/export/starexec/sandbox2/solver/bin/starexec_run_default /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES Problem 1: (VAR M N V1 V2 X X1 X2 X3) (RULES U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ) Problem 1: Dependency Pairs Processor: -> Pairs: U11#(active(X1),X2,X3) -> U11#(X1,X2,X3) U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3) U11#(X1,active(X2),X3) -> U11#(X1,X2,X3) U11#(X1,mark(X2),X3) -> U11#(X1,X2,X3) U11#(X1,X2,active(X3)) -> U11#(X1,X2,X3) U11#(X1,X2,mark(X3)) -> U11#(X1,X2,X3) U12#(active(X1),X2,X3) -> U12#(X1,X2,X3) U12#(mark(X1),X2,X3) -> U12#(X1,X2,X3) U12#(X1,active(X2),X3) -> U12#(X1,X2,X3) U12#(X1,mark(X2),X3) -> U12#(X1,X2,X3) U12#(X1,X2,active(X3)) -> U12#(X1,X2,X3) U12#(X1,X2,mark(X3)) -> U12#(X1,X2,X3) U13#(active(X1),X2,X3) -> U13#(X1,X2,X3) U13#(mark(X1),X2,X3) -> U13#(X1,X2,X3) U13#(X1,active(X2),X3) -> U13#(X1,X2,X3) U13#(X1,mark(X2),X3) -> U13#(X1,X2,X3) U13#(X1,X2,active(X3)) -> U13#(X1,X2,X3) U13#(X1,X2,mark(X3)) -> U13#(X1,X2,X3) U14#(active(X1),X2,X3) -> U14#(X1,X2,X3) U14#(mark(X1),X2,X3) -> U14#(X1,X2,X3) U14#(X1,active(X2),X3) -> U14#(X1,X2,X3) U14#(X1,mark(X2),X3) -> U14#(X1,X2,X3) U14#(X1,X2,active(X3)) -> U14#(X1,X2,X3) U14#(X1,X2,mark(X3)) -> U14#(X1,X2,X3) U15#(active(X1),X2) -> U15#(X1,X2) U15#(mark(X1),X2) -> U15#(X1,X2) U15#(X1,active(X2)) -> U15#(X1,X2) U15#(X1,mark(X2)) -> U15#(X1,X2) U16#(active(X)) -> U16#(X) U16#(mark(X)) -> U16#(X) U21#(active(X1),X2) -> U21#(X1,X2) U21#(mark(X1),X2) -> U21#(X1,X2) U21#(X1,active(X2)) -> U21#(X1,X2) U21#(X1,mark(X2)) -> U21#(X1,X2) U22#(active(X1),X2) -> U22#(X1,X2) U22#(mark(X1),X2) -> U22#(X1,X2) U22#(X1,active(X2)) -> U22#(X1,X2) U22#(X1,mark(X2)) -> U22#(X1,X2) U23#(active(X)) -> U23#(X) U23#(mark(X)) -> U23#(X) U31#(active(X1),X2) -> U31#(X1,X2) U31#(mark(X1),X2) -> U31#(X1,X2) U31#(X1,active(X2)) -> U31#(X1,X2) U31#(X1,mark(X2)) -> U31#(X1,X2) U32#(active(X)) -> U32#(X) U32#(mark(X)) -> U32#(X) U41#(active(X)) -> U41#(X) U41#(mark(X)) -> U41#(X) U51#(active(X1),X2) -> U51#(X1,X2) U51#(mark(X1),X2) -> U51#(X1,X2) U51#(X1,active(X2)) -> U51#(X1,X2) U51#(X1,mark(X2)) -> U51#(X1,X2) U52#(active(X1),X2) -> U52#(X1,X2) U52#(mark(X1),X2) -> U52#(X1,X2) U52#(X1,active(X2)) -> U52#(X1,X2) U52#(X1,mark(X2)) -> U52#(X1,X2) U61#(active(X1),X2,X3) -> U61#(X1,X2,X3) U61#(mark(X1),X2,X3) -> U61#(X1,X2,X3) U61#(X1,active(X2),X3) -> U61#(X1,X2,X3) U61#(X1,mark(X2),X3) -> U61#(X1,X2,X3) U61#(X1,X2,active(X3)) -> U61#(X1,X2,X3) U61#(X1,X2,mark(X3)) -> U61#(X1,X2,X3) U62#(active(X1),X2,X3) -> U62#(X1,X2,X3) U62#(mark(X1),X2,X3) -> U62#(X1,X2,X3) U62#(X1,active(X2),X3) -> U62#(X1,X2,X3) U62#(X1,mark(X2),X3) -> U62#(X1,X2,X3) U62#(X1,X2,active(X3)) -> U62#(X1,X2,X3) U62#(X1,X2,mark(X3)) -> U62#(X1,X2,X3) U63#(active(X1),X2,X3) -> U63#(X1,X2,X3) U63#(mark(X1),X2,X3) -> U63#(X1,X2,X3) U63#(X1,active(X2),X3) -> U63#(X1,X2,X3) U63#(X1,mark(X2),X3) -> U63#(X1,X2,X3) U63#(X1,X2,active(X3)) -> U63#(X1,X2,X3) U63#(X1,X2,mark(X3)) -> U63#(X1,X2,X3) U64#(active(X1),X2,X3) -> U64#(X1,X2,X3) U64#(mark(X1),X2,X3) -> U64#(X1,X2,X3) U64#(X1,active(X2),X3) -> U64#(X1,X2,X3) U64#(X1,mark(X2),X3) -> U64#(X1,X2,X3) U64#(X1,X2,active(X3)) -> U64#(X1,X2,X3) U64#(X1,X2,mark(X3)) -> U64#(X1,X2,X3) ACTIVE(U11(tt,V1,V2)) -> U12#(isNatKind(V1),V1,V2) ACTIVE(U11(tt,V1,V2)) -> ISNATKIND(V1) ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> U13#(isNatKind(V2),V1,V2) ACTIVE(U12(tt,V1,V2)) -> ISNATKIND(V2) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> U14#(isNatKind(V2),V1,V2) ACTIVE(U13(tt,V1,V2)) -> ISNATKIND(V2) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> U15#(isNat(V1),V2) ACTIVE(U14(tt,V1,V2)) -> ISNAT(V1) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> U16#(isNat(V2)) ACTIVE(U15(tt,V2)) -> ISNAT(V2) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U16(tt)) -> MARK(tt) ACTIVE(U21(tt,V1)) -> U22#(isNatKind(V1),V1) ACTIVE(U21(tt,V1)) -> ISNATKIND(V1) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> U23#(isNat(V1)) ACTIVE(U22(tt,V1)) -> ISNAT(V1) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U23(tt)) -> MARK(tt) ACTIVE(U31(tt,V2)) -> U32#(isNatKind(V2)) ACTIVE(U31(tt,V2)) -> ISNATKIND(V2) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(U32(tt)) -> MARK(tt) ACTIVE(U41(tt)) -> MARK(tt) ACTIVE(U51(tt,N)) -> U52#(isNatKind(N),N) ACTIVE(U51(tt,N)) -> ISNATKIND(N) ACTIVE(U51(tt,N)) -> MARK(U52(isNatKind(N),N)) ACTIVE(U52(tt,N)) -> MARK(N) ACTIVE(U61(tt,M,N)) -> U62#(isNatKind(M),M,N) ACTIVE(U61(tt,M,N)) -> ISNATKIND(M) ACTIVE(U61(tt,M,N)) -> MARK(U62(isNatKind(M),M,N)) ACTIVE(U62(tt,M,N)) -> U63#(isNat(N),M,N) ACTIVE(U62(tt,M,N)) -> ISNAT(N) ACTIVE(U62(tt,M,N)) -> MARK(U63(isNat(N),M,N)) ACTIVE(U63(tt,M,N)) -> U64#(isNatKind(N),M,N) ACTIVE(U63(tt,M,N)) -> ISNATKIND(N) ACTIVE(U63(tt,M,N)) -> MARK(U64(isNatKind(N),M,N)) ACTIVE(U64(tt,M,N)) -> MARK(s(plus(N,M))) ACTIVE(U64(tt,M,N)) -> PLUS(N,M) ACTIVE(U64(tt,M,N)) -> S(plus(N,M)) ACTIVE(isNat(plus(V1,V2))) -> U11#(isNatKind(V1),V1,V2) ACTIVE(isNat(plus(V1,V2))) -> ISNATKIND(V1) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> U21#(isNatKind(V1),V1) ACTIVE(isNat(s(V1))) -> ISNATKIND(V1) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNat(0)) -> MARK(tt) ACTIVE(isNatKind(plus(V1,V2))) -> U31#(isNatKind(V1),V2) ACTIVE(isNatKind(plus(V1,V2))) -> ISNATKIND(V1) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> U41#(isNatKind(V1)) ACTIVE(isNatKind(s(V1))) -> ISNATKIND(V1) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) ACTIVE(isNatKind(0)) -> MARK(tt) ACTIVE(plus(N,s(M))) -> U61#(isNat(M),M,N) ACTIVE(plus(N,s(M))) -> ISNAT(M) ACTIVE(plus(N,s(M))) -> MARK(U61(isNat(M),M,N)) ACTIVE(plus(N,0)) -> U51#(isNat(N),N) ACTIVE(plus(N,0)) -> ISNAT(N) ACTIVE(plus(N,0)) -> MARK(U51(isNat(N),N)) ISNAT(active(X)) -> ISNAT(X) ISNAT(mark(X)) -> ISNAT(X) ISNATKIND(active(X)) -> ISNATKIND(X) ISNATKIND(mark(X)) -> ISNATKIND(X) MARK(U11(X1,X2,X3)) -> U11#(mark(X1),X2,X3) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> U12#(mark(X1),X2,X3) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> U13#(mark(X1),X2,X3) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> U14#(mark(X1),X2,X3) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> U15#(mark(X1),X2) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> U16#(mark(X)) MARK(U16(X)) -> ACTIVE(U16(mark(X))) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> U21#(mark(X1),X2) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> U22#(mark(X1),X2) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> U23#(mark(X)) MARK(U23(X)) -> ACTIVE(U23(mark(X))) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> U31#(mark(X1),X2) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> U32#(mark(X)) MARK(U32(X)) -> ACTIVE(U32(mark(X))) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> U41#(mark(X)) MARK(U41(X)) -> ACTIVE(U41(mark(X))) MARK(U41(X)) -> MARK(X) MARK(U51(X1,X2)) -> U51#(mark(X1),X2) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X1,X2)) -> U52#(mark(X1),X2) MARK(U52(X1,X2)) -> ACTIVE(U52(mark(X1),X2)) MARK(U52(X1,X2)) -> MARK(X1) MARK(U61(X1,X2,X3)) -> U61#(mark(X1),X2,X3) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> U62#(mark(X1),X2,X3) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> U63#(mark(X1),X2,X3) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> U64#(mark(X1),X2,X3) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(plus(X1,X2)) -> PLUS(mark(X1),mark(X2)) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(s(X)) -> S(mark(X)) PLUS(active(X1),X2) -> PLUS(X1,X2) PLUS(mark(X1),X2) -> PLUS(X1,X2) PLUS(X1,active(X2)) -> PLUS(X1,X2) PLUS(X1,mark(X2)) -> PLUS(X1,X2) S(active(X)) -> S(X) S(mark(X)) -> S(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1: SCC Processor: -> Pairs: U11#(active(X1),X2,X3) -> U11#(X1,X2,X3) U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3) U11#(X1,active(X2),X3) -> U11#(X1,X2,X3) U11#(X1,mark(X2),X3) -> U11#(X1,X2,X3) U11#(X1,X2,active(X3)) -> U11#(X1,X2,X3) U11#(X1,X2,mark(X3)) -> U11#(X1,X2,X3) U12#(active(X1),X2,X3) -> U12#(X1,X2,X3) U12#(mark(X1),X2,X3) -> U12#(X1,X2,X3) U12#(X1,active(X2),X3) -> U12#(X1,X2,X3) U12#(X1,mark(X2),X3) -> U12#(X1,X2,X3) U12#(X1,X2,active(X3)) -> U12#(X1,X2,X3) U12#(X1,X2,mark(X3)) -> U12#(X1,X2,X3) U13#(active(X1),X2,X3) -> U13#(X1,X2,X3) U13#(mark(X1),X2,X3) -> U13#(X1,X2,X3) U13#(X1,active(X2),X3) -> U13#(X1,X2,X3) U13#(X1,mark(X2),X3) -> U13#(X1,X2,X3) U13#(X1,X2,active(X3)) -> U13#(X1,X2,X3) U13#(X1,X2,mark(X3)) -> U13#(X1,X2,X3) U14#(active(X1),X2,X3) -> U14#(X1,X2,X3) U14#(mark(X1),X2,X3) -> U14#(X1,X2,X3) U14#(X1,active(X2),X3) -> U14#(X1,X2,X3) U14#(X1,mark(X2),X3) -> U14#(X1,X2,X3) U14#(X1,X2,active(X3)) -> U14#(X1,X2,X3) U14#(X1,X2,mark(X3)) -> U14#(X1,X2,X3) U15#(active(X1),X2) -> U15#(X1,X2) U15#(mark(X1),X2) -> U15#(X1,X2) U15#(X1,active(X2)) -> U15#(X1,X2) U15#(X1,mark(X2)) -> U15#(X1,X2) U16#(active(X)) -> U16#(X) U16#(mark(X)) -> U16#(X) U21#(active(X1),X2) -> U21#(X1,X2) U21#(mark(X1),X2) -> U21#(X1,X2) U21#(X1,active(X2)) -> U21#(X1,X2) U21#(X1,mark(X2)) -> U21#(X1,X2) U22#(active(X1),X2) -> U22#(X1,X2) U22#(mark(X1),X2) -> U22#(X1,X2) U22#(X1,active(X2)) -> U22#(X1,X2) U22#(X1,mark(X2)) -> U22#(X1,X2) U23#(active(X)) -> U23#(X) U23#(mark(X)) -> U23#(X) U31#(active(X1),X2) -> U31#(X1,X2) U31#(mark(X1),X2) -> U31#(X1,X2) U31#(X1,active(X2)) -> U31#(X1,X2) U31#(X1,mark(X2)) -> U31#(X1,X2) U32#(active(X)) -> U32#(X) U32#(mark(X)) -> U32#(X) U41#(active(X)) -> U41#(X) U41#(mark(X)) -> U41#(X) U51#(active(X1),X2) -> U51#(X1,X2) U51#(mark(X1),X2) -> U51#(X1,X2) U51#(X1,active(X2)) -> U51#(X1,X2) U51#(X1,mark(X2)) -> U51#(X1,X2) U52#(active(X1),X2) -> U52#(X1,X2) U52#(mark(X1),X2) -> U52#(X1,X2) U52#(X1,active(X2)) -> U52#(X1,X2) U52#(X1,mark(X2)) -> U52#(X1,X2) U61#(active(X1),X2,X3) -> U61#(X1,X2,X3) U61#(mark(X1),X2,X3) -> U61#(X1,X2,X3) U61#(X1,active(X2),X3) -> U61#(X1,X2,X3) U61#(X1,mark(X2),X3) -> U61#(X1,X2,X3) U61#(X1,X2,active(X3)) -> U61#(X1,X2,X3) U61#(X1,X2,mark(X3)) -> U61#(X1,X2,X3) U62#(active(X1),X2,X3) -> U62#(X1,X2,X3) U62#(mark(X1),X2,X3) -> U62#(X1,X2,X3) U62#(X1,active(X2),X3) -> U62#(X1,X2,X3) U62#(X1,mark(X2),X3) -> U62#(X1,X2,X3) U62#(X1,X2,active(X3)) -> U62#(X1,X2,X3) U62#(X1,X2,mark(X3)) -> U62#(X1,X2,X3) U63#(active(X1),X2,X3) -> U63#(X1,X2,X3) U63#(mark(X1),X2,X3) -> U63#(X1,X2,X3) U63#(X1,active(X2),X3) -> U63#(X1,X2,X3) U63#(X1,mark(X2),X3) -> U63#(X1,X2,X3) U63#(X1,X2,active(X3)) -> U63#(X1,X2,X3) U63#(X1,X2,mark(X3)) -> U63#(X1,X2,X3) U64#(active(X1),X2,X3) -> U64#(X1,X2,X3) U64#(mark(X1),X2,X3) -> U64#(X1,X2,X3) U64#(X1,active(X2),X3) -> U64#(X1,X2,X3) U64#(X1,mark(X2),X3) -> U64#(X1,X2,X3) U64#(X1,X2,active(X3)) -> U64#(X1,X2,X3) U64#(X1,X2,mark(X3)) -> U64#(X1,X2,X3) ACTIVE(U11(tt,V1,V2)) -> U12#(isNatKind(V1),V1,V2) ACTIVE(U11(tt,V1,V2)) -> ISNATKIND(V1) ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> U13#(isNatKind(V2),V1,V2) ACTIVE(U12(tt,V1,V2)) -> ISNATKIND(V2) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> U14#(isNatKind(V2),V1,V2) ACTIVE(U13(tt,V1,V2)) -> ISNATKIND(V2) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> U15#(isNat(V1),V2) ACTIVE(U14(tt,V1,V2)) -> ISNAT(V1) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> U16#(isNat(V2)) ACTIVE(U15(tt,V2)) -> ISNAT(V2) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U16(tt)) -> MARK(tt) ACTIVE(U21(tt,V1)) -> U22#(isNatKind(V1),V1) ACTIVE(U21(tt,V1)) -> ISNATKIND(V1) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> U23#(isNat(V1)) ACTIVE(U22(tt,V1)) -> ISNAT(V1) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U23(tt)) -> MARK(tt) ACTIVE(U31(tt,V2)) -> U32#(isNatKind(V2)) ACTIVE(U31(tt,V2)) -> ISNATKIND(V2) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(U32(tt)) -> MARK(tt) ACTIVE(U41(tt)) -> MARK(tt) ACTIVE(U51(tt,N)) -> U52#(isNatKind(N),N) ACTIVE(U51(tt,N)) -> ISNATKIND(N) ACTIVE(U51(tt,N)) -> MARK(U52(isNatKind(N),N)) ACTIVE(U52(tt,N)) -> MARK(N) ACTIVE(U61(tt,M,N)) -> U62#(isNatKind(M),M,N) ACTIVE(U61(tt,M,N)) -> ISNATKIND(M) ACTIVE(U61(tt,M,N)) -> MARK(U62(isNatKind(M),M,N)) ACTIVE(U62(tt,M,N)) -> U63#(isNat(N),M,N) ACTIVE(U62(tt,M,N)) -> ISNAT(N) ACTIVE(U62(tt,M,N)) -> MARK(U63(isNat(N),M,N)) ACTIVE(U63(tt,M,N)) -> U64#(isNatKind(N),M,N) ACTIVE(U63(tt,M,N)) -> ISNATKIND(N) ACTIVE(U63(tt,M,N)) -> MARK(U64(isNatKind(N),M,N)) ACTIVE(U64(tt,M,N)) -> MARK(s(plus(N,M))) ACTIVE(U64(tt,M,N)) -> PLUS(N,M) ACTIVE(U64(tt,M,N)) -> S(plus(N,M)) ACTIVE(isNat(plus(V1,V2))) -> U11#(isNatKind(V1),V1,V2) ACTIVE(isNat(plus(V1,V2))) -> ISNATKIND(V1) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> U21#(isNatKind(V1),V1) ACTIVE(isNat(s(V1))) -> ISNATKIND(V1) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNat(0)) -> MARK(tt) ACTIVE(isNatKind(plus(V1,V2))) -> U31#(isNatKind(V1),V2) ACTIVE(isNatKind(plus(V1,V2))) -> ISNATKIND(V1) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> U41#(isNatKind(V1)) ACTIVE(isNatKind(s(V1))) -> ISNATKIND(V1) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) ACTIVE(isNatKind(0)) -> MARK(tt) ACTIVE(plus(N,s(M))) -> U61#(isNat(M),M,N) ACTIVE(plus(N,s(M))) -> ISNAT(M) ACTIVE(plus(N,s(M))) -> MARK(U61(isNat(M),M,N)) ACTIVE(plus(N,0)) -> U51#(isNat(N),N) ACTIVE(plus(N,0)) -> ISNAT(N) ACTIVE(plus(N,0)) -> MARK(U51(isNat(N),N)) ISNAT(active(X)) -> ISNAT(X) ISNAT(mark(X)) -> ISNAT(X) ISNATKIND(active(X)) -> ISNATKIND(X) ISNATKIND(mark(X)) -> ISNATKIND(X) MARK(U11(X1,X2,X3)) -> U11#(mark(X1),X2,X3) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> U12#(mark(X1),X2,X3) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> U13#(mark(X1),X2,X3) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> U14#(mark(X1),X2,X3) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> U15#(mark(X1),X2) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> U16#(mark(X)) MARK(U16(X)) -> ACTIVE(U16(mark(X))) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> U21#(mark(X1),X2) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> U22#(mark(X1),X2) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> U23#(mark(X)) MARK(U23(X)) -> ACTIVE(U23(mark(X))) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> U31#(mark(X1),X2) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> U32#(mark(X)) MARK(U32(X)) -> ACTIVE(U32(mark(X))) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> U41#(mark(X)) MARK(U41(X)) -> ACTIVE(U41(mark(X))) MARK(U41(X)) -> MARK(X) MARK(U51(X1,X2)) -> U51#(mark(X1),X2) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X1,X2)) -> U52#(mark(X1),X2) MARK(U52(X1,X2)) -> ACTIVE(U52(mark(X1),X2)) MARK(U52(X1,X2)) -> MARK(X1) MARK(U61(X1,X2,X3)) -> U61#(mark(X1),X2,X3) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> U62#(mark(X1),X2,X3) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> U63#(mark(X1),X2,X3) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> U64#(mark(X1),X2,X3) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(plus(X1,X2)) -> PLUS(mark(X1),mark(X2)) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(s(X)) -> S(mark(X)) PLUS(active(X1),X2) -> PLUS(X1,X2) PLUS(mark(X1),X2) -> PLUS(X1,X2) PLUS(X1,active(X2)) -> PLUS(X1,X2) PLUS(X1,mark(X2)) -> PLUS(X1,X2) S(active(X)) -> S(X) S(mark(X)) -> S(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: S(active(X)) -> S(X) S(mark(X)) -> S(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: PLUS(active(X1),X2) -> PLUS(X1,X2) PLUS(mark(X1),X2) -> PLUS(X1,X2) PLUS(X1,active(X2)) -> PLUS(X1,X2) PLUS(X1,mark(X2)) -> PLUS(X1,X2) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: ISNATKIND(active(X)) -> ISNATKIND(X) ISNATKIND(mark(X)) -> ISNATKIND(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: ISNAT(active(X)) -> ISNAT(X) ISNAT(mark(X)) -> ISNAT(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: U64#(active(X1),X2,X3) -> U64#(X1,X2,X3) U64#(mark(X1),X2,X3) -> U64#(X1,X2,X3) U64#(X1,active(X2),X3) -> U64#(X1,X2,X3) U64#(X1,mark(X2),X3) -> U64#(X1,X2,X3) U64#(X1,X2,active(X3)) -> U64#(X1,X2,X3) U64#(X1,X2,mark(X3)) -> U64#(X1,X2,X3) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: U63#(active(X1),X2,X3) -> U63#(X1,X2,X3) U63#(mark(X1),X2,X3) -> U63#(X1,X2,X3) U63#(X1,active(X2),X3) -> U63#(X1,X2,X3) U63#(X1,mark(X2),X3) -> U63#(X1,X2,X3) U63#(X1,X2,active(X3)) -> U63#(X1,X2,X3) U63#(X1,X2,mark(X3)) -> U63#(X1,X2,X3) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: U62#(active(X1),X2,X3) -> U62#(X1,X2,X3) U62#(mark(X1),X2,X3) -> U62#(X1,X2,X3) U62#(X1,active(X2),X3) -> U62#(X1,X2,X3) U62#(X1,mark(X2),X3) -> U62#(X1,X2,X3) U62#(X1,X2,active(X3)) -> U62#(X1,X2,X3) U62#(X1,X2,mark(X3)) -> U62#(X1,X2,X3) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: U61#(active(X1),X2,X3) -> U61#(X1,X2,X3) U61#(mark(X1),X2,X3) -> U61#(X1,X2,X3) U61#(X1,active(X2),X3) -> U61#(X1,X2,X3) U61#(X1,mark(X2),X3) -> U61#(X1,X2,X3) U61#(X1,X2,active(X3)) -> U61#(X1,X2,X3) U61#(X1,X2,mark(X3)) -> U61#(X1,X2,X3) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: U52#(active(X1),X2) -> U52#(X1,X2) U52#(mark(X1),X2) -> U52#(X1,X2) U52#(X1,active(X2)) -> U52#(X1,X2) U52#(X1,mark(X2)) -> U52#(X1,X2) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: U51#(active(X1),X2) -> U51#(X1,X2) U51#(mark(X1),X2) -> U51#(X1,X2) U51#(X1,active(X2)) -> U51#(X1,X2) U51#(X1,mark(X2)) -> U51#(X1,X2) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: U41#(active(X)) -> U41#(X) U41#(mark(X)) -> U41#(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: U32#(active(X)) -> U32#(X) U32#(mark(X)) -> U32#(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: U31#(active(X1),X2) -> U31#(X1,X2) U31#(mark(X1),X2) -> U31#(X1,X2) U31#(X1,active(X2)) -> U31#(X1,X2) U31#(X1,mark(X2)) -> U31#(X1,X2) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: U23#(active(X)) -> U23#(X) U23#(mark(X)) -> U23#(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: U22#(active(X1),X2) -> U22#(X1,X2) U22#(mark(X1),X2) -> U22#(X1,X2) U22#(X1,active(X2)) -> U22#(X1,X2) U22#(X1,mark(X2)) -> U22#(X1,X2) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: U21#(active(X1),X2) -> U21#(X1,X2) U21#(mark(X1),X2) -> U21#(X1,X2) U21#(X1,active(X2)) -> U21#(X1,X2) U21#(X1,mark(X2)) -> U21#(X1,X2) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: U16#(active(X)) -> U16#(X) U16#(mark(X)) -> U16#(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: U15#(active(X1),X2) -> U15#(X1,X2) U15#(mark(X1),X2) -> U15#(X1,X2) U15#(X1,active(X2)) -> U15#(X1,X2) U15#(X1,mark(X2)) -> U15#(X1,X2) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: U14#(active(X1),X2,X3) -> U14#(X1,X2,X3) U14#(mark(X1),X2,X3) -> U14#(X1,X2,X3) U14#(X1,active(X2),X3) -> U14#(X1,X2,X3) U14#(X1,mark(X2),X3) -> U14#(X1,X2,X3) U14#(X1,X2,active(X3)) -> U14#(X1,X2,X3) U14#(X1,X2,mark(X3)) -> U14#(X1,X2,X3) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: U13#(active(X1),X2,X3) -> U13#(X1,X2,X3) U13#(mark(X1),X2,X3) -> U13#(X1,X2,X3) U13#(X1,active(X2),X3) -> U13#(X1,X2,X3) U13#(X1,mark(X2),X3) -> U13#(X1,X2,X3) U13#(X1,X2,active(X3)) -> U13#(X1,X2,X3) U13#(X1,X2,mark(X3)) -> U13#(X1,X2,X3) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: U12#(active(X1),X2,X3) -> U12#(X1,X2,X3) U12#(mark(X1),X2,X3) -> U12#(X1,X2,X3) U12#(X1,active(X2),X3) -> U12#(X1,X2,X3) U12#(X1,mark(X2),X3) -> U12#(X1,X2,X3) U12#(X1,X2,active(X3)) -> U12#(X1,X2,X3) U12#(X1,X2,mark(X3)) -> U12#(X1,X2,X3) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: U11#(active(X1),X2,X3) -> U11#(X1,X2,X3) U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3) U11#(X1,active(X2),X3) -> U11#(X1,X2,X3) U11#(X1,mark(X2),X3) -> U11#(X1,X2,X3) U11#(X1,X2,active(X3)) -> U11#(X1,X2,X3) U11#(X1,X2,mark(X3)) -> U11#(X1,X2,X3) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(U51(tt,N)) -> MARK(U52(isNatKind(N),N)) ACTIVE(U52(tt,N)) -> MARK(N) ACTIVE(U61(tt,M,N)) -> MARK(U62(isNatKind(M),M,N)) ACTIVE(U62(tt,M,N)) -> MARK(U63(isNat(N),M,N)) ACTIVE(U63(tt,M,N)) -> MARK(U64(isNatKind(N),M,N)) ACTIVE(U64(tt,M,N)) -> MARK(s(plus(N,M))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) ACTIVE(plus(N,s(M))) -> MARK(U61(isNat(M),M,N)) ACTIVE(plus(N,0)) -> MARK(U51(isNat(N),N)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> ACTIVE(U16(mark(X))) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> ACTIVE(U23(mark(X))) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> ACTIVE(U32(mark(X))) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> ACTIVE(U41(mark(X))) MARK(U41(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X1,X2)) -> ACTIVE(U52(mark(X1),X2)) MARK(U52(X1,X2)) -> MARK(X1) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) The problem is decomposed in 23 subproblems. Problem 1.1: Subterm Processor: -> Pairs: S(active(X)) -> S(X) S(mark(X)) -> S(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(S) = 1 Problem 1.1: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Subterm Processor: -> Pairs: PLUS(active(X1),X2) -> PLUS(X1,X2) PLUS(mark(X1),X2) -> PLUS(X1,X2) PLUS(X1,active(X2)) -> PLUS(X1,X2) PLUS(X1,mark(X2)) -> PLUS(X1,X2) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(PLUS) = 1 Problem 1.2: SCC Processor: -> Pairs: PLUS(X1,active(X2)) -> PLUS(X1,X2) PLUS(X1,mark(X2)) -> PLUS(X1,X2) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: PLUS(X1,active(X2)) -> PLUS(X1,X2) PLUS(X1,mark(X2)) -> PLUS(X1,X2) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.2: Subterm Processor: -> Pairs: PLUS(X1,active(X2)) -> PLUS(X1,X2) PLUS(X1,mark(X2)) -> PLUS(X1,X2) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(PLUS) = 2 Problem 1.2: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.3: Subterm Processor: -> Pairs: ISNATKIND(active(X)) -> ISNATKIND(X) ISNATKIND(mark(X)) -> ISNATKIND(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(ISNATKIND) = 1 Problem 1.3: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.4: Subterm Processor: -> Pairs: ISNAT(active(X)) -> ISNAT(X) ISNAT(mark(X)) -> ISNAT(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(ISNAT) = 1 Problem 1.4: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.5: Subterm Processor: -> Pairs: U64#(active(X1),X2,X3) -> U64#(X1,X2,X3) U64#(mark(X1),X2,X3) -> U64#(X1,X2,X3) U64#(X1,active(X2),X3) -> U64#(X1,X2,X3) U64#(X1,mark(X2),X3) -> U64#(X1,X2,X3) U64#(X1,X2,active(X3)) -> U64#(X1,X2,X3) U64#(X1,X2,mark(X3)) -> U64#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U64#) = 1 Problem 1.5: SCC Processor: -> Pairs: U64#(X1,active(X2),X3) -> U64#(X1,X2,X3) U64#(X1,mark(X2),X3) -> U64#(X1,X2,X3) U64#(X1,X2,active(X3)) -> U64#(X1,X2,X3) U64#(X1,X2,mark(X3)) -> U64#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U64#(X1,active(X2),X3) -> U64#(X1,X2,X3) U64#(X1,mark(X2),X3) -> U64#(X1,X2,X3) U64#(X1,X2,active(X3)) -> U64#(X1,X2,X3) U64#(X1,X2,mark(X3)) -> U64#(X1,X2,X3) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.5: Subterm Processor: -> Pairs: U64#(X1,active(X2),X3) -> U64#(X1,X2,X3) U64#(X1,mark(X2),X3) -> U64#(X1,X2,X3) U64#(X1,X2,active(X3)) -> U64#(X1,X2,X3) U64#(X1,X2,mark(X3)) -> U64#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U64#) = 2 Problem 1.5: SCC Processor: -> Pairs: U64#(X1,X2,active(X3)) -> U64#(X1,X2,X3) U64#(X1,X2,mark(X3)) -> U64#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U64#(X1,X2,active(X3)) -> U64#(X1,X2,X3) U64#(X1,X2,mark(X3)) -> U64#(X1,X2,X3) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.5: Subterm Processor: -> Pairs: U64#(X1,X2,active(X3)) -> U64#(X1,X2,X3) U64#(X1,X2,mark(X3)) -> U64#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U64#) = 3 Problem 1.5: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.6: Subterm Processor: -> Pairs: U63#(active(X1),X2,X3) -> U63#(X1,X2,X3) U63#(mark(X1),X2,X3) -> U63#(X1,X2,X3) U63#(X1,active(X2),X3) -> U63#(X1,X2,X3) U63#(X1,mark(X2),X3) -> U63#(X1,X2,X3) U63#(X1,X2,active(X3)) -> U63#(X1,X2,X3) U63#(X1,X2,mark(X3)) -> U63#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U63#) = 1 Problem 1.6: SCC Processor: -> Pairs: U63#(X1,active(X2),X3) -> U63#(X1,X2,X3) U63#(X1,mark(X2),X3) -> U63#(X1,X2,X3) U63#(X1,X2,active(X3)) -> U63#(X1,X2,X3) U63#(X1,X2,mark(X3)) -> U63#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U63#(X1,active(X2),X3) -> U63#(X1,X2,X3) U63#(X1,mark(X2),X3) -> U63#(X1,X2,X3) U63#(X1,X2,active(X3)) -> U63#(X1,X2,X3) U63#(X1,X2,mark(X3)) -> U63#(X1,X2,X3) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.6: Subterm Processor: -> Pairs: U63#(X1,active(X2),X3) -> U63#(X1,X2,X3) U63#(X1,mark(X2),X3) -> U63#(X1,X2,X3) U63#(X1,X2,active(X3)) -> U63#(X1,X2,X3) U63#(X1,X2,mark(X3)) -> U63#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U63#) = 2 Problem 1.6: SCC Processor: -> Pairs: U63#(X1,X2,active(X3)) -> U63#(X1,X2,X3) U63#(X1,X2,mark(X3)) -> U63#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U63#(X1,X2,active(X3)) -> U63#(X1,X2,X3) U63#(X1,X2,mark(X3)) -> U63#(X1,X2,X3) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.6: Subterm Processor: -> Pairs: U63#(X1,X2,active(X3)) -> U63#(X1,X2,X3) U63#(X1,X2,mark(X3)) -> U63#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U63#) = 3 Problem 1.6: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.7: Subterm Processor: -> Pairs: U62#(active(X1),X2,X3) -> U62#(X1,X2,X3) U62#(mark(X1),X2,X3) -> U62#(X1,X2,X3) U62#(X1,active(X2),X3) -> U62#(X1,X2,X3) U62#(X1,mark(X2),X3) -> U62#(X1,X2,X3) U62#(X1,X2,active(X3)) -> U62#(X1,X2,X3) U62#(X1,X2,mark(X3)) -> U62#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U62#) = 1 Problem 1.7: SCC Processor: -> Pairs: U62#(X1,active(X2),X3) -> U62#(X1,X2,X3) U62#(X1,mark(X2),X3) -> U62#(X1,X2,X3) U62#(X1,X2,active(X3)) -> U62#(X1,X2,X3) U62#(X1,X2,mark(X3)) -> U62#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U62#(X1,active(X2),X3) -> U62#(X1,X2,X3) U62#(X1,mark(X2),X3) -> U62#(X1,X2,X3) U62#(X1,X2,active(X3)) -> U62#(X1,X2,X3) U62#(X1,X2,mark(X3)) -> U62#(X1,X2,X3) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.7: Subterm Processor: -> Pairs: U62#(X1,active(X2),X3) -> U62#(X1,X2,X3) U62#(X1,mark(X2),X3) -> U62#(X1,X2,X3) U62#(X1,X2,active(X3)) -> U62#(X1,X2,X3) U62#(X1,X2,mark(X3)) -> U62#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U62#) = 2 Problem 1.7: SCC Processor: -> Pairs: U62#(X1,X2,active(X3)) -> U62#(X1,X2,X3) U62#(X1,X2,mark(X3)) -> U62#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U62#(X1,X2,active(X3)) -> U62#(X1,X2,X3) U62#(X1,X2,mark(X3)) -> U62#(X1,X2,X3) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.7: Subterm Processor: -> Pairs: U62#(X1,X2,active(X3)) -> U62#(X1,X2,X3) U62#(X1,X2,mark(X3)) -> U62#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U62#) = 3 Problem 1.7: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.8: Subterm Processor: -> Pairs: U61#(active(X1),X2,X3) -> U61#(X1,X2,X3) U61#(mark(X1),X2,X3) -> U61#(X1,X2,X3) U61#(X1,active(X2),X3) -> U61#(X1,X2,X3) U61#(X1,mark(X2),X3) -> U61#(X1,X2,X3) U61#(X1,X2,active(X3)) -> U61#(X1,X2,X3) U61#(X1,X2,mark(X3)) -> U61#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U61#) = 1 Problem 1.8: SCC Processor: -> Pairs: U61#(X1,active(X2),X3) -> U61#(X1,X2,X3) U61#(X1,mark(X2),X3) -> U61#(X1,X2,X3) U61#(X1,X2,active(X3)) -> U61#(X1,X2,X3) U61#(X1,X2,mark(X3)) -> U61#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U61#(X1,active(X2),X3) -> U61#(X1,X2,X3) U61#(X1,mark(X2),X3) -> U61#(X1,X2,X3) U61#(X1,X2,active(X3)) -> U61#(X1,X2,X3) U61#(X1,X2,mark(X3)) -> U61#(X1,X2,X3) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.8: Subterm Processor: -> Pairs: U61#(X1,active(X2),X3) -> U61#(X1,X2,X3) U61#(X1,mark(X2),X3) -> U61#(X1,X2,X3) U61#(X1,X2,active(X3)) -> U61#(X1,X2,X3) U61#(X1,X2,mark(X3)) -> U61#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U61#) = 2 Problem 1.8: SCC Processor: -> Pairs: U61#(X1,X2,active(X3)) -> U61#(X1,X2,X3) U61#(X1,X2,mark(X3)) -> U61#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U61#(X1,X2,active(X3)) -> U61#(X1,X2,X3) U61#(X1,X2,mark(X3)) -> U61#(X1,X2,X3) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.8: Subterm Processor: -> Pairs: U61#(X1,X2,active(X3)) -> U61#(X1,X2,X3) U61#(X1,X2,mark(X3)) -> U61#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U61#) = 3 Problem 1.8: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.9: Subterm Processor: -> Pairs: U52#(active(X1),X2) -> U52#(X1,X2) U52#(mark(X1),X2) -> U52#(X1,X2) U52#(X1,active(X2)) -> U52#(X1,X2) U52#(X1,mark(X2)) -> U52#(X1,X2) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U52#) = 1 Problem 1.9: SCC Processor: -> Pairs: U52#(X1,active(X2)) -> U52#(X1,X2) U52#(X1,mark(X2)) -> U52#(X1,X2) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U52#(X1,active(X2)) -> U52#(X1,X2) U52#(X1,mark(X2)) -> U52#(X1,X2) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.9: Subterm Processor: -> Pairs: U52#(X1,active(X2)) -> U52#(X1,X2) U52#(X1,mark(X2)) -> U52#(X1,X2) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U52#) = 2 Problem 1.9: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.10: Subterm Processor: -> Pairs: U51#(active(X1),X2) -> U51#(X1,X2) U51#(mark(X1),X2) -> U51#(X1,X2) U51#(X1,active(X2)) -> U51#(X1,X2) U51#(X1,mark(X2)) -> U51#(X1,X2) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U51#) = 1 Problem 1.10: SCC Processor: -> Pairs: U51#(X1,active(X2)) -> U51#(X1,X2) U51#(X1,mark(X2)) -> U51#(X1,X2) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U51#(X1,active(X2)) -> U51#(X1,X2) U51#(X1,mark(X2)) -> U51#(X1,X2) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.10: Subterm Processor: -> Pairs: U51#(X1,active(X2)) -> U51#(X1,X2) U51#(X1,mark(X2)) -> U51#(X1,X2) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U51#) = 2 Problem 1.10: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.11: Subterm Processor: -> Pairs: U41#(active(X)) -> U41#(X) U41#(mark(X)) -> U41#(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U41#) = 1 Problem 1.11: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.12: Subterm Processor: -> Pairs: U32#(active(X)) -> U32#(X) U32#(mark(X)) -> U32#(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U32#) = 1 Problem 1.12: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.13: Subterm Processor: -> Pairs: U31#(active(X1),X2) -> U31#(X1,X2) U31#(mark(X1),X2) -> U31#(X1,X2) U31#(X1,active(X2)) -> U31#(X1,X2) U31#(X1,mark(X2)) -> U31#(X1,X2) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U31#) = 1 Problem 1.13: SCC Processor: -> Pairs: U31#(X1,active(X2)) -> U31#(X1,X2) U31#(X1,mark(X2)) -> U31#(X1,X2) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U31#(X1,active(X2)) -> U31#(X1,X2) U31#(X1,mark(X2)) -> U31#(X1,X2) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.13: Subterm Processor: -> Pairs: U31#(X1,active(X2)) -> U31#(X1,X2) U31#(X1,mark(X2)) -> U31#(X1,X2) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U31#) = 2 Problem 1.13: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.14: Subterm Processor: -> Pairs: U23#(active(X)) -> U23#(X) U23#(mark(X)) -> U23#(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U23#) = 1 Problem 1.14: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.15: Subterm Processor: -> Pairs: U22#(active(X1),X2) -> U22#(X1,X2) U22#(mark(X1),X2) -> U22#(X1,X2) U22#(X1,active(X2)) -> U22#(X1,X2) U22#(X1,mark(X2)) -> U22#(X1,X2) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U22#) = 1 Problem 1.15: SCC Processor: -> Pairs: U22#(X1,active(X2)) -> U22#(X1,X2) U22#(X1,mark(X2)) -> U22#(X1,X2) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U22#(X1,active(X2)) -> U22#(X1,X2) U22#(X1,mark(X2)) -> U22#(X1,X2) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.15: Subterm Processor: -> Pairs: U22#(X1,active(X2)) -> U22#(X1,X2) U22#(X1,mark(X2)) -> U22#(X1,X2) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U22#) = 2 Problem 1.15: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.16: Subterm Processor: -> Pairs: U21#(active(X1),X2) -> U21#(X1,X2) U21#(mark(X1),X2) -> U21#(X1,X2) U21#(X1,active(X2)) -> U21#(X1,X2) U21#(X1,mark(X2)) -> U21#(X1,X2) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U21#) = 1 Problem 1.16: SCC Processor: -> Pairs: U21#(X1,active(X2)) -> U21#(X1,X2) U21#(X1,mark(X2)) -> U21#(X1,X2) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U21#(X1,active(X2)) -> U21#(X1,X2) U21#(X1,mark(X2)) -> U21#(X1,X2) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.16: Subterm Processor: -> Pairs: U21#(X1,active(X2)) -> U21#(X1,X2) U21#(X1,mark(X2)) -> U21#(X1,X2) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U21#) = 2 Problem 1.16: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.17: Subterm Processor: -> Pairs: U16#(active(X)) -> U16#(X) U16#(mark(X)) -> U16#(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U16#) = 1 Problem 1.17: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.18: Subterm Processor: -> Pairs: U15#(active(X1),X2) -> U15#(X1,X2) U15#(mark(X1),X2) -> U15#(X1,X2) U15#(X1,active(X2)) -> U15#(X1,X2) U15#(X1,mark(X2)) -> U15#(X1,X2) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U15#) = 1 Problem 1.18: SCC Processor: -> Pairs: U15#(X1,active(X2)) -> U15#(X1,X2) U15#(X1,mark(X2)) -> U15#(X1,X2) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U15#(X1,active(X2)) -> U15#(X1,X2) U15#(X1,mark(X2)) -> U15#(X1,X2) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.18: Subterm Processor: -> Pairs: U15#(X1,active(X2)) -> U15#(X1,X2) U15#(X1,mark(X2)) -> U15#(X1,X2) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U15#) = 2 Problem 1.18: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.19: Subterm Processor: -> Pairs: U14#(active(X1),X2,X3) -> U14#(X1,X2,X3) U14#(mark(X1),X2,X3) -> U14#(X1,X2,X3) U14#(X1,active(X2),X3) -> U14#(X1,X2,X3) U14#(X1,mark(X2),X3) -> U14#(X1,X2,X3) U14#(X1,X2,active(X3)) -> U14#(X1,X2,X3) U14#(X1,X2,mark(X3)) -> U14#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U14#) = 1 Problem 1.19: SCC Processor: -> Pairs: U14#(X1,active(X2),X3) -> U14#(X1,X2,X3) U14#(X1,mark(X2),X3) -> U14#(X1,X2,X3) U14#(X1,X2,active(X3)) -> U14#(X1,X2,X3) U14#(X1,X2,mark(X3)) -> U14#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U14#(X1,active(X2),X3) -> U14#(X1,X2,X3) U14#(X1,mark(X2),X3) -> U14#(X1,X2,X3) U14#(X1,X2,active(X3)) -> U14#(X1,X2,X3) U14#(X1,X2,mark(X3)) -> U14#(X1,X2,X3) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.19: Subterm Processor: -> Pairs: U14#(X1,active(X2),X3) -> U14#(X1,X2,X3) U14#(X1,mark(X2),X3) -> U14#(X1,X2,X3) U14#(X1,X2,active(X3)) -> U14#(X1,X2,X3) U14#(X1,X2,mark(X3)) -> U14#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U14#) = 2 Problem 1.19: SCC Processor: -> Pairs: U14#(X1,X2,active(X3)) -> U14#(X1,X2,X3) U14#(X1,X2,mark(X3)) -> U14#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U14#(X1,X2,active(X3)) -> U14#(X1,X2,X3) U14#(X1,X2,mark(X3)) -> U14#(X1,X2,X3) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.19: Subterm Processor: -> Pairs: U14#(X1,X2,active(X3)) -> U14#(X1,X2,X3) U14#(X1,X2,mark(X3)) -> U14#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U14#) = 3 Problem 1.19: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.20: Subterm Processor: -> Pairs: U13#(active(X1),X2,X3) -> U13#(X1,X2,X3) U13#(mark(X1),X2,X3) -> U13#(X1,X2,X3) U13#(X1,active(X2),X3) -> U13#(X1,X2,X3) U13#(X1,mark(X2),X3) -> U13#(X1,X2,X3) U13#(X1,X2,active(X3)) -> U13#(X1,X2,X3) U13#(X1,X2,mark(X3)) -> U13#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U13#) = 1 Problem 1.20: SCC Processor: -> Pairs: U13#(X1,active(X2),X3) -> U13#(X1,X2,X3) U13#(X1,mark(X2),X3) -> U13#(X1,X2,X3) U13#(X1,X2,active(X3)) -> U13#(X1,X2,X3) U13#(X1,X2,mark(X3)) -> U13#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U13#(X1,active(X2),X3) -> U13#(X1,X2,X3) U13#(X1,mark(X2),X3) -> U13#(X1,X2,X3) U13#(X1,X2,active(X3)) -> U13#(X1,X2,X3) U13#(X1,X2,mark(X3)) -> U13#(X1,X2,X3) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.20: Subterm Processor: -> Pairs: U13#(X1,active(X2),X3) -> U13#(X1,X2,X3) U13#(X1,mark(X2),X3) -> U13#(X1,X2,X3) U13#(X1,X2,active(X3)) -> U13#(X1,X2,X3) U13#(X1,X2,mark(X3)) -> U13#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U13#) = 2 Problem 1.20: SCC Processor: -> Pairs: U13#(X1,X2,active(X3)) -> U13#(X1,X2,X3) U13#(X1,X2,mark(X3)) -> U13#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U13#(X1,X2,active(X3)) -> U13#(X1,X2,X3) U13#(X1,X2,mark(X3)) -> U13#(X1,X2,X3) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.20: Subterm Processor: -> Pairs: U13#(X1,X2,active(X3)) -> U13#(X1,X2,X3) U13#(X1,X2,mark(X3)) -> U13#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U13#) = 3 Problem 1.20: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.21: Subterm Processor: -> Pairs: U12#(active(X1),X2,X3) -> U12#(X1,X2,X3) U12#(mark(X1),X2,X3) -> U12#(X1,X2,X3) U12#(X1,active(X2),X3) -> U12#(X1,X2,X3) U12#(X1,mark(X2),X3) -> U12#(X1,X2,X3) U12#(X1,X2,active(X3)) -> U12#(X1,X2,X3) U12#(X1,X2,mark(X3)) -> U12#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U12#) = 1 Problem 1.21: SCC Processor: -> Pairs: U12#(X1,active(X2),X3) -> U12#(X1,X2,X3) U12#(X1,mark(X2),X3) -> U12#(X1,X2,X3) U12#(X1,X2,active(X3)) -> U12#(X1,X2,X3) U12#(X1,X2,mark(X3)) -> U12#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U12#(X1,active(X2),X3) -> U12#(X1,X2,X3) U12#(X1,mark(X2),X3) -> U12#(X1,X2,X3) U12#(X1,X2,active(X3)) -> U12#(X1,X2,X3) U12#(X1,X2,mark(X3)) -> U12#(X1,X2,X3) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.21: Subterm Processor: -> Pairs: U12#(X1,active(X2),X3) -> U12#(X1,X2,X3) U12#(X1,mark(X2),X3) -> U12#(X1,X2,X3) U12#(X1,X2,active(X3)) -> U12#(X1,X2,X3) U12#(X1,X2,mark(X3)) -> U12#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U12#) = 2 Problem 1.21: SCC Processor: -> Pairs: U12#(X1,X2,active(X3)) -> U12#(X1,X2,X3) U12#(X1,X2,mark(X3)) -> U12#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U12#(X1,X2,active(X3)) -> U12#(X1,X2,X3) U12#(X1,X2,mark(X3)) -> U12#(X1,X2,X3) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.21: Subterm Processor: -> Pairs: U12#(X1,X2,active(X3)) -> U12#(X1,X2,X3) U12#(X1,X2,mark(X3)) -> U12#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U12#) = 3 Problem 1.21: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.22: Subterm Processor: -> Pairs: U11#(active(X1),X2,X3) -> U11#(X1,X2,X3) U11#(mark(X1),X2,X3) -> U11#(X1,X2,X3) U11#(X1,active(X2),X3) -> U11#(X1,X2,X3) U11#(X1,mark(X2),X3) -> U11#(X1,X2,X3) U11#(X1,X2,active(X3)) -> U11#(X1,X2,X3) U11#(X1,X2,mark(X3)) -> U11#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U11#) = 1 Problem 1.22: SCC Processor: -> Pairs: U11#(X1,active(X2),X3) -> U11#(X1,X2,X3) U11#(X1,mark(X2),X3) -> U11#(X1,X2,X3) U11#(X1,X2,active(X3)) -> U11#(X1,X2,X3) U11#(X1,X2,mark(X3)) -> U11#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U11#(X1,active(X2),X3) -> U11#(X1,X2,X3) U11#(X1,mark(X2),X3) -> U11#(X1,X2,X3) U11#(X1,X2,active(X3)) -> U11#(X1,X2,X3) U11#(X1,X2,mark(X3)) -> U11#(X1,X2,X3) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.22: Subterm Processor: -> Pairs: U11#(X1,active(X2),X3) -> U11#(X1,X2,X3) U11#(X1,mark(X2),X3) -> U11#(X1,X2,X3) U11#(X1,X2,active(X3)) -> U11#(X1,X2,X3) U11#(X1,X2,mark(X3)) -> U11#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U11#) = 2 Problem 1.22: SCC Processor: -> Pairs: U11#(X1,X2,active(X3)) -> U11#(X1,X2,X3) U11#(X1,X2,mark(X3)) -> U11#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: U11#(X1,X2,active(X3)) -> U11#(X1,X2,X3) U11#(X1,X2,mark(X3)) -> U11#(X1,X2,X3) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.22: Subterm Processor: -> Pairs: U11#(X1,X2,active(X3)) -> U11#(X1,X2,X3) U11#(X1,X2,mark(X3)) -> U11#(X1,X2,X3) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(U11#) = 3 Problem 1.22: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(U51(tt,N)) -> MARK(U52(isNatKind(N),N)) ACTIVE(U52(tt,N)) -> MARK(N) ACTIVE(U61(tt,M,N)) -> MARK(U62(isNatKind(M),M,N)) ACTIVE(U62(tt,M,N)) -> MARK(U63(isNat(N),M,N)) ACTIVE(U63(tt,M,N)) -> MARK(U64(isNatKind(N),M,N)) ACTIVE(U64(tt,M,N)) -> MARK(s(plus(N,M))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) ACTIVE(plus(N,s(M))) -> MARK(U61(isNat(M),M,N)) ACTIVE(plus(N,0)) -> MARK(U51(isNat(N),N)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> ACTIVE(U16(mark(X))) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> ACTIVE(U23(mark(X))) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> ACTIVE(U32(mark(X))) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> ACTIVE(U41(mark(X))) MARK(U41(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X1,X2)) -> ACTIVE(U52(mark(X1),X2)) MARK(U52(X1,X2)) -> MARK(X1) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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) = 2.X1 + 2.X2 + 2.X3 + 1 [U63](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [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 [tt] = 2 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(U52(tt,N)) -> MARK(N) ACTIVE(U61(tt,M,N)) -> MARK(U62(isNatKind(M),M,N)) ACTIVE(U62(tt,M,N)) -> MARK(U63(isNat(N),M,N)) ACTIVE(U63(tt,M,N)) -> MARK(U64(isNatKind(N),M,N)) ACTIVE(U64(tt,M,N)) -> MARK(s(plus(N,M))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) ACTIVE(plus(N,s(M))) -> MARK(U61(isNat(M),M,N)) ACTIVE(plus(N,0)) -> MARK(U51(isNat(N),N)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> ACTIVE(U16(mark(X))) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> ACTIVE(U23(mark(X))) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> ACTIVE(U32(mark(X))) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> ACTIVE(U41(mark(X))) MARK(U41(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X1,X2)) -> ACTIVE(U52(mark(X1),X2)) MARK(U52(X1,X2)) -> MARK(X1) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(U52(tt,N)) -> MARK(N) ACTIVE(U61(tt,M,N)) -> MARK(U62(isNatKind(M),M,N)) ACTIVE(U62(tt,M,N)) -> MARK(U63(isNat(N),M,N)) ACTIVE(U63(tt,M,N)) -> MARK(U64(isNatKind(N),M,N)) ACTIVE(U64(tt,M,N)) -> MARK(s(plus(N,M))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) ACTIVE(plus(N,s(M))) -> MARK(U61(isNat(M),M,N)) ACTIVE(plus(N,0)) -> MARK(U51(isNat(N),N)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> ACTIVE(U16(mark(X))) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> ACTIVE(U23(mark(X))) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> ACTIVE(U32(mark(X))) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> ACTIVE(U41(mark(X))) MARK(U41(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X1,X2)) -> ACTIVE(U52(mark(X1),X2)) MARK(U52(X1,X2)) -> MARK(X1) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(U52(tt,N)) -> MARK(N) ACTIVE(U61(tt,M,N)) -> MARK(U62(isNatKind(M),M,N)) ACTIVE(U62(tt,M,N)) -> MARK(U63(isNat(N),M,N)) ACTIVE(U63(tt,M,N)) -> MARK(U64(isNatKind(N),M,N)) ACTIVE(U64(tt,M,N)) -> MARK(s(plus(N,M))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) ACTIVE(plus(N,s(M))) -> MARK(U61(isNat(M),M,N)) ACTIVE(plus(N,0)) -> MARK(U51(isNat(N),N)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> ACTIVE(U16(mark(X))) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> ACTIVE(U23(mark(X))) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> ACTIVE(U32(mark(X))) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> ACTIVE(U41(mark(X))) MARK(U41(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X1,X2)) -> ACTIVE(U52(mark(X1),X2)) MARK(U52(X1,X2)) -> MARK(X1) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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) = X1 + 2.X2 + 2.X3 + 2 [U62](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 [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 [tt] = 2 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(U61(tt,M,N)) -> MARK(U62(isNatKind(M),M,N)) ACTIVE(U62(tt,M,N)) -> MARK(U63(isNat(N),M,N)) ACTIVE(U63(tt,M,N)) -> MARK(U64(isNatKind(N),M,N)) ACTIVE(U64(tt,M,N)) -> MARK(s(plus(N,M))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) ACTIVE(plus(N,s(M))) -> MARK(U61(isNat(M),M,N)) ACTIVE(plus(N,0)) -> MARK(U51(isNat(N),N)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> ACTIVE(U16(mark(X))) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> ACTIVE(U23(mark(X))) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> ACTIVE(U32(mark(X))) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> ACTIVE(U41(mark(X))) MARK(U41(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X1,X2)) -> ACTIVE(U52(mark(X1),X2)) MARK(U52(X1,X2)) -> MARK(X1) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(U61(tt,M,N)) -> MARK(U62(isNatKind(M),M,N)) ACTIVE(U62(tt,M,N)) -> MARK(U63(isNat(N),M,N)) ACTIVE(U63(tt,M,N)) -> MARK(U64(isNatKind(N),M,N)) ACTIVE(U64(tt,M,N)) -> MARK(s(plus(N,M))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) ACTIVE(plus(N,s(M))) -> MARK(U61(isNat(M),M,N)) ACTIVE(plus(N,0)) -> MARK(U51(isNat(N),N)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> ACTIVE(U16(mark(X))) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> ACTIVE(U23(mark(X))) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> ACTIVE(U32(mark(X))) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> ACTIVE(U41(mark(X))) MARK(U41(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X1,X2)) -> ACTIVE(U52(mark(X1),X2)) MARK(U52(X1,X2)) -> MARK(X1) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(U61(tt,M,N)) -> MARK(U62(isNatKind(M),M,N)) ACTIVE(U62(tt,M,N)) -> MARK(U63(isNat(N),M,N)) ACTIVE(U63(tt,M,N)) -> MARK(U64(isNatKind(N),M,N)) ACTIVE(U64(tt,M,N)) -> MARK(s(plus(N,M))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) ACTIVE(plus(N,s(M))) -> MARK(U61(isNat(M),M,N)) ACTIVE(plus(N,0)) -> MARK(U51(isNat(N),N)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> ACTIVE(U16(mark(X))) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> ACTIVE(U23(mark(X))) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> ACTIVE(U32(mark(X))) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> ACTIVE(U41(mark(X))) MARK(U41(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X1,X2)) -> ACTIVE(U52(mark(X1),X2)) MARK(U52(X1,X2)) -> MARK(X1) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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) = 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) = 1 [isNatKind](X) = 1 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X + 1 [0] = 2 [tt] = 1 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(U62(tt,M,N)) -> MARK(U63(isNat(N),M,N)) ACTIVE(U63(tt,M,N)) -> MARK(U64(isNatKind(N),M,N)) ACTIVE(U64(tt,M,N)) -> MARK(s(plus(N,M))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) ACTIVE(plus(N,s(M))) -> MARK(U61(isNat(M),M,N)) ACTIVE(plus(N,0)) -> MARK(U51(isNat(N),N)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> ACTIVE(U16(mark(X))) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> ACTIVE(U23(mark(X))) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> ACTIVE(U32(mark(X))) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> ACTIVE(U41(mark(X))) MARK(U41(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X1,X2)) -> ACTIVE(U52(mark(X1),X2)) MARK(U52(X1,X2)) -> MARK(X1) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(U62(tt,M,N)) -> MARK(U63(isNat(N),M,N)) ACTIVE(U63(tt,M,N)) -> MARK(U64(isNatKind(N),M,N)) ACTIVE(U64(tt,M,N)) -> MARK(s(plus(N,M))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) ACTIVE(plus(N,s(M))) -> MARK(U61(isNat(M),M,N)) ACTIVE(plus(N,0)) -> MARK(U51(isNat(N),N)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> ACTIVE(U16(mark(X))) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> ACTIVE(U23(mark(X))) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> ACTIVE(U32(mark(X))) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> ACTIVE(U41(mark(X))) MARK(U41(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X1,X2)) -> ACTIVE(U52(mark(X1),X2)) MARK(U52(X1,X2)) -> MARK(X1) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(U62(tt,M,N)) -> MARK(U63(isNat(N),M,N)) ACTIVE(U63(tt,M,N)) -> MARK(U64(isNatKind(N),M,N)) ACTIVE(U64(tt,M,N)) -> MARK(s(plus(N,M))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) ACTIVE(plus(N,s(M))) -> MARK(U61(isNat(M),M,N)) ACTIVE(plus(N,0)) -> MARK(U51(isNat(N),N)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> ACTIVE(U16(mark(X))) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> ACTIVE(U23(mark(X))) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> ACTIVE(U32(mark(X))) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> ACTIVE(U41(mark(X))) MARK(U41(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X1,X2)) -> ACTIVE(U52(mark(X1),X2)) MARK(U52(X1,X2)) -> MARK(X1) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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 [U62](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 [U63](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 1 [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 [tt] = 2 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(U63(tt,M,N)) -> MARK(U64(isNatKind(N),M,N)) ACTIVE(U64(tt,M,N)) -> MARK(s(plus(N,M))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) ACTIVE(plus(N,s(M))) -> MARK(U61(isNat(M),M,N)) ACTIVE(plus(N,0)) -> MARK(U51(isNat(N),N)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> ACTIVE(U16(mark(X))) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> ACTIVE(U23(mark(X))) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> ACTIVE(U32(mark(X))) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> ACTIVE(U41(mark(X))) MARK(U41(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X1,X2)) -> ACTIVE(U52(mark(X1),X2)) MARK(U52(X1,X2)) -> MARK(X1) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(U63(tt,M,N)) -> MARK(U64(isNatKind(N),M,N)) ACTIVE(U64(tt,M,N)) -> MARK(s(plus(N,M))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) ACTIVE(plus(N,s(M))) -> MARK(U61(isNat(M),M,N)) ACTIVE(plus(N,0)) -> MARK(U51(isNat(N),N)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> ACTIVE(U16(mark(X))) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> ACTIVE(U23(mark(X))) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> ACTIVE(U32(mark(X))) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> ACTIVE(U41(mark(X))) MARK(U41(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X1,X2)) -> ACTIVE(U52(mark(X1),X2)) MARK(U52(X1,X2)) -> MARK(X1) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(U63(tt,M,N)) -> MARK(U64(isNatKind(N),M,N)) ACTIVE(U64(tt,M,N)) -> MARK(s(plus(N,M))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) ACTIVE(plus(N,s(M))) -> MARK(U61(isNat(M),M,N)) ACTIVE(plus(N,0)) -> MARK(U51(isNat(N),N)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> ACTIVE(U16(mark(X))) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> ACTIVE(U23(mark(X))) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> ACTIVE(U32(mark(X))) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> ACTIVE(U41(mark(X))) MARK(U41(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X1,X2)) -> ACTIVE(U52(mark(X1),X2)) MARK(U52(X1,X2)) -> MARK(X1) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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 + 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 [tt] = 2 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(U64(tt,M,N)) -> MARK(s(plus(N,M))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) ACTIVE(plus(N,s(M))) -> MARK(U61(isNat(M),M,N)) ACTIVE(plus(N,0)) -> MARK(U51(isNat(N),N)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> ACTIVE(U16(mark(X))) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> ACTIVE(U23(mark(X))) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> ACTIVE(U32(mark(X))) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> ACTIVE(U41(mark(X))) MARK(U41(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X1,X2)) -> ACTIVE(U52(mark(X1),X2)) MARK(U52(X1,X2)) -> MARK(X1) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(U64(tt,M,N)) -> MARK(s(plus(N,M))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) ACTIVE(plus(N,s(M))) -> MARK(U61(isNat(M),M,N)) ACTIVE(plus(N,0)) -> MARK(U51(isNat(N),N)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> ACTIVE(U16(mark(X))) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> ACTIVE(U23(mark(X))) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> ACTIVE(U32(mark(X))) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> ACTIVE(U41(mark(X))) MARK(U41(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X1,X2)) -> ACTIVE(U52(mark(X1),X2)) MARK(U52(X1,X2)) -> MARK(X1) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(U64(tt,M,N)) -> MARK(s(plus(N,M))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) ACTIVE(plus(N,s(M))) -> MARK(U61(isNat(M),M,N)) ACTIVE(plus(N,0)) -> MARK(U51(isNat(N),N)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> ACTIVE(U16(mark(X))) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> ACTIVE(U23(mark(X))) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> ACTIVE(U32(mark(X))) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> ACTIVE(U41(mark(X))) MARK(U41(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X1,X2)) -> ACTIVE(U52(mark(X1),X2)) MARK(U52(X1,X2)) -> MARK(X1) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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 + 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) = 2.X1 + 2.X2 + 2.X3 [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 [s](X) = X + 2 [0] = 2 [tt] = 2 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) ACTIVE(plus(N,s(M))) -> MARK(U61(isNat(M),M,N)) ACTIVE(plus(N,0)) -> MARK(U51(isNat(N),N)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> ACTIVE(U16(mark(X))) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> ACTIVE(U23(mark(X))) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> ACTIVE(U32(mark(X))) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> ACTIVE(U41(mark(X))) MARK(U41(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X1,X2)) -> ACTIVE(U52(mark(X1),X2)) MARK(U52(X1,X2)) -> MARK(X1) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) ACTIVE(plus(N,s(M))) -> MARK(U61(isNat(M),M,N)) ACTIVE(plus(N,0)) -> MARK(U51(isNat(N),N)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> ACTIVE(U16(mark(X))) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> ACTIVE(U23(mark(X))) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> ACTIVE(U32(mark(X))) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> ACTIVE(U41(mark(X))) MARK(U41(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X1,X2)) -> ACTIVE(U52(mark(X1),X2)) MARK(U52(X1,X2)) -> MARK(X1) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) ACTIVE(plus(N,s(M))) -> MARK(U61(isNat(M),M,N)) ACTIVE(plus(N,0)) -> MARK(U51(isNat(N),N)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> ACTIVE(U16(mark(X))) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> ACTIVE(U23(mark(X))) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> ACTIVE(U32(mark(X))) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> ACTIVE(U41(mark(X))) MARK(U41(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X1,X2)) -> ACTIVE(U52(mark(X1),X2)) MARK(U52(X1,X2)) -> MARK(X1) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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 + 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 + 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 [tt] = 2 [ACTIVE](X) = X + 2 [MARK](X) = X + 2 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) ACTIVE(plus(N,0)) -> MARK(U51(isNat(N),N)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> ACTIVE(U16(mark(X))) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> ACTIVE(U23(mark(X))) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> ACTIVE(U32(mark(X))) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> ACTIVE(U41(mark(X))) MARK(U41(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X1,X2)) -> ACTIVE(U52(mark(X1),X2)) MARK(U52(X1,X2)) -> MARK(X1) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) ACTIVE(plus(N,0)) -> MARK(U51(isNat(N),N)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> ACTIVE(U16(mark(X))) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> ACTIVE(U23(mark(X))) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> ACTIVE(U32(mark(X))) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> ACTIVE(U41(mark(X))) MARK(U41(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X1,X2)) -> ACTIVE(U52(mark(X1),X2)) MARK(U52(X1,X2)) -> MARK(X1) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) ACTIVE(plus(N,0)) -> MARK(U51(isNat(N),N)) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> ACTIVE(U16(mark(X))) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> ACTIVE(U23(mark(X))) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> ACTIVE(U32(mark(X))) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> ACTIVE(U41(mark(X))) MARK(U41(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X1,X2)) -> ACTIVE(U52(mark(X1),X2)) MARK(U52(X1,X2)) -> MARK(X1) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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) = 2.X1 + 2.X2 + 2.X3 + 2 [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 [tt] = 2 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> ACTIVE(U16(mark(X))) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> ACTIVE(U23(mark(X))) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> ACTIVE(U32(mark(X))) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> ACTIVE(U41(mark(X))) MARK(U41(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X1,X2)) -> ACTIVE(U52(mark(X1),X2)) MARK(U52(X1,X2)) -> MARK(X1) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> ACTIVE(U16(mark(X))) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> ACTIVE(U23(mark(X))) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> ACTIVE(U32(mark(X))) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> ACTIVE(U41(mark(X))) MARK(U41(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X1,X2)) -> ACTIVE(U52(mark(X1),X2)) MARK(U52(X1,X2)) -> MARK(X1) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> ACTIVE(U16(mark(X))) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> ACTIVE(U23(mark(X))) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> ACTIVE(U32(mark(X))) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> ACTIVE(U41(mark(X))) MARK(U41(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X1,X2)) -> ACTIVE(U52(mark(X1),X2)) MARK(U52(X1,X2)) -> MARK(X1) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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) = 1 [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] = 0 [tt] = 0 [ACTIVE](X) = 2.X [MARK](X) = 2 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> ACTIVE(U23(mark(X))) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> ACTIVE(U32(mark(X))) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> ACTIVE(U41(mark(X))) MARK(U41(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X1,X2)) -> ACTIVE(U52(mark(X1),X2)) MARK(U52(X1,X2)) -> MARK(X1) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> ACTIVE(U23(mark(X))) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> ACTIVE(U32(mark(X))) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> ACTIVE(U41(mark(X))) MARK(U41(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X1,X2)) -> ACTIVE(U52(mark(X1),X2)) MARK(U52(X1,X2)) -> MARK(X1) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> ACTIVE(U23(mark(X))) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> ACTIVE(U32(mark(X))) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> ACTIVE(U41(mark(X))) MARK(U41(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X1,X2)) -> ACTIVE(U52(mark(X1),X2)) MARK(U52(X1,X2)) -> MARK(X1) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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) = 1 [U21](X1,X2) = 2 [U22](X1,X2) = 2 [U23](X) = 0 [U31](X1,X2) = 2 [U32](X) = 2 [U41](X) = 2 [U51](X1,X2) = 2 [U52](X1,X2) = 1 [U61](X1,X2,X3) = 2 [U62](X1,X2,X3) = 2 [U63](X1,X2,X3) = 1 [U64](X1,X2,X3) = 2 [active](X) = 2 [isNat](X) = 2 [isNatKind](X) = 2 [mark](X) = 2 [plus](X1,X2) = 2 [s](X) = 2 [0] = 2 [tt] = 2 [ACTIVE](X) = X [MARK](X) = 2 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> ACTIVE(U32(mark(X))) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> ACTIVE(U41(mark(X))) MARK(U41(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X1,X2)) -> ACTIVE(U52(mark(X1),X2)) MARK(U52(X1,X2)) -> MARK(X1) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> ACTIVE(U32(mark(X))) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> ACTIVE(U41(mark(X))) MARK(U41(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X1,X2)) -> ACTIVE(U52(mark(X1),X2)) MARK(U52(X1,X2)) -> MARK(X1) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> ACTIVE(U32(mark(X))) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> ACTIVE(U41(mark(X))) MARK(U41(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X1,X2)) -> ACTIVE(U52(mark(X1),X2)) MARK(U52(X1,X2)) -> MARK(X1) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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) = 0 [U31](X1,X2) = 1 [U32](X) = 0 [U41](X) = 1 [U51](X1,X2) = 0 [U52](X1,X2) = 0 [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] = 0 [tt] = 0 [ACTIVE](X) = 2.X [MARK](X) = 2 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> ACTIVE(U41(mark(X))) MARK(U41(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X1,X2)) -> ACTIVE(U52(mark(X1),X2)) MARK(U52(X1,X2)) -> MARK(X1) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> ACTIVE(U41(mark(X))) MARK(U41(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X1,X2)) -> ACTIVE(U52(mark(X1),X2)) MARK(U52(X1,X2)) -> MARK(X1) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> ACTIVE(U41(mark(X))) MARK(U41(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X1,X2)) -> ACTIVE(U52(mark(X1),X2)) MARK(U52(X1,X2)) -> MARK(X1) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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) = 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] = 0 [tt] = 2 [ACTIVE](X) = 2.X [MARK](X) = 2 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X1,X2)) -> ACTIVE(U52(mark(X1),X2)) MARK(U52(X1,X2)) -> MARK(X1) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X1,X2)) -> ACTIVE(U52(mark(X1),X2)) MARK(U52(X1,X2)) -> MARK(X1) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U51(X1,X2)) -> ACTIVE(U51(mark(X1),X2)) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X1,X2)) -> ACTIVE(U52(mark(X1),X2)) MARK(U52(X1,X2)) -> MARK(X1) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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) = 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 + 1 [U61](X1,X2,X3) = X1 + X2 + 2.X3 + 2 [U62](X1,X2,X3) = X1 + X2 + 2.X3 + 2 [U63](X1,X2,X3) = 2.X1 + X2 + 2.X3 + 2 [U64](X1,X2,X3) = X1 + X2 + 2.X3 + 2 [active](X) = X [isNat](X) = 0 [isNatKind](X) = 0 [mark](X) = X [plus](X1,X2) = 2.X1 + X2 + 1 [s](X) = X + 1 [0] = 2 [tt] = 0 [ACTIVE](X) = 2 [MARK](X) = 2.X + 2 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X1,X2)) -> ACTIVE(U52(mark(X1),X2)) MARK(U52(X1,X2)) -> MARK(X1) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X1,X2)) -> ACTIVE(U52(mark(X1),X2)) MARK(U52(X1,X2)) -> MARK(X1) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U51(X1,X2)) -> MARK(X1) MARK(U52(X1,X2)) -> ACTIVE(U52(mark(X1),X2)) MARK(U52(X1,X2)) -> MARK(X1) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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 [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 + 1 [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 [tt] = 2 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U52(X1,X2)) -> ACTIVE(U52(mark(X1),X2)) MARK(U52(X1,X2)) -> MARK(X1) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U52(X1,X2)) -> ACTIVE(U52(mark(X1),X2)) MARK(U52(X1,X2)) -> MARK(X1) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U52(X1,X2)) -> ACTIVE(U52(mark(X1),X2)) MARK(U52(X1,X2)) -> MARK(X1) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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) = 2.X1 [U14](X1,X2,X3) = 2.X1 [U15](X1,X2) = 2.X1 [U16](X) = 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 + 1 [U52](X1,X2) = 2.X1 + X2 + 1 [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 + 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] = 2 [tt] = 0 [ACTIVE](X) = 2 [MARK](X) = 2.X + 2 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U52(X1,X2)) -> MARK(X1) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U52(X1,X2)) -> MARK(X1) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U52(X1,X2)) -> MARK(X1) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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) = 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 + 1 [0] = 2 [tt] = 2 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> ACTIVE(U61(mark(X1),X2,X3)) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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) = 2 [U52](X1,X2) = 1 [U61](X1,X2,X3) = 0 [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 [tt] = 0 [ACTIVE](X) = 2.X [MARK](X) = 2 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U61(X1,X2,X3)) -> MARK(X1) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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) = X1 + 2.X2 + 2.X3 + 2 [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 + 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 [tt] = 2 [ACTIVE](X) = 2.X + 1 [MARK](X) = 2.X + 1 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U62(X1,X2,X3)) -> ACTIVE(U62(mark(X1),X2,X3)) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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) = 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) = X [U41](X) = X [U51](X1,X2) = 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) = 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] = 2 [tt] = 0 [ACTIVE](X) = 0 [MARK](X) = 2.X Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U62(X1,X2,X3)) -> MARK(X1) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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) = 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 [tt] = 2 [ACTIVE](X) = 2.X [MARK](X) = 2.X Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U63(X1,X2,X3)) -> ACTIVE(U63(mark(X1),X2,X3)) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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) = X1 [U14](X1,X2,X3) = 2.X1 [U15](X1,X2) = X1 [U16](X) = 2.X [U21](X1,X2) = X1 [U22](X1,X2) = X1 [U23](X) = X [U31](X1,X2) = 2.X1 [U32](X) = 2.X [U41](X) = X [U51](X1,X2) = 2.X1 + 2.X2 [U52](X1,X2) = 2.X1 + X2 [U61](X1,X2,X3) = 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) = 0 [isNatKind](X) = 0 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X [0] = 2 [tt] = 0 [ACTIVE](X) = X + 2 [MARK](X) = 2.X + 2 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U63(X1,X2,X3)) -> MARK(X1) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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 + 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) = 2 [isNatKind](X) = 2 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X + 2 [0] = 2 [tt] = 2 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(U11(tt,V1,V2)) -> MARK(U12(isNatKind(V1),V1,V2)) ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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) = 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 [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) = 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 [tt] = 2 [ACTIVE](X) = 2.X + 1 [MARK](X) = 2.X + 1 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(U12(tt,V1,V2)) -> MARK(U13(isNatKind(V2),V1,V2)) ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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) = X1 + 2.X2 + 2.X3 + 2 [U13](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 1 [U14](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 [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 + 1 [U52](X1,X2) = 2.X2 + 1 [U61](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U62](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 [U63](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U64](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 [active](X) = X [isNat](X) = 2.X + 1 [isNatKind](X) = 1 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X [0] = 2 [tt] = 1 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(U13(tt,V1,V2)) -> MARK(U14(isNatKind(V2),V1,V2)) ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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) = 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) = 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] = 1 [tt] = 2 [ACTIVE](X) = 2.X + 1 [MARK](X) = 2.X + 1 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(U14(tt,V1,V2)) -> MARK(U15(isNat(V1),V2)) ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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) = 2.X1 + 2.X2 + 2.X3 [U15](X1,X2) = X1 + 2.X2 + 1 [U16](X) = X + 1 [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) = 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] = 0 [tt] = 2 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(U15(tt,V2)) -> MARK(U16(isNat(V2))) ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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 + 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) = 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 [tt] = 2 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(U21(tt,V1)) -> MARK(U22(isNatKind(V1),V1)) ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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 + 2 [U22](X1,X2) = X1 + 2.X2 + 2 [U23](X) = X + 2 [U31](X1,X2) = X1 [U32](X) = X [U41](X) = X [U51](X1,X2) = X2 + 2 [U52](X1,X2) = X2 + 1 [U61](X1,X2,X3) = 2.X2 + X3 + 2 [U62](X1,X2,X3) = X1 + 2.X2 + X3 [U63](X1,X2,X3) = 2.X2 + X3 + 2 [U64](X1,X2,X3) = X1 + 2.X2 + X3 [active](X) = X [isNat](X) = 2.X + 2 [isNatKind](X) = 2 [mark](X) = X [plus](X1,X2) = X1 + 2.X2 [s](X) = X + 2 [0] = 2 [tt] = 2 [ACTIVE](X) = 2.X + 1 [MARK](X) = 2.X + 1 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(U22(tt,V1)) -> MARK(U23(isNat(V1))) ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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 + 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 + 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 [tt] = 2 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(plus(V1,V2))) -> MARK(U11(isNatKind(V1),V1,V2)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 1 [U12](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 1 [U13](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 1 [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 + 1 [U22](X1,X2) = X1 + 2.X2 + 1 [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 + 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 [tt] = 2 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNatKind(V1),V1)) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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 + 1 [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) = 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 [s](X) = X + 2 [0] = 1 [tt] = 2 [ACTIVE](X) = 2.X + 1 [MARK](X) = 2.X + 1 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> ACTIVE(U11(mark(X1),X2,X3)) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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 + 2 [U15](X1,X2) = X1 + 2.X2 + 1 [U16](X) = X [U21](X1,X2) = 2.X1 + 2.X2 + 2 [U22](X1,X2) = X1 + 2.X2 + 2 [U23](X) = X + 1 [U31](X1,X2) = 2.X1 [U32](X) = X [U41](X) = 2.X [U51](X1,X2) = 2.X2 + 2 [U52](X1,X2) = X1 + 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) = 2.X1 + 2.X2 + 2.X3 + 2 [active](X) = X [isNat](X) = 2.X + 1 [isNatKind](X) = 0 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 1 [s](X) = X + 1 [0] = 2 [tt] = 0 [ACTIVE](X) = 1 [MARK](X) = 2.X + 1 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U11(X1,X2,X3)) -> MARK(X1) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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) = 2.X1 + 2.X2 + 2.X3 [U15](X1,X2) = X1 + 2.X2 + 2 [U16](X) = X + 1 [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) = X2 + 2 [U52](X1,X2) = 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) = 2.X + 2 [isNatKind](X) = 2 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 1 [s](X) = X + 1 [0] = 2 [tt] = 2 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U12(X1,X2,X3)) -> ACTIVE(U12(mark(X1),X2,X3)) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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 + 1 [U16](X) = X [U21](X1,X2) = 2.X1 + 2.X2 + 2 [U22](X1,X2) = 2.X1 + 2.X2 + 2 [U23](X) = X + 1 [U31](X1,X2) = 2.X1 [U32](X) = X [U41](X) = 2.X [U51](X1,X2) = X2 + 1 [U52](X1,X2) = 2.X1 + X2 + 1 [U61](X1,X2,X3) = 2.X2 + X3 + 2 [U62](X1,X2,X3) = 2.X2 + X3 + 2 [U63](X1,X2,X3) = 2.X2 + X3 + 2 [U64](X1,X2,X3) = X1 + 2.X2 + X3 + 2 [active](X) = X [isNat](X) = 2.X + 1 [isNatKind](X) = 0 [mark](X) = X [plus](X1,X2) = X1 + 2.X2 + 1 [s](X) = X + 1 [0] = 0 [tt] = 0 [ACTIVE](X) = 2 [MARK](X) = 2.X + 2 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U12(X1,X2,X3)) -> MARK(X1) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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 [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) = 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 [tt] = 2 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U13(X1,X2,X3)) -> ACTIVE(U13(mark(X1),X2,X3)) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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 + 1 [U13](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 1 [U14](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 1 [U15](X1,X2) = 2.X1 + 2.X2 [U16](X) = 2.X [U21](X1,X2) = 2.X1 + X2 [U22](X1,X2) = 2.X1 + X2 [U23](X) = X [U31](X1,X2) = 2.X1 [U32](X) = X [U41](X) = 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.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) = X [isNatKind](X) = 0 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X [0] = 2 [tt] = 0 [ACTIVE](X) = 2 [MARK](X) = 2.X + 2 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U13(X1,X2,X3)) -> MARK(X1) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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 [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) = 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 + 2 [s](X) = X [0] = 2 [tt] = 2 [ACTIVE](X) = 2.X [MARK](X) = 2.X Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U14(X1,X2,X3)) -> ACTIVE(U14(mark(X1),X2,X3)) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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.X2 + 2.X3 + 2 [U13](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U14](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [U15](X1,X2) = X1 + 2.X2 + 1 [U16](X) = X [U21](X1,X2) = 2.X1 + 2.X2 + 2 [U22](X1,X2) = X1 + 2.X2 + 2 [U23](X) = X + 1 [U31](X1,X2) = 2.X1 [U32](X) = 2.X [U41](X) = X [U51](X1,X2) = X2 + 2 [U52](X1,X2) = X1 + 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 + 2 [active](X) = X [isNat](X) = 2.X + 1 [isNatKind](X) = 0 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 1 [s](X) = X + 1 [0] = 2 [tt] = 0 [ACTIVE](X) = 2 [MARK](X) = 2.X + 2 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U14(X1,X2,X3)) -> MARK(X1) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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) = 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) = 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] = 1 [tt] = 2 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U15(X1,X2)) -> ACTIVE(U15(mark(X1),X2)) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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) = X1 + 2.X2 + 2.X3 + 2 [U13](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [U14](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U15](X1,X2) = X1 + 2.X2 + 2 [U16](X) = X + 1 [U21](X1,X2) = X1 + 2.X2 + 2 [U22](X1,X2) = 2.X1 + 2.X2 [U23](X) = X [U31](X1,X2) = 2.X1 [U32](X) = 2.X [U41](X) = 2.X [U51](X1,X2) = 2.X2 + 2 [U52](X1,X2) = X1 + 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 + 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 [tt] = 0 [ACTIVE](X) = 2 [MARK](X) = 2.X + 2 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U15(X1,X2)) -> MARK(X1) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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 + 1 [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) = 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 [tt] = 2 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U16(X)) -> MARK(X) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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 [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 [tt] = 2 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U21(X1,X2)) -> ACTIVE(U21(mark(X1),X2)) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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.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 [U15](X1,X2) = 2.X2 [U16](X) = 0 [U21](X1,X2) = 2.X1 + X2 + 2 [U22](X1,X2) = 2.X1 + X2 + 2 [U23](X) = X [U31](X1,X2) = 2.X1 [U32](X) = 2.X [U41](X) = X [U51](X1,X2) = X1 + X2 [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) = 2.X1 + 2.X2 + 2.X3 + 1 [active](X) = X [isNat](X) = X + 2 [isNatKind](X) = 0 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 [s](X) = X + 1 [0] = 1 [tt] = 0 [ACTIVE](X) = 2 [MARK](X) = 2.X + 2 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U21(X1,X2)) -> MARK(X1) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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) = 2.X1 + 2.X2 + 2.X3 [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 + 2 [U31](X1,X2) = X1 [U32](X) = X [U41](X) = X [U51](X1,X2) = X2 + 1 [U52](X1,X2) = X2 + 1 [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] = 0 [tt] = 2 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U22(X1,X2)) -> ACTIVE(U22(mark(X1),X2)) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 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 + 2 [U14](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U15](X1,X2) = 2.X2 + 2 [U16](X) = 2 [U21](X1,X2) = 2.X1 + 2.X2 + 2 [U22](X1,X2) = X1 + 2.X2 + 2 [U23](X) = X [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) = X2 + 2.X3 + 2 [U62](X1,X2,X3) = 2.X1 + X2 + 2.X3 + 2 [U63](X1,X2,X3) = X2 + 2.X3 + 2 [U64](X1,X2,X3) = 2.X1 + X2 + 2.X3 + 2 [active](X) = X [isNat](X) = 2.X + 2 [isNatKind](X) = 0 [mark](X) = X [plus](X1,X2) = 2.X1 + X2 + 1 [s](X) = X + 1 [0] = 2 [tt] = 0 [ACTIVE](X) = 2 [MARK](X) = 2.X + 2 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U22(X1,X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 2.X1 + 2.X3 + 2 [U12](X1,X2,X3) = 2.X3 + 2 [U13](X1,X2,X3) = 2.X3 + 2 [U14](X1,X2,X3) = 2.X3 + 2 [U15](X1,X2) = 2.X2 + 2 [U16](X) = X [U21](X1,X2) = 2.X2 + 2 [U22](X1,X2) = 2.X1 + 2.X2 + 2 [U23](X) = X [U31](X1,X2) = 2.X1 [U32](X) = 2.X [U41](X) = 2.X [U51](X1,X2) = X2 + 2 [U52](X1,X2) = 2.X1 + 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] = 1 [tt] = 0 [ACTIVE](X) = 2 [MARK](X) = 2.X + 2 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U23(X)) -> MARK(X) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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) = 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 [tt] = 2 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> ACTIVE(U64(mark(X1),X2,X3)) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 1 [U12](X1,X2,X3) = 2 [U13](X1,X2,X3) = 1 [U14](X1,X2,X3) = 2 [U15](X1,X2) = 2 [U16](X) = 0 [U21](X1,X2) = 0 [U22](X1,X2) = 2 [U23](X) = 2 [U31](X1,X2) = 1 [U32](X) = 0 [U41](X) = 2 [U51](X1,X2) = 0 [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) = 1 [isNatKind](X) = 1 [mark](X) = 2 [plus](X1,X2) = 1 [s](X) = 1 [0] = 2 [tt] = 0 [ACTIVE](X) = X + 1 [MARK](X) = 2 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(U64(X1,X2,X3)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = X1 + 2.X2 + 2 [U12](X1,X2,X3) = 2.X1 + 1 [U13](X1,X2,X3) = 2.X1 + 1 [U14](X1,X2,X3) = 2.X1 [U15](X1,X2) = 2 [U16](X) = 1 [U21](X1,X2) = 2.X2 + 2 [U22](X1,X2) = 2 [U23](X) = 2 [U31](X1,X2) = X1 [U32](X) = X [U41](X) = X [U51](X1,X2) = 2.X2 + 2 [U52](X1,X2) = 2.X1 + 2.X2 [U61](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U62](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 [U63](X1,X2,X3) = 2.X2 + 2.X3 + 2 [U64](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 1 [active](X) = X [isNat](X) = 2.X + 2 [isNatKind](X) = 1 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X [0] = 2 [tt] = 1 [ACTIVE](X) = 2.X + 1 [MARK](X) = 2.X + 1 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(U31(tt,V2)) -> MARK(U32(isNatKind(V2))) ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = 2.X1 + 2.X3 + 2 [U12](X1,X2,X3) = 2.X3 + 2 [U13](X1,X2,X3) = 2.X3 + 2 [U14](X1,X2,X3) = 2.X3 + 2 [U15](X1,X2) = 2.X2 + 2 [U16](X) = 2 [U21](X1,X2) = 2.X2 + 2 [U22](X1,X2) = 2.X2 + 2 [U23](X) = 2 [U31](X1,X2) = 2.X1 + 2.X2 [U32](X) = X + 2 [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) = 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 + 2 [isNatKind](X) = 2.X + 1 [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X [0] = 2 [tt] = 2 [ACTIVE](X) = 2.X + 1 [MARK](X) = 2.X + 1 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(isNatKind(plus(V1,V2))) -> MARK(U31(isNatKind(V1),V2)) ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [U11](X1,X2,X3) = X1 + 2.X2 + 2 [U12](X1,X2,X3) = 2.X2 + 2 [U13](X1,X2,X3) = 2.X2 + 2 [U14](X1,X2,X3) = 2.X2 + 2 [U15](X1,X2) = X1 [U16](X) = 2 [U21](X1,X2) = X1 + 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.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) = 2.X2 + 2.X3 + 2 [active](X) = X [isNat](X) = 2.X + 2 [isNatKind](X) = 2.X [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 1 [s](X) = X + 1 [0] = 2 [tt] = 2 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.23: SCC Processor: -> Pairs: ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Reduction Pair Processor: -> Pairs: ACTIVE(isNatKind(s(V1))) -> MARK(U41(isNatKind(V1))) MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) -> Usable rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->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) = X1 [U16](X) = 2 [U21](X1,X2) = 2 [U22](X1,X2) = 2 [U23](X) = 2 [U31](X1,X2) = 2.X1 + 2.X2 + 2 [U32](X) = X + 2 [U41](X) = X + 1 [U51](X1,X2) = 2.X1 + 2.X2 + 1 [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) = 2.X2 + 2.X3 + 2 [active](X) = X [isNat](X) = 2 [isNatKind](X) = 2.X [mark](X) = X [plus](X1,X2) = 2.X1 + 2.X2 + 1 [s](X) = X + 1 [0] = 2 [tt] = 2 [ACTIVE](X) = 2.X + 1 [MARK](X) = 2.X + 2 Problem 1.23: SCC Processor: -> Pairs: MARK(U31(X1,X2)) -> ACTIVE(U31(mark(X1),X2)) MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(isNatKind(X)) -> ACTIVE(isNatKind(X)) MARK(plus(X1,X2)) -> ACTIVE(plus(mark(X1),mark(X2))) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> MARK(X) ->->-> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) Problem 1.23: Subterm Processor: -> Pairs: MARK(U31(X1,X2)) -> MARK(X1) MARK(U32(X)) -> MARK(X) MARK(U41(X)) -> MARK(X) MARK(plus(X1,X2)) -> MARK(X1) MARK(plus(X1,X2)) -> MARK(X2) MARK(s(X)) -> MARK(X) -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Projection: pi(MARK) = 1 Problem 1.23: SCC Processor: -> Pairs: Empty -> Rules: U11(active(X1),X2,X3) -> U11(X1,X2,X3) U11(mark(X1),X2,X3) -> U11(X1,X2,X3) U11(X1,active(X2),X3) -> U11(X1,X2,X3) U11(X1,mark(X2),X3) -> U11(X1,X2,X3) U11(X1,X2,active(X3)) -> U11(X1,X2,X3) U11(X1,X2,mark(X3)) -> U11(X1,X2,X3) U12(active(X1),X2,X3) -> U12(X1,X2,X3) U12(mark(X1),X2,X3) -> U12(X1,X2,X3) U12(X1,active(X2),X3) -> U12(X1,X2,X3) U12(X1,mark(X2),X3) -> U12(X1,X2,X3) U12(X1,X2,active(X3)) -> U12(X1,X2,X3) U12(X1,X2,mark(X3)) -> U12(X1,X2,X3) U13(active(X1),X2,X3) -> U13(X1,X2,X3) U13(mark(X1),X2,X3) -> U13(X1,X2,X3) U13(X1,active(X2),X3) -> U13(X1,X2,X3) U13(X1,mark(X2),X3) -> U13(X1,X2,X3) U13(X1,X2,active(X3)) -> U13(X1,X2,X3) U13(X1,X2,mark(X3)) -> U13(X1,X2,X3) U14(active(X1),X2,X3) -> U14(X1,X2,X3) U14(mark(X1),X2,X3) -> U14(X1,X2,X3) U14(X1,active(X2),X3) -> U14(X1,X2,X3) U14(X1,mark(X2),X3) -> U14(X1,X2,X3) U14(X1,X2,active(X3)) -> U14(X1,X2,X3) U14(X1,X2,mark(X3)) -> U14(X1,X2,X3) U15(active(X1),X2) -> U15(X1,X2) U15(mark(X1),X2) -> U15(X1,X2) U15(X1,active(X2)) -> U15(X1,X2) U15(X1,mark(X2)) -> U15(X1,X2) U16(active(X)) -> U16(X) U16(mark(X)) -> U16(X) U21(active(X1),X2) -> U21(X1,X2) U21(mark(X1),X2) -> U21(X1,X2) U21(X1,active(X2)) -> U21(X1,X2) U21(X1,mark(X2)) -> U21(X1,X2) U22(active(X1),X2) -> U22(X1,X2) U22(mark(X1),X2) -> U22(X1,X2) U22(X1,active(X2)) -> U22(X1,X2) U22(X1,mark(X2)) -> U22(X1,X2) U23(active(X)) -> U23(X) U23(mark(X)) -> U23(X) U31(active(X1),X2) -> U31(X1,X2) U31(mark(X1),X2) -> U31(X1,X2) U31(X1,active(X2)) -> U31(X1,X2) U31(X1,mark(X2)) -> U31(X1,X2) U32(active(X)) -> U32(X) U32(mark(X)) -> U32(X) U41(active(X)) -> U41(X) U41(mark(X)) -> U41(X) U51(active(X1),X2) -> U51(X1,X2) U51(mark(X1),X2) -> U51(X1,X2) U51(X1,active(X2)) -> U51(X1,X2) U51(X1,mark(X2)) -> U51(X1,X2) U52(active(X1),X2) -> U52(X1,X2) U52(mark(X1),X2) -> U52(X1,X2) U52(X1,active(X2)) -> U52(X1,X2) U52(X1,mark(X2)) -> U52(X1,X2) U61(active(X1),X2,X3) -> U61(X1,X2,X3) U61(mark(X1),X2,X3) -> U61(X1,X2,X3) U61(X1,active(X2),X3) -> U61(X1,X2,X3) U61(X1,mark(X2),X3) -> U61(X1,X2,X3) U61(X1,X2,active(X3)) -> U61(X1,X2,X3) U61(X1,X2,mark(X3)) -> U61(X1,X2,X3) U62(active(X1),X2,X3) -> U62(X1,X2,X3) U62(mark(X1),X2,X3) -> U62(X1,X2,X3) U62(X1,active(X2),X3) -> U62(X1,X2,X3) U62(X1,mark(X2),X3) -> U62(X1,X2,X3) U62(X1,X2,active(X3)) -> U62(X1,X2,X3) U62(X1,X2,mark(X3)) -> U62(X1,X2,X3) U63(active(X1),X2,X3) -> U63(X1,X2,X3) U63(mark(X1),X2,X3) -> U63(X1,X2,X3) U63(X1,active(X2),X3) -> U63(X1,X2,X3) U63(X1,mark(X2),X3) -> U63(X1,X2,X3) U63(X1,X2,active(X3)) -> U63(X1,X2,X3) U63(X1,X2,mark(X3)) -> U63(X1,X2,X3) U64(active(X1),X2,X3) -> U64(X1,X2,X3) U64(mark(X1),X2,X3) -> U64(X1,X2,X3) U64(X1,active(X2),X3) -> U64(X1,X2,X3) U64(X1,mark(X2),X3) -> U64(X1,X2,X3) U64(X1,X2,active(X3)) -> U64(X1,X2,X3) U64(X1,X2,mark(X3)) -> U64(X1,X2,X3) active(U11(tt,V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) active(U12(tt,V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) active(U13(tt,V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) active(U14(tt,V1,V2)) -> mark(U15(isNat(V1),V2)) active(U15(tt,V2)) -> mark(U16(isNat(V2))) active(U16(tt)) -> mark(tt) active(U21(tt,V1)) -> mark(U22(isNatKind(V1),V1)) active(U22(tt,V1)) -> mark(U23(isNat(V1))) active(U23(tt)) -> mark(tt) active(U31(tt,V2)) -> mark(U32(isNatKind(V2))) active(U32(tt)) -> mark(tt) active(U41(tt)) -> mark(tt) active(U51(tt,N)) -> mark(U52(isNatKind(N),N)) active(U52(tt,N)) -> mark(N) active(U61(tt,M,N)) -> mark(U62(isNatKind(M),M,N)) active(U62(tt,M,N)) -> mark(U63(isNat(N),M,N)) active(U63(tt,M,N)) -> mark(U64(isNatKind(N),M,N)) active(U64(tt,M,N)) -> mark(s(plus(N,M))) active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) active(isNat(0)) -> mark(tt) active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) active(isNatKind(0)) -> mark(tt) active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) active(plus(N,0)) -> mark(U51(isNat(N),N)) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) isNatKind(active(X)) -> isNatKind(X) isNatKind(mark(X)) -> isNatKind(X) mark(U11(X1,X2,X3)) -> active(U11(mark(X1),X2,X3)) mark(U12(X1,X2,X3)) -> active(U12(mark(X1),X2,X3)) mark(U13(X1,X2,X3)) -> active(U13(mark(X1),X2,X3)) mark(U14(X1,X2,X3)) -> active(U14(mark(X1),X2,X3)) mark(U15(X1,X2)) -> active(U15(mark(X1),X2)) mark(U16(X)) -> active(U16(mark(X))) mark(U21(X1,X2)) -> active(U21(mark(X1),X2)) mark(U22(X1,X2)) -> active(U22(mark(X1),X2)) mark(U23(X)) -> active(U23(mark(X))) mark(U31(X1,X2)) -> active(U31(mark(X1),X2)) mark(U32(X)) -> active(U32(mark(X))) mark(U41(X)) -> active(U41(mark(X))) mark(U51(X1,X2)) -> active(U51(mark(X1),X2)) mark(U52(X1,X2)) -> active(U52(mark(X1),X2)) mark(U61(X1,X2,X3)) -> active(U61(mark(X1),X2,X3)) mark(U62(X1,X2,X3)) -> active(U62(mark(X1),X2,X3)) mark(U63(X1,X2,X3)) -> active(U63(mark(X1),X2,X3)) mark(U64(X1,X2,X3)) -> active(U64(mark(X1),X2,X3)) mark(isNat(X)) -> active(isNat(X)) mark(isNatKind(X)) -> active(isNatKind(X)) mark(plus(X1,X2)) -> active(plus(mark(X1),mark(X2))) mark(s(X)) -> active(s(mark(X))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1),X2) -> plus(X1,X2) plus(mark(X1),X2) -> plus(X1,X2) plus(X1,active(X2)) -> plus(X1,X2) plus(X1,mark(X2)) -> plus(X1,X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite.