2.X2 + 2 [U52](X1,X2) = 2.X2 + 1 [U61](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U62](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U63](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U64](X1,X2,X3) = X1 + 2.X2 + 2.X3 [active](X) = X [isNat](X) = 2.X + 2 [isNatKind](X) = 2 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 1 [s](X) = X + 1 [0] = 2 [fSNonEmpty] = 0 [tt] = 2 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = 2.X + 1 [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = 2.X + 1 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> ACTIVE(isNat(X:S)) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S)) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 0 [U12](X1,X2,X3) = 1 [U13](X1,X2,X3) = 1 [U14](X1,X2,X3) = 1 [U15](X1,X2) = 1 [U16](X) = 1 [U21](X1,X2) = 1 [U22](X1,X2) = 1 [U23](X) = 1 [U31](X1,X2) = 1 [U32](X) = 1 [U41](X) = 2 [U51](X1,X2) = 1 [U52](X1,X2) = 2 [U61](X1,X2,X3) = 0 [U62](X1,X2,X3) = 2 [U63](X1,X2,X3) = 0 [U64](X1,X2,X3) = 0 [active](X) = 0 [isNat](X) = 0 [isNatKind](X) = 1 [mark](X) = 0 [plus](X1,X2) = 1 [s](X) = 0 [0] = 2 [fSNonEmpty] = 0 [tt] = 1 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = 2.X [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U12](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U13](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 [U14](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U15](X1,X2) = X1 + 2.X2 + 1 [U16](X) = X + 1 [U21](X1,X2) = 2.X1 + 2.X2 [U22](X1,X2) = X1 + 2.X2 + 2 [U23](X) = X + 1 [U31](X1,X2) = X1 [U32](X) = X [U41](X) = X [U51](X1,X2) = 2.X2 + 2 [U52](X1,X2) = 2.X2 + 1 [U61](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U62](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U63](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U64](X1,X2,X3) = X1 + 2.X2 + 2.X3 [active](X) = X [isNat](X) = 2.X + 2 [isNatKind](X) = 2 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 1 [s](X) = X + 1 [0] = 2 [fSNonEmpty] = 0 [tt] = 2 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = 2.X + 2 [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = 2.X + 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U12(X1:S,X2:S,X3:S)) -> ACTIVE(U12(mark(X1:S),X2:S,X3:S)) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U12](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U13](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U14](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U15](X1,X2) = X1 + 2.X2 + 2 [U16](X) = X + 2 [U21](X1,X2) = 2.X1 + 2.X2 + 2 [U22](X1,X2) = 2.X1 + 2.X2 + 2 [U23](X) = X + 2 [U31](X1,X2) = 2.X1 [U32](X) = 2.X [U41](X) = 2.X [U51](X1,X2) = 2.X2 + 2 [U52](X1,X2) = 2.X1 + 2.X2 + 2 [U61](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U62](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U63](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U64](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [active](X) = X [isNat](X) = 2.X [isNatKind](X) = 0 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 1 [s](X) = X + 1 [0] = 2 [fSNonEmpty] = 0 [tt] = 0 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = X + 2 [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = 2.X + 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U12](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U13](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U14](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 [U15](X1,X2) = X1 + 2.X2 + 2 [U16](X) = X [U21](X1,X2) = 2.X1 + 2.X2 [U22](X1,X2) = 2.X1 + 2.X2 [U23](X) = X + 2 [U31](X1,X2) = X1 [U32](X) = X [U41](X) = X [U51](X1,X2) = X2 + 2 [U52](X1,X2) = X2 [U61](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U62](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U63](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U64](X1,X2,X3) = X1 + 2.X2 + 2.X3 [active](X) = X [isNat](X) = 2.X + 2 [isNatKind](X) = 2 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 1 [s](X) = X + 1 [0] = 2 [fSNonEmpty] = 0 [tt] = 2 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = 2.X + 2 [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = 2.X + 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U13(X1:S,X2:S,X3:S)) -> ACTIVE(U13(mark(X1:S),X2:S,X3:S)) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 2 [U12](X1,X2,X3) = 2 [U13](X1,X2,X3) = 0 [U14](X1,X2,X3) = 1 [U15](X1,X2) = 0 [U16](X) = 1 [U21](X1,X2) = 1 [U22](X1,X2) = 1 [U23](X) = 2 [U31](X1,X2) = 1 [U32](X) = 2 [U41](X) = 1 [U51](X1,X2) = 1 [U52](X1,X2) = 1 [U61](X1,X2,X3) = 1 [U62](X1,X2,X3) = 2 [U63](X1,X2,X3) = 2 [U64](X1,X2,X3) = 1 [active](X) = 2 [isNat](X) = 2 [isNatKind](X) = 1 [mark](X) = 2 [plus](X1,X2) = 1 [s](X) = 1 [0] = 1 [fSNonEmpty] = 0 [tt] = 1 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = 2.X [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U12](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 [U13](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U14](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U15](X1,X2) = X1 + 2.X2 + 2 [U16](X) = X [U21](X1,X2) = 2.X1 + 2.X2 [U22](X1,X2) = 2.X1 + 2.X2 [U23](X) = X + 2 [U31](X1,X2) = X1 [U32](X) = X [U41](X) = X [U51](X1,X2) = X2 + 2 [U52](X1,X2) = X2 + 2 [U61](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U62](X1,X2,X3) = X1 + 2.X2 + 2.X3 [U63](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U64](X1,X2,X3) = X1 + 2.X2 + 2.X3 [active](X) = X [isNat](X) = 2.X + 2 [isNatKind](X) = 2 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 1 [s](X) = X + 1 [0] = 2 [fSNonEmpty] = 0 [tt] = 2 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = 2.X [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = 2.X [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U14(X1:S,X2:S,X3:S)) -> ACTIVE(U14(mark(X1:S),X2:S,X3:S)) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 0 [U12](X1,X2,X3) = 0 [U13](X1,X2,X3) = 2 [U14](X1,X2,X3) = 1 [U15](X1,X2) = 1 [U16](X) = 2 [U21](X1,X2) = 2 [U22](X1,X2) = 2 [U23](X) = 0 [U31](X1,X2) = 2 [U32](X) = 0 [U41](X) = 1 [U51](X1,X2) = 0 [U52](X1,X2) = 2 [U61](X1,X2,X3) = 0 [U62](X1,X2,X3) = 0 [U63](X1,X2,X3) = 2 [U64](X1,X2,X3) = 1 [active](X) = 2 [isNat](X) = 2 [isNatKind](X) = 2 [mark](X) = 2 [plus](X1,X2) = 2 [s](X) = 2 [0] = 1 [fSNonEmpty] = 0 [tt] = 2 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = X [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U12](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U13](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U14](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U15](X1,X2) = X1 + 2.X2 + 2 [U16](X) = X + 1 [U21](X1,X2) = X1 + 2.X2 [U22](X1,X2) = X1 + 2.X2 [U23](X) = X [U31](X1,X2) = X1 [U32](X) = X [U41](X) = X [U51](X1,X2) = 2.X2 + 2 [U52](X1,X2) = 2.X2 + 2 [U61](X1,X2,X3) = X2 + 2.X3 + 2 [U62](X1,X2,X3) = X1 + X2 + 2.X3 [U63](X1,X2,X3) = X2 + 2.X3 + 2 [U64](X1,X2,X3) = X1 + X2 + 2.X3 [active](X) = X [isNat](X) = 2.X + 2 [isNatKind](X) = 2 [mark](X) = X [plus](X1,X2) = 2.X1 + X2 + 2 [s](X) = X [0] = 2 [fSNonEmpty] = 0 [tt] = 2 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = 2.X + 2 [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = 2.X + 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U15(X1:S,X2:S)) -> ACTIVE(U15(mark(X1:S),X2:S)) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 0 [U12](X1,X2,X3) = 2 [U13](X1,X2,X3) = 2 [U14](X1,X2,X3) = 0 [U15](X1,X2) = 0 [U16](X) = 0 [U21](X1,X2) = 1 [U22](X1,X2) = 0 [U23](X) = 0 [U31](X1,X2) = 1 [U32](X) = 2 [U41](X) = 0 [U51](X1,X2) = 2 [U52](X1,X2) = 2 [U61](X1,X2,X3) = 0 [U62](X1,X2,X3) = 2 [U63](X1,X2,X3) = 2 [U64](X1,X2,X3) = 0 [active](X) = 2 [isNat](X) = 0 [isNatKind](X) = 1 [mark](X) = 2 [plus](X1,X2) = 0 [s](X) = 0 [0] = 0 [fSNonEmpty] = 0 [tt] = 0 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = 2.X [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U15(X1:S,X2:S)) -> MARK(X1:S) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 2.X1 + X2 + 2.X3 [U12](X1,X2,X3) = 2.X1 + X2 + 2.X3 [U13](X1,X2,X3) = X1 + X2 + 2.X3 + 2 [U14](X1,X2,X3) = 2.X1 + X2 + 2.X3 [U15](X1,X2) = X1 + 2.X2 + 2 [U16](X) = 2.X [U21](X1,X2) = X1 + X2 [U22](X1,X2) = X1 + X2 [U23](X) = X [U31](X1,X2) = X1 [U32](X) = X [U41](X) = X [U51](X1,X2) = X1 + X2 + 2 [U52](X1,X2) = 2.X1 + X2 [U61](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U62](X1,X2,X3) = X1 + 2.X2 + 2.X3 [U63](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U64](X1,X2,X3) = X1 + 2.X2 + 2.X3 [active](X) = X [isNat](X) = X + 2 [isNatKind](X) = 2 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X [0] = 2 [fSNonEmpty] = 0 [tt] = 2 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = 2.X + 1 [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = 2.X + 1 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U16(X:S)) -> MARK(X:S) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U12](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U13](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U14](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 [U15](X1,X2) = X1 + 2.X2 + 2 [U16](X) = X + 2 [U21](X1,X2) = X1 + 2.X2 [U22](X1,X2) = X1 + 2.X2 [U23](X) = X [U31](X1,X2) = X1 [U32](X) = X [U41](X) = X [U51](X1,X2) = 2.X2 + 2 [U52](X1,X2) = X2 + 2 [U61](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U62](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U63](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U64](X1,X2,X3) = X1 + 2.X2 + 2.X3 [active](X) = X [isNat](X) = 2.X + 2 [isNatKind](X) = 2 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X [0] = 0 [fSNonEmpty] = 0 [tt] = 2 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = 2.X [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = 2.X [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S)) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 0 [U12](X1,X2,X3) = 0 [U13](X1,X2,X3) = 0 [U14](X1,X2,X3) = 0 [U15](X1,X2) = 0 [U16](X) = 0 [U21](X1,X2) = 0 [U22](X1,X2) = 0 [U23](X) = 0 [U31](X1,X2) = 2 [U32](X) = 0 [U41](X) = 0 [U51](X1,X2) = 0 [U52](X1,X2) = 0 [U61](X1,X2,X3) = 0 [U62](X1,X2,X3) = 0 [U63](X1,X2,X3) = 2 [U64](X1,X2,X3) = 0 [active](X) = 2 [isNat](X) = 2 [isNatKind](X) = 2 [mark](X) = 2 [plus](X1,X2) = 2 [s](X) = 2 [0] = 0 [fSNonEmpty] = 0 [tt] = 0 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = X [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U21(X1:S,X2:S)) -> MARK(X1:S) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U12](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U13](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U14](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U15](X1,X2) = 2 [U16](X) = 2 [U21](X1,X2) = X1 + X2 + 1 [U22](X1,X2) = X1 + X2 + 1 [U23](X) = X + 1 [U31](X1,X2) = X1 [U32](X) = X [U41](X) = X [U51](X1,X2) = X1 + X2 + 2 [U52](X1,X2) = 2.X1 + X2 [U61](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U62](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U63](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U64](X1,X2,X3) = X1 + 2.X2 + 2.X3 [active](X) = X [isNat](X) = X + 2 [isNatKind](X) = 2 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 1 [s](X) = X + 1 [0] = 2 [fSNonEmpty] = 0 [tt] = 2 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = 2.X + 2 [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = 2.X + 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U22(X1:S,X2:S)) -> ACTIVE(U22(mark(X1:S),X2:S)) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 [U12](X1,X2,X3) = 2.X2 + 2.X3 [U13](X1,X2,X3) = X1 + 2.X2 + 2.X3 [U14](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 [U15](X1,X2) = 2.X2 [U16](X) = 0 [U21](X1,X2) = 2.X2 + 2 [U22](X1,X2) = X1 + 2.X2 + 2 [U23](X) = X [U31](X1,X2) = 2.X1 [U32](X) = X [U41](X) = 2.X [U51](X1,X2) = 2.X2 + 2 [U52](X1,X2) = 2.X2 + 2 [U61](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U62](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U63](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U64](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [active](X) = X [isNat](X) = 2.X + 2 [isNatKind](X) = 0 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X [0] = 2 [fSNonEmpty] = 0 [tt] = 0 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = X + 2 [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = 2.X + 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U22(X1:S,X2:S)) -> MARK(X1:S) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 [U12](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 [U13](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U14](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 [U15](X1,X2) = X1 + 2.X2 + 2 [U16](X) = X + 2 [U21](X1,X2) = 2.X1 + 2.X2 [U22](X1,X2) = X1 + 2.X2 + 2 [U23](X) = X + 2 [U31](X1,X2) = X1 [U32](X) = X [U41](X) = X [U51](X1,X2) = 2.X2 + 2 [U52](X1,X2) = 2.X2 + 2 [U61](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U62](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U63](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U64](X1,X2,X3) = X1 + 2.X2 + 2.X3 [active](X) = X [isNat](X) = 2.X + 2 [isNatKind](X) = 2 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 1 [s](X) = X + 1 [0] = 2 [fSNonEmpty] = 0 [tt] = 2 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = 2.X + 2 [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = 2.X + 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U23(X:S)) -> MARK(X:S) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 [U12](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U13](X1,X2,X3) = X2 + 2.X3 + 2 [U14](X1,X2,X3) = X3 + 2 [U15](X1,X2) = X2 + 2 [U16](X) = 2 [U21](X1,X2) = 2.X1 + 2.X2 [U22](X1,X2) = 2.X1 + 2.X2 [U23](X) = X + 2 [U31](X1,X2) = X1 [U32](X) = X [U41](X) = X [U51](X1,X2) = 2.X2 + 1 [U52](X1,X2) = 2.X2 + 1 [U61](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U62](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U63](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U64](X1,X2,X3) = X1 + 2.X2 + 2.X3 [active](X) = X [isNat](X) = 2.X + 2 [isNatKind](X) = 2 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 1 [s](X) = X + 1 [0] = 2 [fSNonEmpty] = 0 [tt] = 2 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = 2.X + 2 [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = 2.X + 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> ACTIVE(U64(mark(X1:S),X2:S,X3:S)) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 0 [U12](X1,X2,X3) = X1 [U13](X1,X2,X3) = 2.X1 [U14](X1,X2,X3) = 2.X1 [U15](X1,X2) = 0 [U16](X) = 0 [U21](X1,X2) = 0 [U22](X1,X2) = 0 [U23](X) = 0 [U31](X1,X2) = 2.X1 [U32](X) = X [U41](X) = 2.X [U51](X1,X2) = 2.X2 + 2 [U52](X1,X2) = 2.X1 + 2.X2 + 2 [U61](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U62](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U63](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U64](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [active](X) = X [isNat](X) = 0 [isNatKind](X) = 0 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 [s](X) = X + 2 [0] = 2 [fSNonEmpty] = 0 [tt] = 0 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = 2 [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = 2.X + 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 2 [U12](X1,X2,X3) = X1 [U13](X1,X2,X3) = 2 [U14](X1,X2,X3) = 2 [U15](X1,X2) = X1 [U16](X) = X [U21](X1,X2) = X1 [U22](X1,X2) = 2 [U23](X) = 2 [U31](X1,X2) = X1 [U32](X) = X [U41](X) = X [U51](X1,X2) = 2.X1 + 2.X2 + 2 [U52](X1,X2) = 2.X2 + 2 [U61](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U62](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U63](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U64](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [active](X) = X [isNat](X) = 2 [isNatKind](X) = 2 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X + 2 [0] = 2 [fSNonEmpty] = 0 [tt] = 2 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = X + 2 [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = X + 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(U31(tt,V2:S)) -> MARK(U32(isNatKind(V2:S))) ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 2 [U12](X1,X2,X3) = 2 [U13](X1,X2,X3) = 2 [U14](X1,X2,X3) = 2 [U15](X1,X2) = 2 [U16](X) = X [U21](X1,X2) = 2 [U22](X1,X2) = 2 [U23](X) = 2 [U31](X1,X2) = 2.X1 + 2.X2 + 1 [U32](X) = X + 2 [U41](X) = X + 2 [U51](X1,X2) = 2.X1 + 2.X2 [U52](X1,X2) = 2.X2 + 1 [U61](X1,X2,X3) = X1 + 2.X2 + 2.X3 [U62](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U63](X1,X2,X3) = X1 + 2.X2 + 2.X3 [U64](X1,X2,X3) = 2.X2 + 2.X3 + 2 [active](X) = X [isNat](X) = 2 [isNatKind](X) = 2.X + 1 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 1 [s](X) = X + 1 [0] = 2 [fSNonEmpty] = 0 [tt] = 2 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = 2.X + 2 [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = 2.X + 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(U31(isNatKind(V1:S),V2:S)) ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U12](X1,X2,X3) = X1 + 2.X3 + 2 [U13](X1,X2,X3) = X1 + 2 [U14](X1,X2,X3) = 2 [U15](X1,X2) = 2 [U16](X) = 2 [U21](X1,X2) = 2.X2 [U22](X1,X2) = 2.X2 [U23](X) = X [U31](X1,X2) = X1 + 2.X2 + 2 [U32](X) = X + 1 [U41](X) = X [U51](X1,X2) = 2.X2 + 2 [U52](X1,X2) = X2 + 2 [U61](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U62](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U63](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U64](X1,X2,X3) = 2.X2 + 2.X3 + 2 [active](X) = X [isNat](X) = 2.X [isNatKind](X) = 2.X + 2 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X [0] = 2 [fSNonEmpty] = 0 [tt] = 2 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = 2.X + 2 [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = 2.X + 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Reduction Pairs Processor: -> Pairs: ACTIVE(isNatKind(s(V1:S))) -> MARK(U41(isNatKind(V1:S))) MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 2 [U12](X1,X2,X3) = 2 [U13](X1,X2,X3) = 2 [U14](X1,X2,X3) = 2 [U15](X1,X2) = 2 [U16](X) = X [U21](X1,X2) = 2 [U22](X1,X2) = 2 [U23](X) = 2 [U31](X1,X2) = X1 + 2.X2 [U32](X) = X [U41](X) = X [U51](X1,X2) = 2.X1 + 2.X2 [U52](X1,X2) = X1 + X2 + 1 [U61](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 [U62](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U63](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U64](X1,X2,X3) = 2.X2 + 2.X3 + 2 [active](X) = X [isNat](X) = 2 [isNatKind](X) = X + 2 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 [s](X) = X + 2 [0] = 2 [fSNonEmpty] = 0 [tt] = 2 [U11#](X1,X2,X3) = 0 [U12#](X1,X2,X3) = 0 [U13#](X1,X2,X3) = 0 [U14#](X1,X2,X3) = 0 [U15#](X1,X2) = 0 [U16#](X) = 0 [U21#](X1,X2) = 0 [U22#](X1,X2) = 0 [U23#](X) = 0 [U31#](X1,X2) = 0 [U32#](X) = 0 [U41#](X) = 0 [U51#](X1,X2) = 0 [U52#](X1,X2) = 0 [U61#](X1,X2,X3) = 0 [U62#](X1,X2,X3) = 0 [U63#](X1,X2,X3) = 0 [U64#](X1,X2,X3) = 0 [ACTIVE](X) = 2.X [ISNAT](X) = 0 [ISNATKIND](X) = 0 [MARK](X) = 2.X + 2 [PLUS](X1,X2) = 0 [S](X) = 0 Problem 1.23: SCC Processor: -> Pairs: MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S)) MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S)) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.23: Subterm Processor: -> Pairs: MARK(U31(X1:S,X2:S)) -> MARK(X1:S) MARK(U32(X:S)) -> MARK(X:S) MARK(U41(X:S)) -> MARK(X:S) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(MARK) = 1 Problem 1.23: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S) U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S) U12(active(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(mark(X1:S),X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,active(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,mark(X2:S),X3:S) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,active(X3:S)) -> U12(X1:S,X2:S,X3:S) U12(X1:S,X2:S,mark(X3:S)) -> U12(X1:S,X2:S,X3:S) U13(active(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(mark(X1:S),X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,active(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,mark(X2:S),X3:S) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,active(X3:S)) -> U13(X1:S,X2:S,X3:S) U13(X1:S,X2:S,mark(X3:S)) -> U13(X1:S,X2:S,X3:S) U14(active(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(mark(X1:S),X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,active(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,mark(X2:S),X3:S) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,active(X3:S)) -> U14(X1:S,X2:S,X3:S) U14(X1:S,X2:S,mark(X3:S)) -> U14(X1:S,X2:S,X3:S) U15(active(X1:S),X2:S) -> U15(X1:S,X2:S) U15(mark(X1:S),X2:S) -> U15(X1:S,X2:S) U15(X1:S,active(X2:S)) -> U15(X1:S,X2:S) U15(X1:S,mark(X2:S)) -> U15(X1:S,X2:S) U16(active(X:S)) -> U16(X:S) U16(mark(X:S)) -> U16(X:S) U21(active(X1:S),X2:S) -> U21(X1:S,X2:S) U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S) U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S) U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S) U22(active(X1:S),X2:S) -> U22(X1:S,X2:S) U22(mark(X1:S),X2:S) -> U22(X1:S,X2:S) U22(X1:S,active(X2:S)) -> U22(X1:S,X2:S) U22(X1:S,mark(X2:S)) -> U22(X1:S,X2:S) U23(active(X:S)) -> U23(X:S) U23(mark(X:S)) -> U23(X:S) U31(active(X1:S),X2:S) -> U31(X1:S,X2:S) U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S) U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S) U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S) U32(active(X:S)) -> U32(X:S) U32(mark(X:S)) -> U32(X:S) U41(active(X:S)) -> U41(X:S) U41(mark(X:S)) -> U41(X:S) U51(active(X1:S),X2:S) -> U51(X1:S,X2:S) U51(mark(X1:S),X2:S) -> U51(X1:S,X2:S) U51(X1:S,active(X2:S)) -> U51(X1:S,X2:S) U51(X1:S,mark(X2:S)) -> U51(X1:S,X2:S) U52(active(X1:S),X2:S) -> U52(X1:S,X2:S) U52(mark(X1:S),X2:S) -> U52(X1:S,X2:S) U52(X1:S,active(X2:S)) -> U52(X1:S,X2:S) U52(X1:S,mark(X2:S)) -> U52(X1:S,X2:S) U61(active(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(mark(X1:S),X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,active(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,mark(X2:S),X3:S) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,active(X3:S)) -> U61(X1:S,X2:S,X3:S) U61(X1:S,X2:S,mark(X3:S)) -> U61(X1:S,X2:S,X3:S) U62(active(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(mark(X1:S),X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,active(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,mark(X2:S),X3:S) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,active(X3:S)) -> U62(X1:S,X2:S,X3:S) U62(X1:S,X2:S,mark(X3:S)) -> U62(X1:S,X2:S,X3:S) U63(active(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(mark(X1:S),X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,active(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,mark(X2:S),X3:S) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,active(X3:S)) -> U63(X1:S,X2:S,X3:S) U63(X1:S,X2:S,mark(X3:S)) -> U63(X1:S,X2:S,X3:S) U64(active(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(mark(X1:S),X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,active(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,mark(X2:S),X3:S) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,active(X3:S)) -> U64(X1:S,X2:S,X3:S) U64(X1:S,X2:S,mark(X3:S)) -> U64(X1:S,X2:S,X3:S) active(U11(tt,V1:S,V2:S)) -> mark(U12(isNatKind(V1:S),V1:S,V2:S)) active(U12(tt,V1:S,V2:S)) -> mark(U13(isNatKind(V2:S),V1:S,V2:S)) active(U13(tt,V1:S,V2:S)) -> mark(U14(isNatKind(V2:S),V1:S,V2:S)) active(U14(tt,V1:S,V2:S)) -> mark(U15(isNat(V1:S),V2:S)) active(U15(tt,V2:S)) -> mark(U16(isNat(V2:S))) active(U16(tt)) -> mark(tt) active(U21(tt,V1:S)) -> mark(U22(isNatKind(V1:S),V1:S)) active(U22(tt,V1:S)) -> mark(U23(isNat(V1:S))) active(U23(tt)) -> mark(tt) active(U31(tt,V2:S)) -> mark(U32(isNatKind(V2:S))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N:S)) -> mark(U52(isNatKind(N:S),N:S)) active(U52(tt,N:S)) -> mark(N:S) active(U61(tt,M:S,N:S)) -> mark(U62(isNatKind(M:S),M:S,N:S)) active(U62(tt,M:S,N:S)) -> mark(U63(isNat(N:S),M:S,N:S)) active(U63(tt,M:S,N:S)) -> mark(U64(isNatKind(N:S),M:S,N:S)) active(U64(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) active(isNat(plus(V1:S,V2:S))) -> mark(U11(isNatKind(V1:S),V1:S,V2:S)) active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1:S,V2:S))) -> mark(U31(isNatKind(V1:S),V2:S)) active(isNatKind(s(V1:S))) -> mark(U41(isNatKind(V1:S))) active(isNatKind(0)) -> mark(tt) active(plus(N:S,s(M:S))) -> mark(U61(isNat(M:S),M:S,N:S)) active(plus(N:S,0)) -> mark(U51(isNat(N:S),N:S)) isNat(active(X:S)) -> isNat(X:S) isNat(mark(X:S)) -> isNat(X:S) isNatKind(active(X:S)) -> isNatKind(X:S) isNatKind(mark(X:S)) -> isNatKind(X:S) mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S)) mark(U12(X1:S,X2:S,X3:S)) -> active(U12(mark(X1:S),X2:S,X3:S)) mark(U13(X1:S,X2:S,X3:S)) -> active(U13(mark(X1:S),X2:S,X3:S)) mark(U14(X1:S,X2:S,X3:S)) -> active(U14(mark(X1:S),X2:S,X3:S)) mark(U15(X1:S,X2:S)) -> active(U15(mark(X1:S),X2:S)) mark(U16(X:S)) -> active(U16(mark(X:S))) mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S)) mark(U22(X1:S,X2:S)) -> active(U22(mark(X1:S),X2:S)) mark(U23(X:S)) -> active(U23(mark(X:S))) mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S)) mark(U32(X:S)) -> active(U32(mark(X:S))) mark(U41(X:S)) -> active(U41(mark(X:S))) mark(U51(X1:S,X2:S)) -> active(U51(mark(X1:S),X2:S)) mark(U52(X1:S,X2:S)) -> active(U52(mark(X1:S),X2:S)) mark(U61(X1:S,X2:S,X3:S)) -> active(U61(mark(X1:S),X2:S,X3:S)) mark(U62(X1:S,X2:S,X3:S)) -> active(U62(mark(X1:S),X2:S,X3:S)) mark(U63(X1:S,X2:S,X3:S)) -> active(U63(mark(X1:S),X2:S,X3:S)) mark(U64(X1:S,X2:S,X3:S)) -> active(U64(mark(X1:S),X2:S,X3:S)) mark(isNat(X:S)) -> active(isNat(X:S)) mark(isNatKind(X:S)) -> active(isNatKind(X:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite.