1.91/1.92 YES 1.91/1.92 1.91/1.92 Problem 1: 1.91/1.92 1.91/1.92 (VAR v_NonEmpty:S M:S N:S V1:S V2:S X:S X1:S X2:S X3:S) 1.91/1.92 (RULES 1.91/1.92 a__U11(tt,V1:S,V2:S) -> a__U12(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.92 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 1.91/1.92 a__U12(tt,V1:S,V2:S) -> a__U13(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.92 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 1.91/1.92 a__U13(tt,V1:S,V2:S) -> a__U14(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.92 a__U13(X1:S,X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) 1.91/1.92 a__U14(tt,V1:S,V2:S) -> a__U15(a__isNat(V1:S),V2:S) 1.91/1.92 a__U14(X1:S,X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) 1.91/1.92 a__U15(tt,V2:S) -> a__U16(a__isNat(V2:S)) 1.91/1.92 a__U15(X1:S,X2:S) -> U15(X1:S,X2:S) 1.91/1.92 a__U16(tt) -> tt 1.91/1.92 a__U16(X:S) -> U16(X:S) 1.91/1.92 a__U21(tt,V1:S) -> a__U22(a__isNatKind(V1:S),V1:S) 1.91/1.92 a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) 1.91/1.92 a__U22(tt,V1:S) -> a__U23(a__isNat(V1:S)) 1.91/1.92 a__U22(X1:S,X2:S) -> U22(X1:S,X2:S) 1.91/1.92 a__U23(tt) -> tt 1.91/1.92 a__U23(X:S) -> U23(X:S) 1.91/1.92 a__U31(tt,V2:S) -> a__U32(a__isNatKind(V2:S)) 1.91/1.92 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 1.91/1.92 a__U32(tt) -> tt 1.91/1.92 a__U32(X:S) -> U32(X:S) 1.91/1.92 a__U41(tt) -> tt 1.91/1.92 a__U41(X:S) -> U41(X:S) 1.91/1.92 a__U51(tt,N:S) -> a__U52(a__isNatKind(N:S),N:S) 1.91/1.92 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.91/1.92 a__U52(tt,N:S) -> mark(N:S) 1.91/1.92 a__U52(X1:S,X2:S) -> U52(X1:S,X2:S) 1.91/1.92 a__U61(tt,M:S,N:S) -> a__U62(a__isNatKind(M:S),M:S,N:S) 1.91/1.92 a__U61(X1:S,X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) 1.91/1.92 a__U62(tt,M:S,N:S) -> a__U63(a__isNat(N:S),M:S,N:S) 1.91/1.92 a__U62(X1:S,X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) 1.91/1.92 a__U63(tt,M:S,N:S) -> a__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.92 a__U63(X1:S,X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) 1.91/1.92 a__U64(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 1.91/1.92 a__U64(X1:S,X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) 1.91/1.92 a__isNat(0) -> tt 1.91/1.92 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.92 a__isNat(s(V1:S)) -> a__U21(a__isNatKind(V1:S),V1:S) 1.91/1.92 a__isNat(X:S) -> isNat(X:S) 1.91/1.92 a__isNatKind(0) -> tt 1.91/1.92 a__isNatKind(plus(V1:S,V2:S)) -> a__U31(a__isNatKind(V1:S),V2:S) 1.91/1.92 a__isNatKind(s(V1:S)) -> a__U41(a__isNatKind(V1:S)) 1.91/1.92 a__isNatKind(X:S) -> isNatKind(X:S) 1.91/1.92 a__plus(N:S,0) -> a__U51(a__isNat(N:S),N:S) 1.91/1.92 a__plus(N:S,s(M:S)) -> a__U61(a__isNat(M:S),M:S,N:S) 1.91/1.92 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 1.91/1.92 mark(0) -> 0 1.91/1.92 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U13(X1:S,X2:S,X3:S)) -> a__U13(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U14(X1:S,X2:S,X3:S)) -> a__U14(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U15(X1:S,X2:S)) -> a__U15(mark(X1:S),X2:S) 1.91/1.92 mark(U16(X:S)) -> a__U16(mark(X:S)) 1.91/1.92 mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) 1.91/1.92 mark(U22(X1:S,X2:S)) -> a__U22(mark(X1:S),X2:S) 1.91/1.92 mark(U23(X:S)) -> a__U23(mark(X:S)) 1.91/1.92 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 1.91/1.92 mark(U32(X:S)) -> a__U32(mark(X:S)) 1.91/1.92 mark(U41(X:S)) -> a__U41(mark(X:S)) 1.91/1.92 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.91/1.92 mark(U52(X1:S,X2:S)) -> a__U52(mark(X1:S),X2:S) 1.91/1.92 mark(U61(X1:S,X2:S,X3:S)) -> a__U61(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U62(X1:S,X2:S,X3:S)) -> a__U62(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U63(X1:S,X2:S,X3:S)) -> a__U63(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U64(X1:S,X2:S,X3:S)) -> a__U64(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(isNat(X:S)) -> a__isNat(X:S) 1.91/1.92 mark(isNatKind(X:S)) -> a__isNatKind(X:S) 1.91/1.92 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 1.91/1.92 mark(s(X:S)) -> s(mark(X:S)) 1.91/1.92 mark(tt) -> tt 1.91/1.92 ) 1.91/1.92 (STRATEGY INNERMOST) 1.91/1.92 1.91/1.92 Problem 1: 1.91/1.92 1.91/1.92 Dependency Pairs Processor: 1.91/1.92 -> Pairs: 1.91/1.92 A__U11(tt,V1:S,V2:S) -> A__U12(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.92 A__U11(tt,V1:S,V2:S) -> A__ISNATKIND(V1:S) 1.91/1.92 A__U12(tt,V1:S,V2:S) -> A__U13(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.92 A__U12(tt,V1:S,V2:S) -> A__ISNATKIND(V2:S) 1.91/1.92 A__U13(tt,V1:S,V2:S) -> A__U14(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.92 A__U13(tt,V1:S,V2:S) -> A__ISNATKIND(V2:S) 1.91/1.92 A__U14(tt,V1:S,V2:S) -> A__U15(a__isNat(V1:S),V2:S) 1.91/1.92 A__U14(tt,V1:S,V2:S) -> A__ISNAT(V1:S) 1.91/1.92 A__U15(tt,V2:S) -> A__U16(a__isNat(V2:S)) 1.91/1.92 A__U15(tt,V2:S) -> A__ISNAT(V2:S) 1.91/1.92 A__U21(tt,V1:S) -> A__U22(a__isNatKind(V1:S),V1:S) 1.91/1.92 A__U21(tt,V1:S) -> A__ISNATKIND(V1:S) 1.91/1.92 A__U22(tt,V1:S) -> A__U23(a__isNat(V1:S)) 1.91/1.92 A__U22(tt,V1:S) -> A__ISNAT(V1:S) 1.91/1.92 A__U31(tt,V2:S) -> A__U32(a__isNatKind(V2:S)) 1.91/1.92 A__U31(tt,V2:S) -> A__ISNATKIND(V2:S) 1.91/1.92 A__U51(tt,N:S) -> A__U52(a__isNatKind(N:S),N:S) 1.91/1.92 A__U51(tt,N:S) -> A__ISNATKIND(N:S) 1.91/1.92 A__U52(tt,N:S) -> MARK(N:S) 1.91/1.92 A__U61(tt,M:S,N:S) -> A__U62(a__isNatKind(M:S),M:S,N:S) 1.91/1.92 A__U61(tt,M:S,N:S) -> A__ISNATKIND(M:S) 1.91/1.92 A__U62(tt,M:S,N:S) -> A__U63(a__isNat(N:S),M:S,N:S) 1.91/1.92 A__U62(tt,M:S,N:S) -> A__ISNAT(N:S) 1.91/1.92 A__U63(tt,M:S,N:S) -> A__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.92 A__U63(tt,M:S,N:S) -> A__ISNATKIND(N:S) 1.91/1.92 A__U64(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S)) 1.91/1.92 A__U64(tt,M:S,N:S) -> MARK(M:S) 1.91/1.92 A__U64(tt,M:S,N:S) -> MARK(N:S) 1.91/1.92 A__ISNAT(plus(V1:S,V2:S)) -> A__U11(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.92 A__ISNAT(plus(V1:S,V2:S)) -> A__ISNATKIND(V1:S) 1.91/1.92 A__ISNAT(s(V1:S)) -> A__U21(a__isNatKind(V1:S),V1:S) 1.91/1.92 A__ISNAT(s(V1:S)) -> A__ISNATKIND(V1:S) 1.91/1.92 A__ISNATKIND(plus(V1:S,V2:S)) -> A__U31(a__isNatKind(V1:S),V2:S) 1.91/1.92 A__ISNATKIND(plus(V1:S,V2:S)) -> A__ISNATKIND(V1:S) 1.91/1.92 A__ISNATKIND(s(V1:S)) -> A__U41(a__isNatKind(V1:S)) 1.91/1.92 A__ISNATKIND(s(V1:S)) -> A__ISNATKIND(V1:S) 1.91/1.92 A__PLUS(N:S,0) -> A__U51(a__isNat(N:S),N:S) 1.91/1.92 A__PLUS(N:S,0) -> A__ISNAT(N:S) 1.91/1.92 A__PLUS(N:S,s(M:S)) -> A__U61(a__isNat(M:S),M:S,N:S) 1.91/1.92 A__PLUS(N:S,s(M:S)) -> A__ISNAT(M:S) 1.91/1.92 MARK(U11(X1:S,X2:S,X3:S)) -> A__U11(mark(X1:S),X2:S,X3:S) 1.91/1.92 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.92 MARK(U12(X1:S,X2:S,X3:S)) -> A__U12(mark(X1:S),X2:S,X3:S) 1.91/1.92 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.92 MARK(U13(X1:S,X2:S,X3:S)) -> A__U13(mark(X1:S),X2:S,X3:S) 1.91/1.92 MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.92 MARK(U14(X1:S,X2:S,X3:S)) -> A__U14(mark(X1:S),X2:S,X3:S) 1.91/1.92 MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.92 MARK(U15(X1:S,X2:S)) -> A__U15(mark(X1:S),X2:S) 1.91/1.92 MARK(U15(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.92 MARK(U16(X:S)) -> A__U16(mark(X:S)) 1.91/1.92 MARK(U16(X:S)) -> MARK(X:S) 1.91/1.92 MARK(U21(X1:S,X2:S)) -> A__U21(mark(X1:S),X2:S) 1.91/1.92 MARK(U21(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.92 MARK(U22(X1:S,X2:S)) -> A__U22(mark(X1:S),X2:S) 1.91/1.92 MARK(U22(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.92 MARK(U23(X:S)) -> A__U23(mark(X:S)) 1.91/1.92 MARK(U23(X:S)) -> MARK(X:S) 1.91/1.92 MARK(U31(X1:S,X2:S)) -> A__U31(mark(X1:S),X2:S) 1.91/1.92 MARK(U31(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.92 MARK(U32(X:S)) -> A__U32(mark(X:S)) 1.91/1.92 MARK(U32(X:S)) -> MARK(X:S) 1.91/1.92 MARK(U41(X:S)) -> A__U41(mark(X:S)) 1.91/1.92 MARK(U41(X:S)) -> MARK(X:S) 1.91/1.92 MARK(U51(X1:S,X2:S)) -> A__U51(mark(X1:S),X2:S) 1.91/1.92 MARK(U51(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.92 MARK(U52(X1:S,X2:S)) -> A__U52(mark(X1:S),X2:S) 1.91/1.92 MARK(U52(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.92 MARK(U61(X1:S,X2:S,X3:S)) -> A__U61(mark(X1:S),X2:S,X3:S) 1.91/1.92 MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.92 MARK(U62(X1:S,X2:S,X3:S)) -> A__U62(mark(X1:S),X2:S,X3:S) 1.91/1.92 MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.92 MARK(U63(X1:S,X2:S,X3:S)) -> A__U63(mark(X1:S),X2:S,X3:S) 1.91/1.92 MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.92 MARK(U64(X1:S,X2:S,X3:S)) -> A__U64(mark(X1:S),X2:S,X3:S) 1.91/1.92 MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.92 MARK(isNat(X:S)) -> A__ISNAT(X:S) 1.91/1.92 MARK(isNatKind(X:S)) -> A__ISNATKIND(X:S) 1.91/1.92 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 1.91/1.92 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.92 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.91/1.92 MARK(s(X:S)) -> MARK(X:S) 1.91/1.92 -> Rules: 1.91/1.92 a__U11(tt,V1:S,V2:S) -> a__U12(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.92 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 1.91/1.92 a__U12(tt,V1:S,V2:S) -> a__U13(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.92 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 1.91/1.92 a__U13(tt,V1:S,V2:S) -> a__U14(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.92 a__U13(X1:S,X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) 1.91/1.92 a__U14(tt,V1:S,V2:S) -> a__U15(a__isNat(V1:S),V2:S) 1.91/1.92 a__U14(X1:S,X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) 1.91/1.92 a__U15(tt,V2:S) -> a__U16(a__isNat(V2:S)) 1.91/1.92 a__U15(X1:S,X2:S) -> U15(X1:S,X2:S) 1.91/1.92 a__U16(tt) -> tt 1.91/1.92 a__U16(X:S) -> U16(X:S) 1.91/1.92 a__U21(tt,V1:S) -> a__U22(a__isNatKind(V1:S),V1:S) 1.91/1.92 a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) 1.91/1.92 a__U22(tt,V1:S) -> a__U23(a__isNat(V1:S)) 1.91/1.92 a__U22(X1:S,X2:S) -> U22(X1:S,X2:S) 1.91/1.92 a__U23(tt) -> tt 1.91/1.92 a__U23(X:S) -> U23(X:S) 1.91/1.92 a__U31(tt,V2:S) -> a__U32(a__isNatKind(V2:S)) 1.91/1.92 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 1.91/1.92 a__U32(tt) -> tt 1.91/1.92 a__U32(X:S) -> U32(X:S) 1.91/1.92 a__U41(tt) -> tt 1.91/1.92 a__U41(X:S) -> U41(X:S) 1.91/1.92 a__U51(tt,N:S) -> a__U52(a__isNatKind(N:S),N:S) 1.91/1.92 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.91/1.92 a__U52(tt,N:S) -> mark(N:S) 1.91/1.92 a__U52(X1:S,X2:S) -> U52(X1:S,X2:S) 1.91/1.92 a__U61(tt,M:S,N:S) -> a__U62(a__isNatKind(M:S),M:S,N:S) 1.91/1.92 a__U61(X1:S,X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) 1.91/1.92 a__U62(tt,M:S,N:S) -> a__U63(a__isNat(N:S),M:S,N:S) 1.91/1.92 a__U62(X1:S,X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) 1.91/1.92 a__U63(tt,M:S,N:S) -> a__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.92 a__U63(X1:S,X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) 1.91/1.92 a__U64(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 1.91/1.92 a__U64(X1:S,X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) 1.91/1.92 a__isNat(0) -> tt 1.91/1.92 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.92 a__isNat(s(V1:S)) -> a__U21(a__isNatKind(V1:S),V1:S) 1.91/1.92 a__isNat(X:S) -> isNat(X:S) 1.91/1.92 a__isNatKind(0) -> tt 1.91/1.92 a__isNatKind(plus(V1:S,V2:S)) -> a__U31(a__isNatKind(V1:S),V2:S) 1.91/1.92 a__isNatKind(s(V1:S)) -> a__U41(a__isNatKind(V1:S)) 1.91/1.92 a__isNatKind(X:S) -> isNatKind(X:S) 1.91/1.92 a__plus(N:S,0) -> a__U51(a__isNat(N:S),N:S) 1.91/1.92 a__plus(N:S,s(M:S)) -> a__U61(a__isNat(M:S),M:S,N:S) 1.91/1.92 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 1.91/1.92 mark(0) -> 0 1.91/1.92 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U13(X1:S,X2:S,X3:S)) -> a__U13(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U14(X1:S,X2:S,X3:S)) -> a__U14(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U15(X1:S,X2:S)) -> a__U15(mark(X1:S),X2:S) 1.91/1.92 mark(U16(X:S)) -> a__U16(mark(X:S)) 1.91/1.92 mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) 1.91/1.92 mark(U22(X1:S,X2:S)) -> a__U22(mark(X1:S),X2:S) 1.91/1.92 mark(U23(X:S)) -> a__U23(mark(X:S)) 1.91/1.92 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 1.91/1.92 mark(U32(X:S)) -> a__U32(mark(X:S)) 1.91/1.92 mark(U41(X:S)) -> a__U41(mark(X:S)) 1.91/1.92 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.91/1.92 mark(U52(X1:S,X2:S)) -> a__U52(mark(X1:S),X2:S) 1.91/1.92 mark(U61(X1:S,X2:S,X3:S)) -> a__U61(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U62(X1:S,X2:S,X3:S)) -> a__U62(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U63(X1:S,X2:S,X3:S)) -> a__U63(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U64(X1:S,X2:S,X3:S)) -> a__U64(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(isNat(X:S)) -> a__isNat(X:S) 1.91/1.92 mark(isNatKind(X:S)) -> a__isNatKind(X:S) 1.91/1.92 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 1.91/1.92 mark(s(X:S)) -> s(mark(X:S)) 1.91/1.92 mark(tt) -> tt 1.91/1.92 1.91/1.92 Problem 1: 1.91/1.92 1.91/1.92 SCC Processor: 1.91/1.92 -> Pairs: 1.91/1.92 A__U11(tt,V1:S,V2:S) -> A__U12(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.92 A__U11(tt,V1:S,V2:S) -> A__ISNATKIND(V1:S) 1.91/1.92 A__U12(tt,V1:S,V2:S) -> A__U13(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.92 A__U12(tt,V1:S,V2:S) -> A__ISNATKIND(V2:S) 1.91/1.92 A__U13(tt,V1:S,V2:S) -> A__U14(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.92 A__U13(tt,V1:S,V2:S) -> A__ISNATKIND(V2:S) 1.91/1.92 A__U14(tt,V1:S,V2:S) -> A__U15(a__isNat(V1:S),V2:S) 1.91/1.92 A__U14(tt,V1:S,V2:S) -> A__ISNAT(V1:S) 1.91/1.92 A__U15(tt,V2:S) -> A__U16(a__isNat(V2:S)) 1.91/1.92 A__U15(tt,V2:S) -> A__ISNAT(V2:S) 1.91/1.92 A__U21(tt,V1:S) -> A__U22(a__isNatKind(V1:S),V1:S) 1.91/1.92 A__U21(tt,V1:S) -> A__ISNATKIND(V1:S) 1.91/1.92 A__U22(tt,V1:S) -> A__U23(a__isNat(V1:S)) 1.91/1.92 A__U22(tt,V1:S) -> A__ISNAT(V1:S) 1.91/1.92 A__U31(tt,V2:S) -> A__U32(a__isNatKind(V2:S)) 1.91/1.92 A__U31(tt,V2:S) -> A__ISNATKIND(V2:S) 1.91/1.92 A__U51(tt,N:S) -> A__U52(a__isNatKind(N:S),N:S) 1.91/1.92 A__U51(tt,N:S) -> A__ISNATKIND(N:S) 1.91/1.92 A__U52(tt,N:S) -> MARK(N:S) 1.91/1.92 A__U61(tt,M:S,N:S) -> A__U62(a__isNatKind(M:S),M:S,N:S) 1.91/1.92 A__U61(tt,M:S,N:S) -> A__ISNATKIND(M:S) 1.91/1.92 A__U62(tt,M:S,N:S) -> A__U63(a__isNat(N:S),M:S,N:S) 1.91/1.92 A__U62(tt,M:S,N:S) -> A__ISNAT(N:S) 1.91/1.92 A__U63(tt,M:S,N:S) -> A__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.92 A__U63(tt,M:S,N:S) -> A__ISNATKIND(N:S) 1.91/1.92 A__U64(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S)) 1.91/1.92 A__U64(tt,M:S,N:S) -> MARK(M:S) 1.91/1.92 A__U64(tt,M:S,N:S) -> MARK(N:S) 1.91/1.92 A__ISNAT(plus(V1:S,V2:S)) -> A__U11(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.92 A__ISNAT(plus(V1:S,V2:S)) -> A__ISNATKIND(V1:S) 1.91/1.92 A__ISNAT(s(V1:S)) -> A__U21(a__isNatKind(V1:S),V1:S) 1.91/1.92 A__ISNAT(s(V1:S)) -> A__ISNATKIND(V1:S) 1.91/1.92 A__ISNATKIND(plus(V1:S,V2:S)) -> A__U31(a__isNatKind(V1:S),V2:S) 1.91/1.92 A__ISNATKIND(plus(V1:S,V2:S)) -> A__ISNATKIND(V1:S) 1.91/1.92 A__ISNATKIND(s(V1:S)) -> A__U41(a__isNatKind(V1:S)) 1.91/1.92 A__ISNATKIND(s(V1:S)) -> A__ISNATKIND(V1:S) 1.91/1.92 A__PLUS(N:S,0) -> A__U51(a__isNat(N:S),N:S) 1.91/1.92 A__PLUS(N:S,0) -> A__ISNAT(N:S) 1.91/1.92 A__PLUS(N:S,s(M:S)) -> A__U61(a__isNat(M:S),M:S,N:S) 1.91/1.92 A__PLUS(N:S,s(M:S)) -> A__ISNAT(M:S) 1.91/1.92 MARK(U11(X1:S,X2:S,X3:S)) -> A__U11(mark(X1:S),X2:S,X3:S) 1.91/1.92 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.92 MARK(U12(X1:S,X2:S,X3:S)) -> A__U12(mark(X1:S),X2:S,X3:S) 1.91/1.92 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.92 MARK(U13(X1:S,X2:S,X3:S)) -> A__U13(mark(X1:S),X2:S,X3:S) 1.91/1.92 MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.92 MARK(U14(X1:S,X2:S,X3:S)) -> A__U14(mark(X1:S),X2:S,X3:S) 1.91/1.92 MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.92 MARK(U15(X1:S,X2:S)) -> A__U15(mark(X1:S),X2:S) 1.91/1.92 MARK(U15(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.92 MARK(U16(X:S)) -> A__U16(mark(X:S)) 1.91/1.92 MARK(U16(X:S)) -> MARK(X:S) 1.91/1.92 MARK(U21(X1:S,X2:S)) -> A__U21(mark(X1:S),X2:S) 1.91/1.92 MARK(U21(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.92 MARK(U22(X1:S,X2:S)) -> A__U22(mark(X1:S),X2:S) 1.91/1.92 MARK(U22(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.92 MARK(U23(X:S)) -> A__U23(mark(X:S)) 1.91/1.92 MARK(U23(X:S)) -> MARK(X:S) 1.91/1.92 MARK(U31(X1:S,X2:S)) -> A__U31(mark(X1:S),X2:S) 1.91/1.92 MARK(U31(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.92 MARK(U32(X:S)) -> A__U32(mark(X:S)) 1.91/1.92 MARK(U32(X:S)) -> MARK(X:S) 1.91/1.92 MARK(U41(X:S)) -> A__U41(mark(X:S)) 1.91/1.92 MARK(U41(X:S)) -> MARK(X:S) 1.91/1.92 MARK(U51(X1:S,X2:S)) -> A__U51(mark(X1:S),X2:S) 1.91/1.92 MARK(U51(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.92 MARK(U52(X1:S,X2:S)) -> A__U52(mark(X1:S),X2:S) 1.91/1.92 MARK(U52(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.92 MARK(U61(X1:S,X2:S,X3:S)) -> A__U61(mark(X1:S),X2:S,X3:S) 1.91/1.92 MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.92 MARK(U62(X1:S,X2:S,X3:S)) -> A__U62(mark(X1:S),X2:S,X3:S) 1.91/1.92 MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.92 MARK(U63(X1:S,X2:S,X3:S)) -> A__U63(mark(X1:S),X2:S,X3:S) 1.91/1.92 MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.92 MARK(U64(X1:S,X2:S,X3:S)) -> A__U64(mark(X1:S),X2:S,X3:S) 1.91/1.92 MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.92 MARK(isNat(X:S)) -> A__ISNAT(X:S) 1.91/1.92 MARK(isNatKind(X:S)) -> A__ISNATKIND(X:S) 1.91/1.92 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 1.91/1.92 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.92 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.91/1.92 MARK(s(X:S)) -> MARK(X:S) 1.91/1.92 -> Rules: 1.91/1.92 a__U11(tt,V1:S,V2:S) -> a__U12(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.92 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 1.91/1.92 a__U12(tt,V1:S,V2:S) -> a__U13(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.92 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 1.91/1.92 a__U13(tt,V1:S,V2:S) -> a__U14(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.92 a__U13(X1:S,X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) 1.91/1.92 a__U14(tt,V1:S,V2:S) -> a__U15(a__isNat(V1:S),V2:S) 1.91/1.92 a__U14(X1:S,X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) 1.91/1.92 a__U15(tt,V2:S) -> a__U16(a__isNat(V2:S)) 1.91/1.92 a__U15(X1:S,X2:S) -> U15(X1:S,X2:S) 1.91/1.92 a__U16(tt) -> tt 1.91/1.92 a__U16(X:S) -> U16(X:S) 1.91/1.92 a__U21(tt,V1:S) -> a__U22(a__isNatKind(V1:S),V1:S) 1.91/1.92 a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) 1.91/1.92 a__U22(tt,V1:S) -> a__U23(a__isNat(V1:S)) 1.91/1.92 a__U22(X1:S,X2:S) -> U22(X1:S,X2:S) 1.91/1.92 a__U23(tt) -> tt 1.91/1.92 a__U23(X:S) -> U23(X:S) 1.91/1.92 a__U31(tt,V2:S) -> a__U32(a__isNatKind(V2:S)) 1.91/1.92 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 1.91/1.92 a__U32(tt) -> tt 1.91/1.92 a__U32(X:S) -> U32(X:S) 1.91/1.92 a__U41(tt) -> tt 1.91/1.92 a__U41(X:S) -> U41(X:S) 1.91/1.92 a__U51(tt,N:S) -> a__U52(a__isNatKind(N:S),N:S) 1.91/1.92 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.91/1.92 a__U52(tt,N:S) -> mark(N:S) 1.91/1.92 a__U52(X1:S,X2:S) -> U52(X1:S,X2:S) 1.91/1.92 a__U61(tt,M:S,N:S) -> a__U62(a__isNatKind(M:S),M:S,N:S) 1.91/1.92 a__U61(X1:S,X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) 1.91/1.92 a__U62(tt,M:S,N:S) -> a__U63(a__isNat(N:S),M:S,N:S) 1.91/1.92 a__U62(X1:S,X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) 1.91/1.92 a__U63(tt,M:S,N:S) -> a__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.92 a__U63(X1:S,X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) 1.91/1.92 a__U64(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 1.91/1.92 a__U64(X1:S,X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) 1.91/1.92 a__isNat(0) -> tt 1.91/1.92 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.92 a__isNat(s(V1:S)) -> a__U21(a__isNatKind(V1:S),V1:S) 1.91/1.92 a__isNat(X:S) -> isNat(X:S) 1.91/1.92 a__isNatKind(0) -> tt 1.91/1.92 a__isNatKind(plus(V1:S,V2:S)) -> a__U31(a__isNatKind(V1:S),V2:S) 1.91/1.92 a__isNatKind(s(V1:S)) -> a__U41(a__isNatKind(V1:S)) 1.91/1.92 a__isNatKind(X:S) -> isNatKind(X:S) 1.91/1.92 a__plus(N:S,0) -> a__U51(a__isNat(N:S),N:S) 1.91/1.92 a__plus(N:S,s(M:S)) -> a__U61(a__isNat(M:S),M:S,N:S) 1.91/1.92 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 1.91/1.92 mark(0) -> 0 1.91/1.92 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U13(X1:S,X2:S,X3:S)) -> a__U13(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U14(X1:S,X2:S,X3:S)) -> a__U14(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U15(X1:S,X2:S)) -> a__U15(mark(X1:S),X2:S) 1.91/1.92 mark(U16(X:S)) -> a__U16(mark(X:S)) 1.91/1.92 mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) 1.91/1.92 mark(U22(X1:S,X2:S)) -> a__U22(mark(X1:S),X2:S) 1.91/1.92 mark(U23(X:S)) -> a__U23(mark(X:S)) 1.91/1.92 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 1.91/1.92 mark(U32(X:S)) -> a__U32(mark(X:S)) 1.91/1.92 mark(U41(X:S)) -> a__U41(mark(X:S)) 1.91/1.92 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.91/1.92 mark(U52(X1:S,X2:S)) -> a__U52(mark(X1:S),X2:S) 1.91/1.92 mark(U61(X1:S,X2:S,X3:S)) -> a__U61(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U62(X1:S,X2:S,X3:S)) -> a__U62(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U63(X1:S,X2:S,X3:S)) -> a__U63(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U64(X1:S,X2:S,X3:S)) -> a__U64(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(isNat(X:S)) -> a__isNat(X:S) 1.91/1.92 mark(isNatKind(X:S)) -> a__isNatKind(X:S) 1.91/1.92 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 1.91/1.92 mark(s(X:S)) -> s(mark(X:S)) 1.91/1.92 mark(tt) -> tt 1.91/1.92 ->Strongly Connected Components: 1.91/1.92 ->->Cycle: 1.91/1.92 ->->-> Pairs: 1.91/1.92 A__U31(tt,V2:S) -> A__ISNATKIND(V2:S) 1.91/1.92 A__ISNATKIND(plus(V1:S,V2:S)) -> A__U31(a__isNatKind(V1:S),V2:S) 1.91/1.92 A__ISNATKIND(plus(V1:S,V2:S)) -> A__ISNATKIND(V1:S) 1.91/1.92 A__ISNATKIND(s(V1:S)) -> A__ISNATKIND(V1:S) 1.91/1.92 ->->-> Rules: 1.91/1.92 a__U11(tt,V1:S,V2:S) -> a__U12(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.92 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 1.91/1.92 a__U12(tt,V1:S,V2:S) -> a__U13(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.92 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 1.91/1.92 a__U13(tt,V1:S,V2:S) -> a__U14(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.92 a__U13(X1:S,X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) 1.91/1.92 a__U14(tt,V1:S,V2:S) -> a__U15(a__isNat(V1:S),V2:S) 1.91/1.92 a__U14(X1:S,X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) 1.91/1.92 a__U15(tt,V2:S) -> a__U16(a__isNat(V2:S)) 1.91/1.92 a__U15(X1:S,X2:S) -> U15(X1:S,X2:S) 1.91/1.92 a__U16(tt) -> tt 1.91/1.92 a__U16(X:S) -> U16(X:S) 1.91/1.92 a__U21(tt,V1:S) -> a__U22(a__isNatKind(V1:S),V1:S) 1.91/1.92 a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) 1.91/1.92 a__U22(tt,V1:S) -> a__U23(a__isNat(V1:S)) 1.91/1.92 a__U22(X1:S,X2:S) -> U22(X1:S,X2:S) 1.91/1.92 a__U23(tt) -> tt 1.91/1.92 a__U23(X:S) -> U23(X:S) 1.91/1.92 a__U31(tt,V2:S) -> a__U32(a__isNatKind(V2:S)) 1.91/1.92 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 1.91/1.92 a__U32(tt) -> tt 1.91/1.92 a__U32(X:S) -> U32(X:S) 1.91/1.92 a__U41(tt) -> tt 1.91/1.92 a__U41(X:S) -> U41(X:S) 1.91/1.92 a__U51(tt,N:S) -> a__U52(a__isNatKind(N:S),N:S) 1.91/1.92 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.91/1.92 a__U52(tt,N:S) -> mark(N:S) 1.91/1.92 a__U52(X1:S,X2:S) -> U52(X1:S,X2:S) 1.91/1.92 a__U61(tt,M:S,N:S) -> a__U62(a__isNatKind(M:S),M:S,N:S) 1.91/1.92 a__U61(X1:S,X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) 1.91/1.92 a__U62(tt,M:S,N:S) -> a__U63(a__isNat(N:S),M:S,N:S) 1.91/1.92 a__U62(X1:S,X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) 1.91/1.92 a__U63(tt,M:S,N:S) -> a__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.92 a__U63(X1:S,X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) 1.91/1.92 a__U64(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 1.91/1.92 a__U64(X1:S,X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) 1.91/1.92 a__isNat(0) -> tt 1.91/1.92 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.92 a__isNat(s(V1:S)) -> a__U21(a__isNatKind(V1:S),V1:S) 1.91/1.92 a__isNat(X:S) -> isNat(X:S) 1.91/1.92 a__isNatKind(0) -> tt 1.91/1.92 a__isNatKind(plus(V1:S,V2:S)) -> a__U31(a__isNatKind(V1:S),V2:S) 1.91/1.92 a__isNatKind(s(V1:S)) -> a__U41(a__isNatKind(V1:S)) 1.91/1.92 a__isNatKind(X:S) -> isNatKind(X:S) 1.91/1.92 a__plus(N:S,0) -> a__U51(a__isNat(N:S),N:S) 1.91/1.92 a__plus(N:S,s(M:S)) -> a__U61(a__isNat(M:S),M:S,N:S) 1.91/1.92 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 1.91/1.92 mark(0) -> 0 1.91/1.92 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U13(X1:S,X2:S,X3:S)) -> a__U13(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U14(X1:S,X2:S,X3:S)) -> a__U14(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U15(X1:S,X2:S)) -> a__U15(mark(X1:S),X2:S) 1.91/1.92 mark(U16(X:S)) -> a__U16(mark(X:S)) 1.91/1.92 mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) 1.91/1.92 mark(U22(X1:S,X2:S)) -> a__U22(mark(X1:S),X2:S) 1.91/1.92 mark(U23(X:S)) -> a__U23(mark(X:S)) 1.91/1.92 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 1.91/1.92 mark(U32(X:S)) -> a__U32(mark(X:S)) 1.91/1.92 mark(U41(X:S)) -> a__U41(mark(X:S)) 1.91/1.92 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.91/1.92 mark(U52(X1:S,X2:S)) -> a__U52(mark(X1:S),X2:S) 1.91/1.92 mark(U61(X1:S,X2:S,X3:S)) -> a__U61(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U62(X1:S,X2:S,X3:S)) -> a__U62(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U63(X1:S,X2:S,X3:S)) -> a__U63(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U64(X1:S,X2:S,X3:S)) -> a__U64(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(isNat(X:S)) -> a__isNat(X:S) 1.91/1.92 mark(isNatKind(X:S)) -> a__isNatKind(X:S) 1.91/1.92 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 1.91/1.92 mark(s(X:S)) -> s(mark(X:S)) 1.91/1.92 mark(tt) -> tt 1.91/1.92 ->->Cycle: 1.91/1.92 ->->-> Pairs: 1.91/1.92 A__U11(tt,V1:S,V2:S) -> A__U12(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.92 A__U12(tt,V1:S,V2:S) -> A__U13(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.92 A__U13(tt,V1:S,V2:S) -> A__U14(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.92 A__U14(tt,V1:S,V2:S) -> A__U15(a__isNat(V1:S),V2:S) 1.91/1.92 A__U14(tt,V1:S,V2:S) -> A__ISNAT(V1:S) 1.91/1.92 A__U15(tt,V2:S) -> A__ISNAT(V2:S) 1.91/1.92 A__U21(tt,V1:S) -> A__U22(a__isNatKind(V1:S),V1:S) 1.91/1.92 A__U22(tt,V1:S) -> A__ISNAT(V1:S) 1.91/1.92 A__ISNAT(plus(V1:S,V2:S)) -> A__U11(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.92 A__ISNAT(s(V1:S)) -> A__U21(a__isNatKind(V1:S),V1:S) 1.91/1.92 ->->-> Rules: 1.91/1.92 a__U11(tt,V1:S,V2:S) -> a__U12(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.92 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 1.91/1.92 a__U12(tt,V1:S,V2:S) -> a__U13(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.92 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 1.91/1.92 a__U13(tt,V1:S,V2:S) -> a__U14(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.92 a__U13(X1:S,X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) 1.91/1.92 a__U14(tt,V1:S,V2:S) -> a__U15(a__isNat(V1:S),V2:S) 1.91/1.92 a__U14(X1:S,X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) 1.91/1.92 a__U15(tt,V2:S) -> a__U16(a__isNat(V2:S)) 1.91/1.92 a__U15(X1:S,X2:S) -> U15(X1:S,X2:S) 1.91/1.92 a__U16(tt) -> tt 1.91/1.92 a__U16(X:S) -> U16(X:S) 1.91/1.92 a__U21(tt,V1:S) -> a__U22(a__isNatKind(V1:S),V1:S) 1.91/1.92 a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) 1.91/1.92 a__U22(tt,V1:S) -> a__U23(a__isNat(V1:S)) 1.91/1.92 a__U22(X1:S,X2:S) -> U22(X1:S,X2:S) 1.91/1.92 a__U23(tt) -> tt 1.91/1.92 a__U23(X:S) -> U23(X:S) 1.91/1.92 a__U31(tt,V2:S) -> a__U32(a__isNatKind(V2:S)) 1.91/1.92 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 1.91/1.92 a__U32(tt) -> tt 1.91/1.92 a__U32(X:S) -> U32(X:S) 1.91/1.92 a__U41(tt) -> tt 1.91/1.92 a__U41(X:S) -> U41(X:S) 1.91/1.92 a__U51(tt,N:S) -> a__U52(a__isNatKind(N:S),N:S) 1.91/1.92 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.91/1.92 a__U52(tt,N:S) -> mark(N:S) 1.91/1.92 a__U52(X1:S,X2:S) -> U52(X1:S,X2:S) 1.91/1.92 a__U61(tt,M:S,N:S) -> a__U62(a__isNatKind(M:S),M:S,N:S) 1.91/1.92 a__U61(X1:S,X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) 1.91/1.92 a__U62(tt,M:S,N:S) -> a__U63(a__isNat(N:S),M:S,N:S) 1.91/1.92 a__U62(X1:S,X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) 1.91/1.92 a__U63(tt,M:S,N:S) -> a__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.92 a__U63(X1:S,X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) 1.91/1.92 a__U64(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 1.91/1.92 a__U64(X1:S,X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) 1.91/1.92 a__isNat(0) -> tt 1.91/1.92 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.92 a__isNat(s(V1:S)) -> a__U21(a__isNatKind(V1:S),V1:S) 1.91/1.92 a__isNat(X:S) -> isNat(X:S) 1.91/1.92 a__isNatKind(0) -> tt 1.91/1.92 a__isNatKind(plus(V1:S,V2:S)) -> a__U31(a__isNatKind(V1:S),V2:S) 1.91/1.92 a__isNatKind(s(V1:S)) -> a__U41(a__isNatKind(V1:S)) 1.91/1.92 a__isNatKind(X:S) -> isNatKind(X:S) 1.91/1.92 a__plus(N:S,0) -> a__U51(a__isNat(N:S),N:S) 1.91/1.92 a__plus(N:S,s(M:S)) -> a__U61(a__isNat(M:S),M:S,N:S) 1.91/1.92 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 1.91/1.92 mark(0) -> 0 1.91/1.92 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U13(X1:S,X2:S,X3:S)) -> a__U13(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U14(X1:S,X2:S,X3:S)) -> a__U14(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U15(X1:S,X2:S)) -> a__U15(mark(X1:S),X2:S) 1.91/1.92 mark(U16(X:S)) -> a__U16(mark(X:S)) 1.91/1.92 mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) 1.91/1.92 mark(U22(X1:S,X2:S)) -> a__U22(mark(X1:S),X2:S) 1.91/1.92 mark(U23(X:S)) -> a__U23(mark(X:S)) 1.91/1.92 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 1.91/1.92 mark(U32(X:S)) -> a__U32(mark(X:S)) 1.91/1.92 mark(U41(X:S)) -> a__U41(mark(X:S)) 1.91/1.92 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.91/1.92 mark(U52(X1:S,X2:S)) -> a__U52(mark(X1:S),X2:S) 1.91/1.92 mark(U61(X1:S,X2:S,X3:S)) -> a__U61(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U62(X1:S,X2:S,X3:S)) -> a__U62(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U63(X1:S,X2:S,X3:S)) -> a__U63(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U64(X1:S,X2:S,X3:S)) -> a__U64(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(isNat(X:S)) -> a__isNat(X:S) 1.91/1.92 mark(isNatKind(X:S)) -> a__isNatKind(X:S) 1.91/1.92 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 1.91/1.92 mark(s(X:S)) -> s(mark(X:S)) 1.91/1.92 mark(tt) -> tt 1.91/1.92 ->->Cycle: 1.91/1.92 ->->-> Pairs: 1.91/1.92 A__U51(tt,N:S) -> A__U52(a__isNatKind(N:S),N:S) 1.91/1.92 A__U52(tt,N:S) -> MARK(N:S) 1.91/1.92 A__U61(tt,M:S,N:S) -> A__U62(a__isNatKind(M:S),M:S,N:S) 1.91/1.92 A__U62(tt,M:S,N:S) -> A__U63(a__isNat(N:S),M:S,N:S) 1.91/1.92 A__U63(tt,M:S,N:S) -> A__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.92 A__U64(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S)) 1.91/1.92 A__U64(tt,M:S,N:S) -> MARK(M:S) 1.91/1.92 A__U64(tt,M:S,N:S) -> MARK(N:S) 1.91/1.92 A__PLUS(N:S,0) -> A__U51(a__isNat(N:S),N:S) 1.91/1.92 A__PLUS(N:S,s(M:S)) -> A__U61(a__isNat(M:S),M:S,N:S) 1.91/1.92 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.92 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.92 MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.92 MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.92 MARK(U15(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.92 MARK(U16(X:S)) -> MARK(X:S) 1.91/1.92 MARK(U21(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.92 MARK(U22(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.92 MARK(U23(X:S)) -> MARK(X:S) 1.91/1.92 MARK(U31(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.92 MARK(U32(X:S)) -> MARK(X:S) 1.91/1.92 MARK(U41(X:S)) -> MARK(X:S) 1.91/1.92 MARK(U51(X1:S,X2:S)) -> A__U51(mark(X1:S),X2:S) 1.91/1.92 MARK(U51(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.92 MARK(U52(X1:S,X2:S)) -> A__U52(mark(X1:S),X2:S) 1.91/1.92 MARK(U52(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.92 MARK(U61(X1:S,X2:S,X3:S)) -> A__U61(mark(X1:S),X2:S,X3:S) 1.91/1.92 MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.92 MARK(U62(X1:S,X2:S,X3:S)) -> A__U62(mark(X1:S),X2:S,X3:S) 1.91/1.92 MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.92 MARK(U63(X1:S,X2:S,X3:S)) -> A__U63(mark(X1:S),X2:S,X3:S) 1.91/1.92 MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.92 MARK(U64(X1:S,X2:S,X3:S)) -> A__U64(mark(X1:S),X2:S,X3:S) 1.91/1.92 MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.92 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 1.91/1.92 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.92 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.91/1.92 MARK(s(X:S)) -> MARK(X:S) 1.91/1.92 ->->-> Rules: 1.91/1.92 a__U11(tt,V1:S,V2:S) -> a__U12(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.92 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 1.91/1.92 a__U12(tt,V1:S,V2:S) -> a__U13(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.92 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 1.91/1.92 a__U13(tt,V1:S,V2:S) -> a__U14(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.92 a__U13(X1:S,X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) 1.91/1.92 a__U14(tt,V1:S,V2:S) -> a__U15(a__isNat(V1:S),V2:S) 1.91/1.92 a__U14(X1:S,X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) 1.91/1.92 a__U15(tt,V2:S) -> a__U16(a__isNat(V2:S)) 1.91/1.92 a__U15(X1:S,X2:S) -> U15(X1:S,X2:S) 1.91/1.92 a__U16(tt) -> tt 1.91/1.92 a__U16(X:S) -> U16(X:S) 1.91/1.92 a__U21(tt,V1:S) -> a__U22(a__isNatKind(V1:S),V1:S) 1.91/1.92 a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) 1.91/1.92 a__U22(tt,V1:S) -> a__U23(a__isNat(V1:S)) 1.91/1.92 a__U22(X1:S,X2:S) -> U22(X1:S,X2:S) 1.91/1.92 a__U23(tt) -> tt 1.91/1.92 a__U23(X:S) -> U23(X:S) 1.91/1.92 a__U31(tt,V2:S) -> a__U32(a__isNatKind(V2:S)) 1.91/1.92 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 1.91/1.92 a__U32(tt) -> tt 1.91/1.92 a__U32(X:S) -> U32(X:S) 1.91/1.92 a__U41(tt) -> tt 1.91/1.92 a__U41(X:S) -> U41(X:S) 1.91/1.92 a__U51(tt,N:S) -> a__U52(a__isNatKind(N:S),N:S) 1.91/1.92 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.91/1.92 a__U52(tt,N:S) -> mark(N:S) 1.91/1.92 a__U52(X1:S,X2:S) -> U52(X1:S,X2:S) 1.91/1.92 a__U61(tt,M:S,N:S) -> a__U62(a__isNatKind(M:S),M:S,N:S) 1.91/1.92 a__U61(X1:S,X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) 1.91/1.92 a__U62(tt,M:S,N:S) -> a__U63(a__isNat(N:S),M:S,N:S) 1.91/1.92 a__U62(X1:S,X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) 1.91/1.92 a__U63(tt,M:S,N:S) -> a__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.92 a__U63(X1:S,X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) 1.91/1.92 a__U64(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 1.91/1.92 a__U64(X1:S,X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) 1.91/1.92 a__isNat(0) -> tt 1.91/1.92 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.92 a__isNat(s(V1:S)) -> a__U21(a__isNatKind(V1:S),V1:S) 1.91/1.92 a__isNat(X:S) -> isNat(X:S) 1.91/1.92 a__isNatKind(0) -> tt 1.91/1.92 a__isNatKind(plus(V1:S,V2:S)) -> a__U31(a__isNatKind(V1:S),V2:S) 1.91/1.92 a__isNatKind(s(V1:S)) -> a__U41(a__isNatKind(V1:S)) 1.91/1.92 a__isNatKind(X:S) -> isNatKind(X:S) 1.91/1.92 a__plus(N:S,0) -> a__U51(a__isNat(N:S),N:S) 1.91/1.92 a__plus(N:S,s(M:S)) -> a__U61(a__isNat(M:S),M:S,N:S) 1.91/1.92 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 1.91/1.92 mark(0) -> 0 1.91/1.92 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U13(X1:S,X2:S,X3:S)) -> a__U13(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U14(X1:S,X2:S,X3:S)) -> a__U14(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U15(X1:S,X2:S)) -> a__U15(mark(X1:S),X2:S) 1.91/1.92 mark(U16(X:S)) -> a__U16(mark(X:S)) 1.91/1.92 mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) 1.91/1.92 mark(U22(X1:S,X2:S)) -> a__U22(mark(X1:S),X2:S) 1.91/1.92 mark(U23(X:S)) -> a__U23(mark(X:S)) 1.91/1.92 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 1.91/1.92 mark(U32(X:S)) -> a__U32(mark(X:S)) 1.91/1.92 mark(U41(X:S)) -> a__U41(mark(X:S)) 1.91/1.92 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.91/1.92 mark(U52(X1:S,X2:S)) -> a__U52(mark(X1:S),X2:S) 1.91/1.92 mark(U61(X1:S,X2:S,X3:S)) -> a__U61(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U62(X1:S,X2:S,X3:S)) -> a__U62(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U63(X1:S,X2:S,X3:S)) -> a__U63(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U64(X1:S,X2:S,X3:S)) -> a__U64(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(isNat(X:S)) -> a__isNat(X:S) 1.91/1.92 mark(isNatKind(X:S)) -> a__isNatKind(X:S) 1.91/1.92 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 1.91/1.92 mark(s(X:S)) -> s(mark(X:S)) 1.91/1.92 mark(tt) -> tt 1.91/1.92 1.91/1.92 1.91/1.92 The problem is decomposed in 3 subproblems. 1.91/1.92 1.91/1.92 Problem 1.1: 1.91/1.92 1.91/1.92 Subterm Processor: 1.91/1.92 -> Pairs: 1.91/1.92 A__U31(tt,V2:S) -> A__ISNATKIND(V2:S) 1.91/1.92 A__ISNATKIND(plus(V1:S,V2:S)) -> A__U31(a__isNatKind(V1:S),V2:S) 1.91/1.92 A__ISNATKIND(plus(V1:S,V2:S)) -> A__ISNATKIND(V1:S) 1.91/1.92 A__ISNATKIND(s(V1:S)) -> A__ISNATKIND(V1:S) 1.91/1.92 -> Rules: 1.91/1.92 a__U11(tt,V1:S,V2:S) -> a__U12(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.92 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 1.91/1.92 a__U12(tt,V1:S,V2:S) -> a__U13(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.92 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 1.91/1.92 a__U13(tt,V1:S,V2:S) -> a__U14(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.92 a__U13(X1:S,X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) 1.91/1.92 a__U14(tt,V1:S,V2:S) -> a__U15(a__isNat(V1:S),V2:S) 1.91/1.92 a__U14(X1:S,X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) 1.91/1.92 a__U15(tt,V2:S) -> a__U16(a__isNat(V2:S)) 1.91/1.92 a__U15(X1:S,X2:S) -> U15(X1:S,X2:S) 1.91/1.92 a__U16(tt) -> tt 1.91/1.92 a__U16(X:S) -> U16(X:S) 1.91/1.92 a__U21(tt,V1:S) -> a__U22(a__isNatKind(V1:S),V1:S) 1.91/1.92 a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) 1.91/1.92 a__U22(tt,V1:S) -> a__U23(a__isNat(V1:S)) 1.91/1.92 a__U22(X1:S,X2:S) -> U22(X1:S,X2:S) 1.91/1.92 a__U23(tt) -> tt 1.91/1.92 a__U23(X:S) -> U23(X:S) 1.91/1.92 a__U31(tt,V2:S) -> a__U32(a__isNatKind(V2:S)) 1.91/1.92 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 1.91/1.92 a__U32(tt) -> tt 1.91/1.92 a__U32(X:S) -> U32(X:S) 1.91/1.92 a__U41(tt) -> tt 1.91/1.92 a__U41(X:S) -> U41(X:S) 1.91/1.92 a__U51(tt,N:S) -> a__U52(a__isNatKind(N:S),N:S) 1.91/1.92 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.91/1.92 a__U52(tt,N:S) -> mark(N:S) 1.91/1.92 a__U52(X1:S,X2:S) -> U52(X1:S,X2:S) 1.91/1.92 a__U61(tt,M:S,N:S) -> a__U62(a__isNatKind(M:S),M:S,N:S) 1.91/1.92 a__U61(X1:S,X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) 1.91/1.92 a__U62(tt,M:S,N:S) -> a__U63(a__isNat(N:S),M:S,N:S) 1.91/1.92 a__U62(X1:S,X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) 1.91/1.92 a__U63(tt,M:S,N:S) -> a__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.92 a__U63(X1:S,X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) 1.91/1.92 a__U64(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 1.91/1.92 a__U64(X1:S,X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) 1.91/1.92 a__isNat(0) -> tt 1.91/1.92 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.92 a__isNat(s(V1:S)) -> a__U21(a__isNatKind(V1:S),V1:S) 1.91/1.92 a__isNat(X:S) -> isNat(X:S) 1.91/1.92 a__isNatKind(0) -> tt 1.91/1.92 a__isNatKind(plus(V1:S,V2:S)) -> a__U31(a__isNatKind(V1:S),V2:S) 1.91/1.92 a__isNatKind(s(V1:S)) -> a__U41(a__isNatKind(V1:S)) 1.91/1.92 a__isNatKind(X:S) -> isNatKind(X:S) 1.91/1.92 a__plus(N:S,0) -> a__U51(a__isNat(N:S),N:S) 1.91/1.92 a__plus(N:S,s(M:S)) -> a__U61(a__isNat(M:S),M:S,N:S) 1.91/1.92 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 1.91/1.92 mark(0) -> 0 1.91/1.92 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U13(X1:S,X2:S,X3:S)) -> a__U13(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U14(X1:S,X2:S,X3:S)) -> a__U14(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U15(X1:S,X2:S)) -> a__U15(mark(X1:S),X2:S) 1.91/1.92 mark(U16(X:S)) -> a__U16(mark(X:S)) 1.91/1.92 mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) 1.91/1.92 mark(U22(X1:S,X2:S)) -> a__U22(mark(X1:S),X2:S) 1.91/1.92 mark(U23(X:S)) -> a__U23(mark(X:S)) 1.91/1.92 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 1.91/1.92 mark(U32(X:S)) -> a__U32(mark(X:S)) 1.91/1.92 mark(U41(X:S)) -> a__U41(mark(X:S)) 1.91/1.92 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.91/1.92 mark(U52(X1:S,X2:S)) -> a__U52(mark(X1:S),X2:S) 1.91/1.92 mark(U61(X1:S,X2:S,X3:S)) -> a__U61(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U62(X1:S,X2:S,X3:S)) -> a__U62(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U63(X1:S,X2:S,X3:S)) -> a__U63(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U64(X1:S,X2:S,X3:S)) -> a__U64(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(isNat(X:S)) -> a__isNat(X:S) 1.91/1.92 mark(isNatKind(X:S)) -> a__isNatKind(X:S) 1.91/1.92 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 1.91/1.92 mark(s(X:S)) -> s(mark(X:S)) 1.91/1.92 mark(tt) -> tt 1.91/1.92 ->Projection: 1.91/1.92 pi(A__U31) = 2 1.91/1.92 pi(A__ISNATKIND) = 1 1.91/1.92 1.91/1.92 Problem 1.1: 1.91/1.92 1.91/1.92 SCC Processor: 1.91/1.92 -> Pairs: 1.91/1.92 A__U31(tt,V2:S) -> A__ISNATKIND(V2:S) 1.91/1.92 -> Rules: 1.91/1.92 a__U11(tt,V1:S,V2:S) -> a__U12(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.92 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 1.91/1.92 a__U12(tt,V1:S,V2:S) -> a__U13(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.92 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 1.91/1.92 a__U13(tt,V1:S,V2:S) -> a__U14(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.92 a__U13(X1:S,X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) 1.91/1.92 a__U14(tt,V1:S,V2:S) -> a__U15(a__isNat(V1:S),V2:S) 1.91/1.92 a__U14(X1:S,X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) 1.91/1.92 a__U15(tt,V2:S) -> a__U16(a__isNat(V2:S)) 1.91/1.92 a__U15(X1:S,X2:S) -> U15(X1:S,X2:S) 1.91/1.92 a__U16(tt) -> tt 1.91/1.92 a__U16(X:S) -> U16(X:S) 1.91/1.92 a__U21(tt,V1:S) -> a__U22(a__isNatKind(V1:S),V1:S) 1.91/1.92 a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) 1.91/1.92 a__U22(tt,V1:S) -> a__U23(a__isNat(V1:S)) 1.91/1.92 a__U22(X1:S,X2:S) -> U22(X1:S,X2:S) 1.91/1.92 a__U23(tt) -> tt 1.91/1.92 a__U23(X:S) -> U23(X:S) 1.91/1.92 a__U31(tt,V2:S) -> a__U32(a__isNatKind(V2:S)) 1.91/1.92 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 1.91/1.92 a__U32(tt) -> tt 1.91/1.92 a__U32(X:S) -> U32(X:S) 1.91/1.92 a__U41(tt) -> tt 1.91/1.92 a__U41(X:S) -> U41(X:S) 1.91/1.92 a__U51(tt,N:S) -> a__U52(a__isNatKind(N:S),N:S) 1.91/1.92 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.91/1.92 a__U52(tt,N:S) -> mark(N:S) 1.91/1.92 a__U52(X1:S,X2:S) -> U52(X1:S,X2:S) 1.91/1.92 a__U61(tt,M:S,N:S) -> a__U62(a__isNatKind(M:S),M:S,N:S) 1.91/1.92 a__U61(X1:S,X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) 1.91/1.92 a__U62(tt,M:S,N:S) -> a__U63(a__isNat(N:S),M:S,N:S) 1.91/1.92 a__U62(X1:S,X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) 1.91/1.92 a__U63(tt,M:S,N:S) -> a__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.92 a__U63(X1:S,X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) 1.91/1.92 a__U64(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 1.91/1.92 a__U64(X1:S,X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) 1.91/1.92 a__isNat(0) -> tt 1.91/1.92 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.92 a__isNat(s(V1:S)) -> a__U21(a__isNatKind(V1:S),V1:S) 1.91/1.92 a__isNat(X:S) -> isNat(X:S) 1.91/1.92 a__isNatKind(0) -> tt 1.91/1.92 a__isNatKind(plus(V1:S,V2:S)) -> a__U31(a__isNatKind(V1:S),V2:S) 1.91/1.92 a__isNatKind(s(V1:S)) -> a__U41(a__isNatKind(V1:S)) 1.91/1.92 a__isNatKind(X:S) -> isNatKind(X:S) 1.91/1.92 a__plus(N:S,0) -> a__U51(a__isNat(N:S),N:S) 1.91/1.92 a__plus(N:S,s(M:S)) -> a__U61(a__isNat(M:S),M:S,N:S) 1.91/1.92 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 1.91/1.92 mark(0) -> 0 1.91/1.92 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U13(X1:S,X2:S,X3:S)) -> a__U13(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U14(X1:S,X2:S,X3:S)) -> a__U14(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U15(X1:S,X2:S)) -> a__U15(mark(X1:S),X2:S) 1.91/1.92 mark(U16(X:S)) -> a__U16(mark(X:S)) 1.91/1.92 mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) 1.91/1.92 mark(U22(X1:S,X2:S)) -> a__U22(mark(X1:S),X2:S) 1.91/1.92 mark(U23(X:S)) -> a__U23(mark(X:S)) 1.91/1.92 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 1.91/1.92 mark(U32(X:S)) -> a__U32(mark(X:S)) 1.91/1.92 mark(U41(X:S)) -> a__U41(mark(X:S)) 1.91/1.92 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.91/1.92 mark(U52(X1:S,X2:S)) -> a__U52(mark(X1:S),X2:S) 1.91/1.92 mark(U61(X1:S,X2:S,X3:S)) -> a__U61(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U62(X1:S,X2:S,X3:S)) -> a__U62(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U63(X1:S,X2:S,X3:S)) -> a__U63(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U64(X1:S,X2:S,X3:S)) -> a__U64(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(isNat(X:S)) -> a__isNat(X:S) 1.91/1.92 mark(isNatKind(X:S)) -> a__isNatKind(X:S) 1.91/1.92 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 1.91/1.92 mark(s(X:S)) -> s(mark(X:S)) 1.91/1.92 mark(tt) -> tt 1.91/1.92 ->Strongly Connected Components: 1.91/1.92 There is no strongly connected component 1.91/1.92 1.91/1.92 The problem is finite. 1.91/1.92 1.91/1.92 Problem 1.2: 1.91/1.92 1.91/1.92 Reduction Pairs Processor: 1.91/1.92 -> Pairs: 1.91/1.92 A__U11(tt,V1:S,V2:S) -> A__U12(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.92 A__U12(tt,V1:S,V2:S) -> A__U13(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.92 A__U13(tt,V1:S,V2:S) -> A__U14(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.92 A__U14(tt,V1:S,V2:S) -> A__U15(a__isNat(V1:S),V2:S) 1.91/1.92 A__U14(tt,V1:S,V2:S) -> A__ISNAT(V1:S) 1.91/1.92 A__U15(tt,V2:S) -> A__ISNAT(V2:S) 1.91/1.92 A__U21(tt,V1:S) -> A__U22(a__isNatKind(V1:S),V1:S) 1.91/1.92 A__U22(tt,V1:S) -> A__ISNAT(V1:S) 1.91/1.92 A__ISNAT(plus(V1:S,V2:S)) -> A__U11(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.92 A__ISNAT(s(V1:S)) -> A__U21(a__isNatKind(V1:S),V1:S) 1.91/1.92 -> Rules: 1.91/1.92 a__U11(tt,V1:S,V2:S) -> a__U12(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.92 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 1.91/1.92 a__U12(tt,V1:S,V2:S) -> a__U13(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.92 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 1.91/1.92 a__U13(tt,V1:S,V2:S) -> a__U14(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.92 a__U13(X1:S,X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) 1.91/1.92 a__U14(tt,V1:S,V2:S) -> a__U15(a__isNat(V1:S),V2:S) 1.91/1.92 a__U14(X1:S,X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) 1.91/1.92 a__U15(tt,V2:S) -> a__U16(a__isNat(V2:S)) 1.91/1.92 a__U15(X1:S,X2:S) -> U15(X1:S,X2:S) 1.91/1.92 a__U16(tt) -> tt 1.91/1.92 a__U16(X:S) -> U16(X:S) 1.91/1.92 a__U21(tt,V1:S) -> a__U22(a__isNatKind(V1:S),V1:S) 1.91/1.92 a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) 1.91/1.92 a__U22(tt,V1:S) -> a__U23(a__isNat(V1:S)) 1.91/1.92 a__U22(X1:S,X2:S) -> U22(X1:S,X2:S) 1.91/1.92 a__U23(tt) -> tt 1.91/1.92 a__U23(X:S) -> U23(X:S) 1.91/1.92 a__U31(tt,V2:S) -> a__U32(a__isNatKind(V2:S)) 1.91/1.92 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 1.91/1.92 a__U32(tt) -> tt 1.91/1.92 a__U32(X:S) -> U32(X:S) 1.91/1.92 a__U41(tt) -> tt 1.91/1.92 a__U41(X:S) -> U41(X:S) 1.91/1.92 a__U51(tt,N:S) -> a__U52(a__isNatKind(N:S),N:S) 1.91/1.92 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.91/1.92 a__U52(tt,N:S) -> mark(N:S) 1.91/1.92 a__U52(X1:S,X2:S) -> U52(X1:S,X2:S) 1.91/1.92 a__U61(tt,M:S,N:S) -> a__U62(a__isNatKind(M:S),M:S,N:S) 1.91/1.92 a__U61(X1:S,X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) 1.91/1.92 a__U62(tt,M:S,N:S) -> a__U63(a__isNat(N:S),M:S,N:S) 1.91/1.92 a__U62(X1:S,X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) 1.91/1.92 a__U63(tt,M:S,N:S) -> a__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.92 a__U63(X1:S,X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) 1.91/1.92 a__U64(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 1.91/1.92 a__U64(X1:S,X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) 1.91/1.92 a__isNat(0) -> tt 1.91/1.92 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.92 a__isNat(s(V1:S)) -> a__U21(a__isNatKind(V1:S),V1:S) 1.91/1.92 a__isNat(X:S) -> isNat(X:S) 1.91/1.92 a__isNatKind(0) -> tt 1.91/1.92 a__isNatKind(plus(V1:S,V2:S)) -> a__U31(a__isNatKind(V1:S),V2:S) 1.91/1.92 a__isNatKind(s(V1:S)) -> a__U41(a__isNatKind(V1:S)) 1.91/1.92 a__isNatKind(X:S) -> isNatKind(X:S) 1.91/1.92 a__plus(N:S,0) -> a__U51(a__isNat(N:S),N:S) 1.91/1.92 a__plus(N:S,s(M:S)) -> a__U61(a__isNat(M:S),M:S,N:S) 1.91/1.92 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 1.91/1.92 mark(0) -> 0 1.91/1.92 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U13(X1:S,X2:S,X3:S)) -> a__U13(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U14(X1:S,X2:S,X3:S)) -> a__U14(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U15(X1:S,X2:S)) -> a__U15(mark(X1:S),X2:S) 1.91/1.92 mark(U16(X:S)) -> a__U16(mark(X:S)) 1.91/1.92 mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) 1.91/1.92 mark(U22(X1:S,X2:S)) -> a__U22(mark(X1:S),X2:S) 1.91/1.92 mark(U23(X:S)) -> a__U23(mark(X:S)) 1.91/1.92 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 1.91/1.92 mark(U32(X:S)) -> a__U32(mark(X:S)) 1.91/1.92 mark(U41(X:S)) -> a__U41(mark(X:S)) 1.91/1.92 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.91/1.92 mark(U52(X1:S,X2:S)) -> a__U52(mark(X1:S),X2:S) 1.91/1.92 mark(U61(X1:S,X2:S,X3:S)) -> a__U61(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U62(X1:S,X2:S,X3:S)) -> a__U62(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U63(X1:S,X2:S,X3:S)) -> a__U63(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U64(X1:S,X2:S,X3:S)) -> a__U64(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(isNat(X:S)) -> a__isNat(X:S) 1.91/1.92 mark(isNatKind(X:S)) -> a__isNatKind(X:S) 1.91/1.92 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 1.91/1.92 mark(s(X:S)) -> s(mark(X:S)) 1.91/1.92 mark(tt) -> tt 1.91/1.92 -> Usable rules: 1.91/1.92 a__U11(tt,V1:S,V2:S) -> a__U12(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.92 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 1.91/1.92 a__U12(tt,V1:S,V2:S) -> a__U13(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.92 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 1.91/1.92 a__U13(tt,V1:S,V2:S) -> a__U14(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.92 a__U13(X1:S,X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) 1.91/1.92 a__U14(tt,V1:S,V2:S) -> a__U15(a__isNat(V1:S),V2:S) 1.91/1.92 a__U14(X1:S,X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) 1.91/1.92 a__U15(tt,V2:S) -> a__U16(a__isNat(V2:S)) 1.91/1.92 a__U15(X1:S,X2:S) -> U15(X1:S,X2:S) 1.91/1.92 a__U16(tt) -> tt 1.91/1.92 a__U16(X:S) -> U16(X:S) 1.91/1.92 a__U21(tt,V1:S) -> a__U22(a__isNatKind(V1:S),V1:S) 1.91/1.92 a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) 1.91/1.92 a__U22(tt,V1:S) -> a__U23(a__isNat(V1:S)) 1.91/1.92 a__U22(X1:S,X2:S) -> U22(X1:S,X2:S) 1.91/1.92 a__U23(tt) -> tt 1.91/1.92 a__U23(X:S) -> U23(X:S) 1.91/1.92 a__U31(tt,V2:S) -> a__U32(a__isNatKind(V2:S)) 1.91/1.92 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 1.91/1.92 a__U32(tt) -> tt 1.91/1.92 a__U32(X:S) -> U32(X:S) 1.91/1.92 a__U41(tt) -> tt 1.91/1.92 a__U41(X:S) -> U41(X:S) 1.91/1.92 a__isNat(0) -> tt 1.91/1.92 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.92 a__isNat(s(V1:S)) -> a__U21(a__isNatKind(V1:S),V1:S) 1.91/1.92 a__isNat(X:S) -> isNat(X:S) 1.91/1.92 a__isNatKind(0) -> tt 1.91/1.92 a__isNatKind(plus(V1:S,V2:S)) -> a__U31(a__isNatKind(V1:S),V2:S) 1.91/1.92 a__isNatKind(s(V1:S)) -> a__U41(a__isNatKind(V1:S)) 1.91/1.92 a__isNatKind(X:S) -> isNatKind(X:S) 1.91/1.92 ->Interpretation type: 1.91/1.92 Linear 1.91/1.92 ->Coefficients: 1.91/1.92 Natural Numbers 1.91/1.92 ->Dimension: 1.91/1.92 1 1.91/1.92 ->Bound: 1.91/1.92 2 1.91/1.92 ->Interpretation: 1.91/1.92 1.91/1.92 [a__U11](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 1.91/1.92 [a__U12](X1,X2,X3) = X1 + 2.X3 + 2 1.91/1.92 [a__U13](X1,X2,X3) = X1 + 2.X3 + 2 1.91/1.92 [a__U14](X1,X2,X3) = X1 + 2.X3 + 1 1.91/1.92 [a__U15](X1,X2) = 2.X2 + 2 1.91/1.92 [a__U16](X) = X 1.91/1.92 [a__U21](X1,X2) = 2.X1 + 2.X2 + 2 1.91/1.92 [a__U22](X1,X2) = 2.X1 + 2.X2 + 2 1.91/1.92 [a__U23](X) = X + 2 1.91/1.92 [a__U31](X1,X2) = 2 1.91/1.92 [a__U32](X) = 2 1.91/1.92 [a__U41](X) = X 1.91/1.92 [a__U51](X1,X2) = 0 1.91/1.92 [a__U52](X1,X2) = 0 1.91/1.92 [a__U61](X1,X2,X3) = 0 1.91/1.92 [a__U62](X1,X2,X3) = 0 1.91/1.92 [a__U63](X1,X2,X3) = 0 1.91/1.92 [a__U64](X1,X2,X3) = 0 1.91/1.92 [a__isNat](X) = 2.X + 2 1.91/1.92 [a__isNatKind](X) = 2 1.91/1.92 [a__plus](X1,X2) = 0 1.91/1.92 [mark](X) = 0 1.91/1.92 [0] = 2 1.91/1.92 [U11](X1,X2,X3) = X1 1.91/1.92 [U12](X1,X2,X3) = X1 + 2.X3 + 1 1.91/1.92 [U13](X1,X2,X3) = 2.X3 + 2 1.91/1.92 [U14](X1,X2,X3) = X1 + X3 1.91/1.92 [U15](X1,X2) = 2.X2 1.91/1.92 [U16](X) = X 1.91/1.92 [U21](X1,X2) = 2.X1 + X2 1.91/1.92 [U22](X1,X2) = 2.X2 + 2 1.91/1.92 [U23](X) = X + 2 1.91/1.92 [U31](X1,X2) = 1 1.91/1.92 [U32](X) = 2 1.91/1.92 [U41](X) = X 1.91/1.92 [U51](X1,X2) = 0 1.91/1.92 [U52](X1,X2) = 0 1.91/1.92 [U61](X1,X2,X3) = 0 1.91/1.92 [U62](X1,X2,X3) = 0 1.91/1.92 [U63](X1,X2,X3) = 0 1.91/1.92 [U64](X1,X2,X3) = 0 1.91/1.92 [fSNonEmpty] = 0 1.91/1.92 [isNat](X) = 2.X + 2 1.91/1.92 [isNatKind](X) = 2 1.91/1.92 [plus](X1,X2) = 2.X1 + 2.X2 + 2 1.91/1.92 [s](X) = 2.X + 2 1.91/1.92 [tt] = 2 1.91/1.92 [A__U11](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 1.91/1.92 [A__U12](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 1 1.91/1.92 [A__U13](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 1 1.91/1.92 [A__U14](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 1 1.91/1.92 [A__U15](X1,X2) = X1 + 2.X2 + 2 1.91/1.92 [A__U16](X) = 0 1.91/1.92 [A__U21](X1,X2) = 2.X1 + 2.X2 + 2 1.91/1.92 [A__U22](X1,X2) = X1 + 2.X2 + 2 1.91/1.92 [A__U23](X) = 0 1.91/1.92 [A__U31](X1,X2) = 0 1.91/1.92 [A__U32](X) = 0 1.91/1.92 [A__U41](X) = 0 1.91/1.92 [A__U51](X1,X2) = 0 1.91/1.92 [A__U52](X1,X2) = 0 1.91/1.92 [A__U61](X1,X2,X3) = 0 1.91/1.92 [A__U62](X1,X2,X3) = 0 1.91/1.92 [A__U63](X1,X2,X3) = 0 1.91/1.92 [A__U64](X1,X2,X3) = 0 1.91/1.92 [A__ISNAT](X) = 2.X + 2 1.91/1.92 [A__ISNATKIND](X) = 0 1.91/1.92 [A__PLUS](X1,X2) = 0 1.91/1.92 [MARK](X) = 0 1.91/1.92 1.91/1.92 Problem 1.2: 1.91/1.92 1.91/1.92 SCC Processor: 1.91/1.92 -> Pairs: 1.91/1.92 A__U12(tt,V1:S,V2:S) -> A__U13(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.92 A__U13(tt,V1:S,V2:S) -> A__U14(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.92 A__U14(tt,V1:S,V2:S) -> A__U15(a__isNat(V1:S),V2:S) 1.91/1.92 A__U14(tt,V1:S,V2:S) -> A__ISNAT(V1:S) 1.91/1.92 A__U15(tt,V2:S) -> A__ISNAT(V2:S) 1.91/1.92 A__U21(tt,V1:S) -> A__U22(a__isNatKind(V1:S),V1:S) 1.91/1.92 A__U22(tt,V1:S) -> A__ISNAT(V1:S) 1.91/1.92 A__ISNAT(plus(V1:S,V2:S)) -> A__U11(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.92 A__ISNAT(s(V1:S)) -> A__U21(a__isNatKind(V1:S),V1:S) 1.91/1.92 -> Rules: 1.91/1.92 a__U11(tt,V1:S,V2:S) -> a__U12(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.92 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 1.91/1.92 a__U12(tt,V1:S,V2:S) -> a__U13(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.92 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 1.91/1.92 a__U13(tt,V1:S,V2:S) -> a__U14(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.92 a__U13(X1:S,X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) 1.91/1.92 a__U14(tt,V1:S,V2:S) -> a__U15(a__isNat(V1:S),V2:S) 1.91/1.92 a__U14(X1:S,X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) 1.91/1.92 a__U15(tt,V2:S) -> a__U16(a__isNat(V2:S)) 1.91/1.92 a__U15(X1:S,X2:S) -> U15(X1:S,X2:S) 1.91/1.92 a__U16(tt) -> tt 1.91/1.92 a__U16(X:S) -> U16(X:S) 1.91/1.92 a__U21(tt,V1:S) -> a__U22(a__isNatKind(V1:S),V1:S) 1.91/1.92 a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) 1.91/1.92 a__U22(tt,V1:S) -> a__U23(a__isNat(V1:S)) 1.91/1.92 a__U22(X1:S,X2:S) -> U22(X1:S,X2:S) 1.91/1.92 a__U23(tt) -> tt 1.91/1.92 a__U23(X:S) -> U23(X:S) 1.91/1.92 a__U31(tt,V2:S) -> a__U32(a__isNatKind(V2:S)) 1.91/1.92 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 1.91/1.92 a__U32(tt) -> tt 1.91/1.92 a__U32(X:S) -> U32(X:S) 1.91/1.92 a__U41(tt) -> tt 1.91/1.92 a__U41(X:S) -> U41(X:S) 1.91/1.92 a__U51(tt,N:S) -> a__U52(a__isNatKind(N:S),N:S) 1.91/1.92 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.91/1.92 a__U52(tt,N:S) -> mark(N:S) 1.91/1.92 a__U52(X1:S,X2:S) -> U52(X1:S,X2:S) 1.91/1.92 a__U61(tt,M:S,N:S) -> a__U62(a__isNatKind(M:S),M:S,N:S) 1.91/1.92 a__U61(X1:S,X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) 1.91/1.92 a__U62(tt,M:S,N:S) -> a__U63(a__isNat(N:S),M:S,N:S) 1.91/1.92 a__U62(X1:S,X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) 1.91/1.92 a__U63(tt,M:S,N:S) -> a__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.92 a__U63(X1:S,X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) 1.91/1.92 a__U64(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 1.91/1.92 a__U64(X1:S,X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) 1.91/1.92 a__isNat(0) -> tt 1.91/1.92 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.92 a__isNat(s(V1:S)) -> a__U21(a__isNatKind(V1:S),V1:S) 1.91/1.92 a__isNat(X:S) -> isNat(X:S) 1.91/1.92 a__isNatKind(0) -> tt 1.91/1.92 a__isNatKind(plus(V1:S,V2:S)) -> a__U31(a__isNatKind(V1:S),V2:S) 1.91/1.92 a__isNatKind(s(V1:S)) -> a__U41(a__isNatKind(V1:S)) 1.91/1.92 a__isNatKind(X:S) -> isNatKind(X:S) 1.91/1.92 a__plus(N:S,0) -> a__U51(a__isNat(N:S),N:S) 1.91/1.92 a__plus(N:S,s(M:S)) -> a__U61(a__isNat(M:S),M:S,N:S) 1.91/1.92 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 1.91/1.92 mark(0) -> 0 1.91/1.92 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U13(X1:S,X2:S,X3:S)) -> a__U13(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U14(X1:S,X2:S,X3:S)) -> a__U14(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U15(X1:S,X2:S)) -> a__U15(mark(X1:S),X2:S) 1.91/1.92 mark(U16(X:S)) -> a__U16(mark(X:S)) 1.91/1.92 mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) 1.91/1.92 mark(U22(X1:S,X2:S)) -> a__U22(mark(X1:S),X2:S) 1.91/1.92 mark(U23(X:S)) -> a__U23(mark(X:S)) 1.91/1.92 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 1.91/1.92 mark(U32(X:S)) -> a__U32(mark(X:S)) 1.91/1.92 mark(U41(X:S)) -> a__U41(mark(X:S)) 1.91/1.92 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.91/1.92 mark(U52(X1:S,X2:S)) -> a__U52(mark(X1:S),X2:S) 1.91/1.92 mark(U61(X1:S,X2:S,X3:S)) -> a__U61(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U62(X1:S,X2:S,X3:S)) -> a__U62(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U63(X1:S,X2:S,X3:S)) -> a__U63(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U64(X1:S,X2:S,X3:S)) -> a__U64(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(isNat(X:S)) -> a__isNat(X:S) 1.91/1.92 mark(isNatKind(X:S)) -> a__isNatKind(X:S) 1.91/1.92 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 1.91/1.92 mark(s(X:S)) -> s(mark(X:S)) 1.91/1.92 mark(tt) -> tt 1.91/1.92 ->Strongly Connected Components: 1.91/1.92 ->->Cycle: 1.91/1.92 ->->-> Pairs: 1.91/1.92 A__U21(tt,V1:S) -> A__U22(a__isNatKind(V1:S),V1:S) 1.91/1.92 A__U22(tt,V1:S) -> A__ISNAT(V1:S) 1.91/1.92 A__ISNAT(s(V1:S)) -> A__U21(a__isNatKind(V1:S),V1:S) 1.91/1.92 ->->-> Rules: 1.91/1.92 a__U11(tt,V1:S,V2:S) -> a__U12(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.92 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 1.91/1.92 a__U12(tt,V1:S,V2:S) -> a__U13(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.92 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 1.91/1.92 a__U13(tt,V1:S,V2:S) -> a__U14(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.92 a__U13(X1:S,X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) 1.91/1.92 a__U14(tt,V1:S,V2:S) -> a__U15(a__isNat(V1:S),V2:S) 1.91/1.92 a__U14(X1:S,X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) 1.91/1.92 a__U15(tt,V2:S) -> a__U16(a__isNat(V2:S)) 1.91/1.92 a__U15(X1:S,X2:S) -> U15(X1:S,X2:S) 1.91/1.92 a__U16(tt) -> tt 1.91/1.92 a__U16(X:S) -> U16(X:S) 1.91/1.92 a__U21(tt,V1:S) -> a__U22(a__isNatKind(V1:S),V1:S) 1.91/1.92 a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) 1.91/1.92 a__U22(tt,V1:S) -> a__U23(a__isNat(V1:S)) 1.91/1.92 a__U22(X1:S,X2:S) -> U22(X1:S,X2:S) 1.91/1.92 a__U23(tt) -> tt 1.91/1.92 a__U23(X:S) -> U23(X:S) 1.91/1.92 a__U31(tt,V2:S) -> a__U32(a__isNatKind(V2:S)) 1.91/1.92 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 1.91/1.92 a__U32(tt) -> tt 1.91/1.92 a__U32(X:S) -> U32(X:S) 1.91/1.92 a__U41(tt) -> tt 1.91/1.92 a__U41(X:S) -> U41(X:S) 1.91/1.92 a__U51(tt,N:S) -> a__U52(a__isNatKind(N:S),N:S) 1.91/1.92 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.91/1.92 a__U52(tt,N:S) -> mark(N:S) 1.91/1.92 a__U52(X1:S,X2:S) -> U52(X1:S,X2:S) 1.91/1.92 a__U61(tt,M:S,N:S) -> a__U62(a__isNatKind(M:S),M:S,N:S) 1.91/1.92 a__U61(X1:S,X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) 1.91/1.92 a__U62(tt,M:S,N:S) -> a__U63(a__isNat(N:S),M:S,N:S) 1.91/1.92 a__U62(X1:S,X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) 1.91/1.92 a__U63(tt,M:S,N:S) -> a__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.92 a__U63(X1:S,X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) 1.91/1.92 a__U64(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 1.91/1.92 a__U64(X1:S,X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) 1.91/1.92 a__isNat(0) -> tt 1.91/1.92 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.92 a__isNat(s(V1:S)) -> a__U21(a__isNatKind(V1:S),V1:S) 1.91/1.92 a__isNat(X:S) -> isNat(X:S) 1.91/1.92 a__isNatKind(0) -> tt 1.91/1.92 a__isNatKind(plus(V1:S,V2:S)) -> a__U31(a__isNatKind(V1:S),V2:S) 1.91/1.92 a__isNatKind(s(V1:S)) -> a__U41(a__isNatKind(V1:S)) 1.91/1.92 a__isNatKind(X:S) -> isNatKind(X:S) 1.91/1.92 a__plus(N:S,0) -> a__U51(a__isNat(N:S),N:S) 1.91/1.92 a__plus(N:S,s(M:S)) -> a__U61(a__isNat(M:S),M:S,N:S) 1.91/1.92 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 1.91/1.92 mark(0) -> 0 1.91/1.92 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U13(X1:S,X2:S,X3:S)) -> a__U13(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U14(X1:S,X2:S,X3:S)) -> a__U14(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U15(X1:S,X2:S)) -> a__U15(mark(X1:S),X2:S) 1.91/1.92 mark(U16(X:S)) -> a__U16(mark(X:S)) 1.91/1.92 mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) 1.91/1.92 mark(U22(X1:S,X2:S)) -> a__U22(mark(X1:S),X2:S) 1.91/1.92 mark(U23(X:S)) -> a__U23(mark(X:S)) 1.91/1.92 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 1.91/1.92 mark(U32(X:S)) -> a__U32(mark(X:S)) 1.91/1.92 mark(U41(X:S)) -> a__U41(mark(X:S)) 1.91/1.92 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.91/1.92 mark(U52(X1:S,X2:S)) -> a__U52(mark(X1:S),X2:S) 1.91/1.92 mark(U61(X1:S,X2:S,X3:S)) -> a__U61(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U62(X1:S,X2:S,X3:S)) -> a__U62(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U63(X1:S,X2:S,X3:S)) -> a__U63(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(U64(X1:S,X2:S,X3:S)) -> a__U64(mark(X1:S),X2:S,X3:S) 1.91/1.92 mark(isNat(X:S)) -> a__isNat(X:S) 1.91/1.92 mark(isNatKind(X:S)) -> a__isNatKind(X:S) 1.91/1.92 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 1.91/1.92 mark(s(X:S)) -> s(mark(X:S)) 1.91/1.92 mark(tt) -> tt 1.91/1.92 1.91/1.92 Problem 1.2: 1.91/1.92 1.91/1.92 Subterm Processor: 1.91/1.92 -> Pairs: 1.91/1.92 A__U21(tt,V1:S) -> A__U22(a__isNatKind(V1:S),V1:S) 1.91/1.92 A__U22(tt,V1:S) -> A__ISNAT(V1:S) 1.91/1.92 A__ISNAT(s(V1:S)) -> A__U21(a__isNatKind(V1:S),V1:S) 1.91/1.92 -> Rules: 1.91/1.92 a__U11(tt,V1:S,V2:S) -> a__U12(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.92 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 1.91/1.92 a__U12(tt,V1:S,V2:S) -> a__U13(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.93 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 1.91/1.93 a__U13(tt,V1:S,V2:S) -> a__U14(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.93 a__U13(X1:S,X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) 1.91/1.93 a__U14(tt,V1:S,V2:S) -> a__U15(a__isNat(V1:S),V2:S) 1.91/1.93 a__U14(X1:S,X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) 1.91/1.93 a__U15(tt,V2:S) -> a__U16(a__isNat(V2:S)) 1.91/1.93 a__U15(X1:S,X2:S) -> U15(X1:S,X2:S) 1.91/1.93 a__U16(tt) -> tt 1.91/1.93 a__U16(X:S) -> U16(X:S) 1.91/1.93 a__U21(tt,V1:S) -> a__U22(a__isNatKind(V1:S),V1:S) 1.91/1.93 a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) 1.91/1.93 a__U22(tt,V1:S) -> a__U23(a__isNat(V1:S)) 1.91/1.93 a__U22(X1:S,X2:S) -> U22(X1:S,X2:S) 1.91/1.93 a__U23(tt) -> tt 1.91/1.93 a__U23(X:S) -> U23(X:S) 1.91/1.93 a__U31(tt,V2:S) -> a__U32(a__isNatKind(V2:S)) 1.91/1.93 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 1.91/1.93 a__U32(tt) -> tt 1.91/1.93 a__U32(X:S) -> U32(X:S) 1.91/1.93 a__U41(tt) -> tt 1.91/1.93 a__U41(X:S) -> U41(X:S) 1.91/1.93 a__U51(tt,N:S) -> a__U52(a__isNatKind(N:S),N:S) 1.91/1.93 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.91/1.93 a__U52(tt,N:S) -> mark(N:S) 1.91/1.93 a__U52(X1:S,X2:S) -> U52(X1:S,X2:S) 1.91/1.93 a__U61(tt,M:S,N:S) -> a__U62(a__isNatKind(M:S),M:S,N:S) 1.91/1.93 a__U61(X1:S,X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) 1.91/1.93 a__U62(tt,M:S,N:S) -> a__U63(a__isNat(N:S),M:S,N:S) 1.91/1.93 a__U62(X1:S,X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) 1.91/1.93 a__U63(tt,M:S,N:S) -> a__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.93 a__U63(X1:S,X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) 1.91/1.93 a__U64(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 1.91/1.93 a__U64(X1:S,X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) 1.91/1.93 a__isNat(0) -> tt 1.91/1.93 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.93 a__isNat(s(V1:S)) -> a__U21(a__isNatKind(V1:S),V1:S) 1.91/1.93 a__isNat(X:S) -> isNat(X:S) 1.91/1.93 a__isNatKind(0) -> tt 1.91/1.93 a__isNatKind(plus(V1:S,V2:S)) -> a__U31(a__isNatKind(V1:S),V2:S) 1.91/1.93 a__isNatKind(s(V1:S)) -> a__U41(a__isNatKind(V1:S)) 1.91/1.93 a__isNatKind(X:S) -> isNatKind(X:S) 1.91/1.93 a__plus(N:S,0) -> a__U51(a__isNat(N:S),N:S) 1.91/1.93 a__plus(N:S,s(M:S)) -> a__U61(a__isNat(M:S),M:S,N:S) 1.91/1.93 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 1.91/1.93 mark(0) -> 0 1.91/1.93 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U13(X1:S,X2:S,X3:S)) -> a__U13(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U14(X1:S,X2:S,X3:S)) -> a__U14(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U15(X1:S,X2:S)) -> a__U15(mark(X1:S),X2:S) 1.91/1.93 mark(U16(X:S)) -> a__U16(mark(X:S)) 1.91/1.93 mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) 1.91/1.93 mark(U22(X1:S,X2:S)) -> a__U22(mark(X1:S),X2:S) 1.91/1.93 mark(U23(X:S)) -> a__U23(mark(X:S)) 1.91/1.93 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 1.91/1.93 mark(U32(X:S)) -> a__U32(mark(X:S)) 1.91/1.93 mark(U41(X:S)) -> a__U41(mark(X:S)) 1.91/1.93 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.91/1.93 mark(U52(X1:S,X2:S)) -> a__U52(mark(X1:S),X2:S) 1.91/1.93 mark(U61(X1:S,X2:S,X3:S)) -> a__U61(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U62(X1:S,X2:S,X3:S)) -> a__U62(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U63(X1:S,X2:S,X3:S)) -> a__U63(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U64(X1:S,X2:S,X3:S)) -> a__U64(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(isNat(X:S)) -> a__isNat(X:S) 1.91/1.93 mark(isNatKind(X:S)) -> a__isNatKind(X:S) 1.91/1.93 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 1.91/1.93 mark(s(X:S)) -> s(mark(X:S)) 1.91/1.93 mark(tt) -> tt 1.91/1.93 ->Projection: 1.91/1.93 pi(A__U21) = 2 1.91/1.93 pi(A__U22) = 2 1.91/1.93 pi(A__ISNAT) = 1 1.91/1.93 1.91/1.93 Problem 1.2: 1.91/1.93 1.91/1.93 SCC Processor: 1.91/1.93 -> Pairs: 1.91/1.93 A__U21(tt,V1:S) -> A__U22(a__isNatKind(V1:S),V1:S) 1.91/1.93 A__U22(tt,V1:S) -> A__ISNAT(V1:S) 1.91/1.93 -> Rules: 1.91/1.93 a__U11(tt,V1:S,V2:S) -> a__U12(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.93 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 1.91/1.93 a__U12(tt,V1:S,V2:S) -> a__U13(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.93 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 1.91/1.93 a__U13(tt,V1:S,V2:S) -> a__U14(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.93 a__U13(X1:S,X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) 1.91/1.93 a__U14(tt,V1:S,V2:S) -> a__U15(a__isNat(V1:S),V2:S) 1.91/1.93 a__U14(X1:S,X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) 1.91/1.93 a__U15(tt,V2:S) -> a__U16(a__isNat(V2:S)) 1.91/1.93 a__U15(X1:S,X2:S) -> U15(X1:S,X2:S) 1.91/1.93 a__U16(tt) -> tt 1.91/1.93 a__U16(X:S) -> U16(X:S) 1.91/1.93 a__U21(tt,V1:S) -> a__U22(a__isNatKind(V1:S),V1:S) 1.91/1.93 a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) 1.91/1.93 a__U22(tt,V1:S) -> a__U23(a__isNat(V1:S)) 1.91/1.93 a__U22(X1:S,X2:S) -> U22(X1:S,X2:S) 1.91/1.93 a__U23(tt) -> tt 1.91/1.93 a__U23(X:S) -> U23(X:S) 1.91/1.93 a__U31(tt,V2:S) -> a__U32(a__isNatKind(V2:S)) 1.91/1.93 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 1.91/1.93 a__U32(tt) -> tt 1.91/1.93 a__U32(X:S) -> U32(X:S) 1.91/1.93 a__U41(tt) -> tt 1.91/1.93 a__U41(X:S) -> U41(X:S) 1.91/1.93 a__U51(tt,N:S) -> a__U52(a__isNatKind(N:S),N:S) 1.91/1.93 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.91/1.93 a__U52(tt,N:S) -> mark(N:S) 1.91/1.93 a__U52(X1:S,X2:S) -> U52(X1:S,X2:S) 1.91/1.93 a__U61(tt,M:S,N:S) -> a__U62(a__isNatKind(M:S),M:S,N:S) 1.91/1.93 a__U61(X1:S,X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) 1.91/1.93 a__U62(tt,M:S,N:S) -> a__U63(a__isNat(N:S),M:S,N:S) 1.91/1.93 a__U62(X1:S,X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) 1.91/1.93 a__U63(tt,M:S,N:S) -> a__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.93 a__U63(X1:S,X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) 1.91/1.93 a__U64(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 1.91/1.93 a__U64(X1:S,X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) 1.91/1.93 a__isNat(0) -> tt 1.91/1.93 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.93 a__isNat(s(V1:S)) -> a__U21(a__isNatKind(V1:S),V1:S) 1.91/1.93 a__isNat(X:S) -> isNat(X:S) 1.91/1.93 a__isNatKind(0) -> tt 1.91/1.93 a__isNatKind(plus(V1:S,V2:S)) -> a__U31(a__isNatKind(V1:S),V2:S) 1.91/1.93 a__isNatKind(s(V1:S)) -> a__U41(a__isNatKind(V1:S)) 1.91/1.93 a__isNatKind(X:S) -> isNatKind(X:S) 1.91/1.93 a__plus(N:S,0) -> a__U51(a__isNat(N:S),N:S) 1.91/1.93 a__plus(N:S,s(M:S)) -> a__U61(a__isNat(M:S),M:S,N:S) 1.91/1.93 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 1.91/1.93 mark(0) -> 0 1.91/1.93 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U13(X1:S,X2:S,X3:S)) -> a__U13(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U14(X1:S,X2:S,X3:S)) -> a__U14(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U15(X1:S,X2:S)) -> a__U15(mark(X1:S),X2:S) 1.91/1.93 mark(U16(X:S)) -> a__U16(mark(X:S)) 1.91/1.93 mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) 1.91/1.93 mark(U22(X1:S,X2:S)) -> a__U22(mark(X1:S),X2:S) 1.91/1.93 mark(U23(X:S)) -> a__U23(mark(X:S)) 1.91/1.93 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 1.91/1.93 mark(U32(X:S)) -> a__U32(mark(X:S)) 1.91/1.93 mark(U41(X:S)) -> a__U41(mark(X:S)) 1.91/1.93 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.91/1.93 mark(U52(X1:S,X2:S)) -> a__U52(mark(X1:S),X2:S) 1.91/1.93 mark(U61(X1:S,X2:S,X3:S)) -> a__U61(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U62(X1:S,X2:S,X3:S)) -> a__U62(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U63(X1:S,X2:S,X3:S)) -> a__U63(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U64(X1:S,X2:S,X3:S)) -> a__U64(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(isNat(X:S)) -> a__isNat(X:S) 1.91/1.93 mark(isNatKind(X:S)) -> a__isNatKind(X:S) 1.91/1.93 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 1.91/1.93 mark(s(X:S)) -> s(mark(X:S)) 1.91/1.93 mark(tt) -> tt 1.91/1.93 ->Strongly Connected Components: 1.91/1.93 There is no strongly connected component 1.91/1.93 1.91/1.93 The problem is finite. 1.91/1.93 1.91/1.93 Problem 1.3: 1.91/1.93 1.91/1.93 Reduction Pairs Processor: 1.91/1.93 -> Pairs: 1.91/1.93 A__U51(tt,N:S) -> A__U52(a__isNatKind(N:S),N:S) 1.91/1.93 A__U52(tt,N:S) -> MARK(N:S) 1.91/1.93 A__U61(tt,M:S,N:S) -> A__U62(a__isNatKind(M:S),M:S,N:S) 1.91/1.93 A__U62(tt,M:S,N:S) -> A__U63(a__isNat(N:S),M:S,N:S) 1.91/1.93 A__U63(tt,M:S,N:S) -> A__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.93 A__U64(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S)) 1.91/1.93 A__U64(tt,M:S,N:S) -> MARK(M:S) 1.91/1.93 A__U64(tt,M:S,N:S) -> MARK(N:S) 1.91/1.93 A__PLUS(N:S,0) -> A__U51(a__isNat(N:S),N:S) 1.91/1.93 A__PLUS(N:S,s(M:S)) -> A__U61(a__isNat(M:S),M:S,N:S) 1.91/1.93 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U15(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U16(X:S)) -> MARK(X:S) 1.91/1.93 MARK(U21(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U22(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U23(X:S)) -> MARK(X:S) 1.91/1.93 MARK(U31(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U32(X:S)) -> MARK(X:S) 1.91/1.93 MARK(U41(X:S)) -> MARK(X:S) 1.91/1.93 MARK(U51(X1:S,X2:S)) -> A__U51(mark(X1:S),X2:S) 1.91/1.93 MARK(U51(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U52(X1:S,X2:S)) -> A__U52(mark(X1:S),X2:S) 1.91/1.93 MARK(U52(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U61(X1:S,X2:S,X3:S)) -> A__U61(mark(X1:S),X2:S,X3:S) 1.91/1.93 MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U62(X1:S,X2:S,X3:S)) -> A__U62(mark(X1:S),X2:S,X3:S) 1.91/1.93 MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U63(X1:S,X2:S,X3:S)) -> A__U63(mark(X1:S),X2:S,X3:S) 1.91/1.93 MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U64(X1:S,X2:S,X3:S)) -> A__U64(mark(X1:S),X2:S,X3:S) 1.91/1.93 MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 1.91/1.93 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.91/1.93 MARK(s(X:S)) -> MARK(X:S) 1.91/1.93 -> Rules: 1.91/1.93 a__U11(tt,V1:S,V2:S) -> a__U12(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.93 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 1.91/1.93 a__U12(tt,V1:S,V2:S) -> a__U13(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.93 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 1.91/1.93 a__U13(tt,V1:S,V2:S) -> a__U14(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.93 a__U13(X1:S,X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) 1.91/1.93 a__U14(tt,V1:S,V2:S) -> a__U15(a__isNat(V1:S),V2:S) 1.91/1.93 a__U14(X1:S,X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) 1.91/1.93 a__U15(tt,V2:S) -> a__U16(a__isNat(V2:S)) 1.91/1.93 a__U15(X1:S,X2:S) -> U15(X1:S,X2:S) 1.91/1.93 a__U16(tt) -> tt 1.91/1.93 a__U16(X:S) -> U16(X:S) 1.91/1.93 a__U21(tt,V1:S) -> a__U22(a__isNatKind(V1:S),V1:S) 1.91/1.93 a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) 1.91/1.93 a__U22(tt,V1:S) -> a__U23(a__isNat(V1:S)) 1.91/1.93 a__U22(X1:S,X2:S) -> U22(X1:S,X2:S) 1.91/1.93 a__U23(tt) -> tt 1.91/1.93 a__U23(X:S) -> U23(X:S) 1.91/1.93 a__U31(tt,V2:S) -> a__U32(a__isNatKind(V2:S)) 1.91/1.93 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 1.91/1.93 a__U32(tt) -> tt 1.91/1.93 a__U32(X:S) -> U32(X:S) 1.91/1.93 a__U41(tt) -> tt 1.91/1.93 a__U41(X:S) -> U41(X:S) 1.91/1.93 a__U51(tt,N:S) -> a__U52(a__isNatKind(N:S),N:S) 1.91/1.93 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.91/1.93 a__U52(tt,N:S) -> mark(N:S) 1.91/1.93 a__U52(X1:S,X2:S) -> U52(X1:S,X2:S) 1.91/1.93 a__U61(tt,M:S,N:S) -> a__U62(a__isNatKind(M:S),M:S,N:S) 1.91/1.93 a__U61(X1:S,X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) 1.91/1.93 a__U62(tt,M:S,N:S) -> a__U63(a__isNat(N:S),M:S,N:S) 1.91/1.93 a__U62(X1:S,X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) 1.91/1.93 a__U63(tt,M:S,N:S) -> a__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.93 a__U63(X1:S,X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) 1.91/1.93 a__U64(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 1.91/1.93 a__U64(X1:S,X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) 1.91/1.93 a__isNat(0) -> tt 1.91/1.93 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.93 a__isNat(s(V1:S)) -> a__U21(a__isNatKind(V1:S),V1:S) 1.91/1.93 a__isNat(X:S) -> isNat(X:S) 1.91/1.93 a__isNatKind(0) -> tt 1.91/1.93 a__isNatKind(plus(V1:S,V2:S)) -> a__U31(a__isNatKind(V1:S),V2:S) 1.91/1.93 a__isNatKind(s(V1:S)) -> a__U41(a__isNatKind(V1:S)) 1.91/1.93 a__isNatKind(X:S) -> isNatKind(X:S) 1.91/1.93 a__plus(N:S,0) -> a__U51(a__isNat(N:S),N:S) 1.91/1.93 a__plus(N:S,s(M:S)) -> a__U61(a__isNat(M:S),M:S,N:S) 1.91/1.93 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 1.91/1.93 mark(0) -> 0 1.91/1.93 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U13(X1:S,X2:S,X3:S)) -> a__U13(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U14(X1:S,X2:S,X3:S)) -> a__U14(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U15(X1:S,X2:S)) -> a__U15(mark(X1:S),X2:S) 1.91/1.93 mark(U16(X:S)) -> a__U16(mark(X:S)) 1.91/1.93 mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) 1.91/1.93 mark(U22(X1:S,X2:S)) -> a__U22(mark(X1:S),X2:S) 1.91/1.93 mark(U23(X:S)) -> a__U23(mark(X:S)) 1.91/1.93 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 1.91/1.93 mark(U32(X:S)) -> a__U32(mark(X:S)) 1.91/1.93 mark(U41(X:S)) -> a__U41(mark(X:S)) 1.91/1.93 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.91/1.93 mark(U52(X1:S,X2:S)) -> a__U52(mark(X1:S),X2:S) 1.91/1.93 mark(U61(X1:S,X2:S,X3:S)) -> a__U61(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U62(X1:S,X2:S,X3:S)) -> a__U62(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U63(X1:S,X2:S,X3:S)) -> a__U63(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U64(X1:S,X2:S,X3:S)) -> a__U64(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(isNat(X:S)) -> a__isNat(X:S) 1.91/1.93 mark(isNatKind(X:S)) -> a__isNatKind(X:S) 1.91/1.93 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 1.91/1.93 mark(s(X:S)) -> s(mark(X:S)) 1.91/1.93 mark(tt) -> tt 1.91/1.93 -> Usable rules: 1.91/1.93 a__U11(tt,V1:S,V2:S) -> a__U12(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.93 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 1.91/1.93 a__U12(tt,V1:S,V2:S) -> a__U13(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.93 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 1.91/1.93 a__U13(tt,V1:S,V2:S) -> a__U14(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.93 a__U13(X1:S,X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) 1.91/1.93 a__U14(tt,V1:S,V2:S) -> a__U15(a__isNat(V1:S),V2:S) 1.91/1.93 a__U14(X1:S,X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) 1.91/1.93 a__U15(tt,V2:S) -> a__U16(a__isNat(V2:S)) 1.91/1.93 a__U15(X1:S,X2:S) -> U15(X1:S,X2:S) 1.91/1.93 a__U16(tt) -> tt 1.91/1.93 a__U16(X:S) -> U16(X:S) 1.91/1.93 a__U21(tt,V1:S) -> a__U22(a__isNatKind(V1:S),V1:S) 1.91/1.93 a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) 1.91/1.93 a__U22(tt,V1:S) -> a__U23(a__isNat(V1:S)) 1.91/1.93 a__U22(X1:S,X2:S) -> U22(X1:S,X2:S) 1.91/1.93 a__U23(tt) -> tt 1.91/1.93 a__U23(X:S) -> U23(X:S) 1.91/1.93 a__U31(tt,V2:S) -> a__U32(a__isNatKind(V2:S)) 1.91/1.93 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 1.91/1.93 a__U32(tt) -> tt 1.91/1.93 a__U32(X:S) -> U32(X:S) 1.91/1.93 a__U41(tt) -> tt 1.91/1.93 a__U41(X:S) -> U41(X:S) 1.91/1.93 a__U51(tt,N:S) -> a__U52(a__isNatKind(N:S),N:S) 1.91/1.93 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.91/1.93 a__U52(tt,N:S) -> mark(N:S) 1.91/1.93 a__U52(X1:S,X2:S) -> U52(X1:S,X2:S) 1.91/1.93 a__U61(tt,M:S,N:S) -> a__U62(a__isNatKind(M:S),M:S,N:S) 1.91/1.93 a__U61(X1:S,X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) 1.91/1.93 a__U62(tt,M:S,N:S) -> a__U63(a__isNat(N:S),M:S,N:S) 1.91/1.93 a__U62(X1:S,X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) 1.91/1.93 a__U63(tt,M:S,N:S) -> a__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.93 a__U63(X1:S,X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) 1.91/1.93 a__U64(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 1.91/1.93 a__U64(X1:S,X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) 1.91/1.93 a__isNat(0) -> tt 1.91/1.93 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.93 a__isNat(s(V1:S)) -> a__U21(a__isNatKind(V1:S),V1:S) 1.91/1.93 a__isNat(X:S) -> isNat(X:S) 1.91/1.93 a__isNatKind(0) -> tt 1.91/1.93 a__isNatKind(plus(V1:S,V2:S)) -> a__U31(a__isNatKind(V1:S),V2:S) 1.91/1.93 a__isNatKind(s(V1:S)) -> a__U41(a__isNatKind(V1:S)) 1.91/1.93 a__isNatKind(X:S) -> isNatKind(X:S) 1.91/1.93 a__plus(N:S,0) -> a__U51(a__isNat(N:S),N:S) 1.91/1.93 a__plus(N:S,s(M:S)) -> a__U61(a__isNat(M:S),M:S,N:S) 1.91/1.93 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 1.91/1.93 mark(0) -> 0 1.91/1.93 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U13(X1:S,X2:S,X3:S)) -> a__U13(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U14(X1:S,X2:S,X3:S)) -> a__U14(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U15(X1:S,X2:S)) -> a__U15(mark(X1:S),X2:S) 1.91/1.93 mark(U16(X:S)) -> a__U16(mark(X:S)) 1.91/1.93 mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) 1.91/1.93 mark(U22(X1:S,X2:S)) -> a__U22(mark(X1:S),X2:S) 1.91/1.93 mark(U23(X:S)) -> a__U23(mark(X:S)) 1.91/1.93 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 1.91/1.93 mark(U32(X:S)) -> a__U32(mark(X:S)) 1.91/1.93 mark(U41(X:S)) -> a__U41(mark(X:S)) 1.91/1.93 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.91/1.93 mark(U52(X1:S,X2:S)) -> a__U52(mark(X1:S),X2:S) 1.91/1.93 mark(U61(X1:S,X2:S,X3:S)) -> a__U61(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U62(X1:S,X2:S,X3:S)) -> a__U62(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U63(X1:S,X2:S,X3:S)) -> a__U63(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U64(X1:S,X2:S,X3:S)) -> a__U64(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(isNat(X:S)) -> a__isNat(X:S) 1.91/1.93 mark(isNatKind(X:S)) -> a__isNatKind(X:S) 1.91/1.93 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 1.91/1.93 mark(s(X:S)) -> s(mark(X:S)) 1.91/1.93 mark(tt) -> tt 1.91/1.93 ->Interpretation type: 1.91/1.93 Linear 1.91/1.93 ->Coefficients: 1.91/1.93 Natural Numbers 1.91/1.93 ->Dimension: 1.91/1.93 1 1.91/1.93 ->Bound: 1.91/1.93 2 1.91/1.93 ->Interpretation: 1.91/1.93 1.91/1.93 [a__U11](X1,X2,X3) = X1 1.91/1.93 [a__U12](X1,X2,X3) = X1 1.91/1.93 [a__U13](X1,X2,X3) = X1 1.91/1.93 [a__U14](X1,X2,X3) = X1 1.91/1.93 [a__U15](X1,X2) = X1 1.91/1.93 [a__U16](X) = X 1.91/1.93 [a__U21](X1,X2) = X1 1.91/1.93 [a__U22](X1,X2) = X1 1.91/1.93 [a__U23](X) = X 1.91/1.93 [a__U31](X1,X2) = X1 1.91/1.93 [a__U32](X) = X 1.91/1.93 [a__U41](X) = X 1.91/1.93 [a__U51](X1,X2) = 2.X1 + 2.X2 1.91/1.93 [a__U52](X1,X2) = X1 + X2 + 2 1.91/1.93 [a__U61](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 1.91/1.93 [a__U62](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 1.91/1.93 [a__U63](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 1.91/1.93 [a__U64](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 1.91/1.93 [a__isNat](X) = 2 1.91/1.93 [a__isNatKind](X) = 2 1.91/1.93 [a__plus](X1,X2) = 2.X1 + 2.X2 + 2 1.91/1.93 [mark](X) = X 1.91/1.93 [0] = 2 1.91/1.93 [U11](X1,X2,X3) = X1 1.91/1.93 [U12](X1,X2,X3) = X1 1.91/1.93 [U13](X1,X2,X3) = X1 1.91/1.93 [U14](X1,X2,X3) = X1 1.91/1.93 [U15](X1,X2) = X1 1.91/1.93 [U16](X) = X 1.91/1.93 [U21](X1,X2) = X1 1.91/1.93 [U22](X1,X2) = X1 1.91/1.93 [U23](X) = X 1.91/1.93 [U31](X1,X2) = X1 1.91/1.93 [U32](X) = X 1.91/1.93 [U41](X) = X 1.91/1.93 [U51](X1,X2) = 2.X1 + 2.X2 1.91/1.93 [U52](X1,X2) = X1 + X2 + 2 1.91/1.93 [U61](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 1.91/1.93 [U62](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 1.91/1.93 [U63](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 1.91/1.93 [U64](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 1.91/1.93 [fSNonEmpty] = 0 1.91/1.93 [isNat](X) = 2 1.91/1.93 [isNatKind](X) = 2 1.91/1.93 [plus](X1,X2) = 2.X1 + 2.X2 + 2 1.91/1.93 [s](X) = X + 2 1.91/1.93 [tt] = 2 1.91/1.93 [A__U11](X1,X2,X3) = 0 1.91/1.93 [A__U12](X1,X2,X3) = 0 1.91/1.93 [A__U13](X1,X2,X3) = 0 1.91/1.93 [A__U14](X1,X2,X3) = 0 1.91/1.93 [A__U15](X1,X2) = 0 1.91/1.93 [A__U16](X) = 0 1.91/1.93 [A__U21](X1,X2) = 0 1.91/1.93 [A__U22](X1,X2) = 0 1.91/1.93 [A__U23](X) = 0 1.91/1.93 [A__U31](X1,X2) = 0 1.91/1.93 [A__U32](X) = 0 1.91/1.93 [A__U41](X) = 0 1.91/1.93 [A__U51](X1,X2) = 2.X1 + 2.X2 + 2 1.91/1.93 [A__U52](X1,X2) = X1 + 2.X2 + 2 1.91/1.93 [A__U61](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 1.91/1.93 [A__U62](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 1.91/1.93 [A__U63](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 1.91/1.93 [A__U64](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 1.91/1.93 [A__ISNAT](X) = 0 1.91/1.93 [A__ISNATKIND](X) = 0 1.91/1.93 [A__PLUS](X1,X2) = 2.X1 + 2.X2 + 2 1.91/1.93 [MARK](X) = 2.X + 2 1.91/1.93 1.91/1.93 Problem 1.3: 1.91/1.93 1.91/1.93 SCC Processor: 1.91/1.93 -> Pairs: 1.91/1.93 A__U52(tt,N:S) -> MARK(N:S) 1.91/1.93 A__U61(tt,M:S,N:S) -> A__U62(a__isNatKind(M:S),M:S,N:S) 1.91/1.93 A__U62(tt,M:S,N:S) -> A__U63(a__isNat(N:S),M:S,N:S) 1.91/1.93 A__U63(tt,M:S,N:S) -> A__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.93 A__U64(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S)) 1.91/1.93 A__U64(tt,M:S,N:S) -> MARK(M:S) 1.91/1.93 A__U64(tt,M:S,N:S) -> MARK(N:S) 1.91/1.93 A__PLUS(N:S,0) -> A__U51(a__isNat(N:S),N:S) 1.91/1.93 A__PLUS(N:S,s(M:S)) -> A__U61(a__isNat(M:S),M:S,N:S) 1.91/1.93 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U15(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U16(X:S)) -> MARK(X:S) 1.91/1.93 MARK(U21(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U22(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U23(X:S)) -> MARK(X:S) 1.91/1.93 MARK(U31(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U32(X:S)) -> MARK(X:S) 1.91/1.93 MARK(U41(X:S)) -> MARK(X:S) 1.91/1.93 MARK(U51(X1:S,X2:S)) -> A__U51(mark(X1:S),X2:S) 1.91/1.93 MARK(U51(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U52(X1:S,X2:S)) -> A__U52(mark(X1:S),X2:S) 1.91/1.93 MARK(U52(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U61(X1:S,X2:S,X3:S)) -> A__U61(mark(X1:S),X2:S,X3:S) 1.91/1.93 MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U62(X1:S,X2:S,X3:S)) -> A__U62(mark(X1:S),X2:S,X3:S) 1.91/1.93 MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U63(X1:S,X2:S,X3:S)) -> A__U63(mark(X1:S),X2:S,X3:S) 1.91/1.93 MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U64(X1:S,X2:S,X3:S)) -> A__U64(mark(X1:S),X2:S,X3:S) 1.91/1.93 MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 1.91/1.93 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.91/1.93 MARK(s(X:S)) -> MARK(X:S) 1.91/1.93 -> Rules: 1.91/1.93 a__U11(tt,V1:S,V2:S) -> a__U12(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.93 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 1.91/1.93 a__U12(tt,V1:S,V2:S) -> a__U13(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.93 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 1.91/1.93 a__U13(tt,V1:S,V2:S) -> a__U14(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.93 a__U13(X1:S,X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) 1.91/1.93 a__U14(tt,V1:S,V2:S) -> a__U15(a__isNat(V1:S),V2:S) 1.91/1.93 a__U14(X1:S,X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) 1.91/1.93 a__U15(tt,V2:S) -> a__U16(a__isNat(V2:S)) 1.91/1.93 a__U15(X1:S,X2:S) -> U15(X1:S,X2:S) 1.91/1.93 a__U16(tt) -> tt 1.91/1.93 a__U16(X:S) -> U16(X:S) 1.91/1.93 a__U21(tt,V1:S) -> a__U22(a__isNatKind(V1:S),V1:S) 1.91/1.93 a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) 1.91/1.93 a__U22(tt,V1:S) -> a__U23(a__isNat(V1:S)) 1.91/1.93 a__U22(X1:S,X2:S) -> U22(X1:S,X2:S) 1.91/1.93 a__U23(tt) -> tt 1.91/1.93 a__U23(X:S) -> U23(X:S) 1.91/1.93 a__U31(tt,V2:S) -> a__U32(a__isNatKind(V2:S)) 1.91/1.93 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 1.91/1.93 a__U32(tt) -> tt 1.91/1.93 a__U32(X:S) -> U32(X:S) 1.91/1.93 a__U41(tt) -> tt 1.91/1.93 a__U41(X:S) -> U41(X:S) 1.91/1.93 a__U51(tt,N:S) -> a__U52(a__isNatKind(N:S),N:S) 1.91/1.93 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.91/1.93 a__U52(tt,N:S) -> mark(N:S) 1.91/1.93 a__U52(X1:S,X2:S) -> U52(X1:S,X2:S) 1.91/1.93 a__U61(tt,M:S,N:S) -> a__U62(a__isNatKind(M:S),M:S,N:S) 1.91/1.93 a__U61(X1:S,X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) 1.91/1.93 a__U62(tt,M:S,N:S) -> a__U63(a__isNat(N:S),M:S,N:S) 1.91/1.93 a__U62(X1:S,X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) 1.91/1.93 a__U63(tt,M:S,N:S) -> a__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.93 a__U63(X1:S,X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) 1.91/1.93 a__U64(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 1.91/1.93 a__U64(X1:S,X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) 1.91/1.93 a__isNat(0) -> tt 1.91/1.93 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.93 a__isNat(s(V1:S)) -> a__U21(a__isNatKind(V1:S),V1:S) 1.91/1.93 a__isNat(X:S) -> isNat(X:S) 1.91/1.93 a__isNatKind(0) -> tt 1.91/1.93 a__isNatKind(plus(V1:S,V2:S)) -> a__U31(a__isNatKind(V1:S),V2:S) 1.91/1.93 a__isNatKind(s(V1:S)) -> a__U41(a__isNatKind(V1:S)) 1.91/1.93 a__isNatKind(X:S) -> isNatKind(X:S) 1.91/1.93 a__plus(N:S,0) -> a__U51(a__isNat(N:S),N:S) 1.91/1.93 a__plus(N:S,s(M:S)) -> a__U61(a__isNat(M:S),M:S,N:S) 1.91/1.93 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 1.91/1.93 mark(0) -> 0 1.91/1.93 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U13(X1:S,X2:S,X3:S)) -> a__U13(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U14(X1:S,X2:S,X3:S)) -> a__U14(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U15(X1:S,X2:S)) -> a__U15(mark(X1:S),X2:S) 1.91/1.93 mark(U16(X:S)) -> a__U16(mark(X:S)) 1.91/1.93 mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) 1.91/1.93 mark(U22(X1:S,X2:S)) -> a__U22(mark(X1:S),X2:S) 1.91/1.93 mark(U23(X:S)) -> a__U23(mark(X:S)) 1.91/1.93 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 1.91/1.93 mark(U32(X:S)) -> a__U32(mark(X:S)) 1.91/1.93 mark(U41(X:S)) -> a__U41(mark(X:S)) 1.91/1.93 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.91/1.93 mark(U52(X1:S,X2:S)) -> a__U52(mark(X1:S),X2:S) 1.91/1.93 mark(U61(X1:S,X2:S,X3:S)) -> a__U61(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U62(X1:S,X2:S,X3:S)) -> a__U62(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U63(X1:S,X2:S,X3:S)) -> a__U63(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U64(X1:S,X2:S,X3:S)) -> a__U64(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(isNat(X:S)) -> a__isNat(X:S) 1.91/1.93 mark(isNatKind(X:S)) -> a__isNatKind(X:S) 1.91/1.93 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 1.91/1.93 mark(s(X:S)) -> s(mark(X:S)) 1.91/1.93 mark(tt) -> tt 1.91/1.93 ->Strongly Connected Components: 1.91/1.93 ->->Cycle: 1.91/1.93 ->->-> Pairs: 1.91/1.93 A__U52(tt,N:S) -> MARK(N:S) 1.91/1.93 A__U61(tt,M:S,N:S) -> A__U62(a__isNatKind(M:S),M:S,N:S) 1.91/1.93 A__U62(tt,M:S,N:S) -> A__U63(a__isNat(N:S),M:S,N:S) 1.91/1.93 A__U63(tt,M:S,N:S) -> A__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.93 A__U64(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S)) 1.91/1.93 A__U64(tt,M:S,N:S) -> MARK(M:S) 1.91/1.93 A__U64(tt,M:S,N:S) -> MARK(N:S) 1.91/1.93 A__PLUS(N:S,s(M:S)) -> A__U61(a__isNat(M:S),M:S,N:S) 1.91/1.93 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U15(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U16(X:S)) -> MARK(X:S) 1.91/1.93 MARK(U21(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U22(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U23(X:S)) -> MARK(X:S) 1.91/1.93 MARK(U31(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U32(X:S)) -> MARK(X:S) 1.91/1.93 MARK(U41(X:S)) -> MARK(X:S) 1.91/1.93 MARK(U51(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U52(X1:S,X2:S)) -> A__U52(mark(X1:S),X2:S) 1.91/1.93 MARK(U52(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U61(X1:S,X2:S,X3:S)) -> A__U61(mark(X1:S),X2:S,X3:S) 1.91/1.93 MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U62(X1:S,X2:S,X3:S)) -> A__U62(mark(X1:S),X2:S,X3:S) 1.91/1.93 MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U63(X1:S,X2:S,X3:S)) -> A__U63(mark(X1:S),X2:S,X3:S) 1.91/1.93 MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U64(X1:S,X2:S,X3:S)) -> A__U64(mark(X1:S),X2:S,X3:S) 1.91/1.93 MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 1.91/1.93 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.91/1.93 MARK(s(X:S)) -> MARK(X:S) 1.91/1.93 ->->-> Rules: 1.91/1.93 a__U11(tt,V1:S,V2:S) -> a__U12(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.93 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 1.91/1.93 a__U12(tt,V1:S,V2:S) -> a__U13(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.93 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 1.91/1.93 a__U13(tt,V1:S,V2:S) -> a__U14(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.93 a__U13(X1:S,X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) 1.91/1.93 a__U14(tt,V1:S,V2:S) -> a__U15(a__isNat(V1:S),V2:S) 1.91/1.93 a__U14(X1:S,X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) 1.91/1.93 a__U15(tt,V2:S) -> a__U16(a__isNat(V2:S)) 1.91/1.93 a__U15(X1:S,X2:S) -> U15(X1:S,X2:S) 1.91/1.93 a__U16(tt) -> tt 1.91/1.93 a__U16(X:S) -> U16(X:S) 1.91/1.93 a__U21(tt,V1:S) -> a__U22(a__isNatKind(V1:S),V1:S) 1.91/1.93 a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) 1.91/1.93 a__U22(tt,V1:S) -> a__U23(a__isNat(V1:S)) 1.91/1.93 a__U22(X1:S,X2:S) -> U22(X1:S,X2:S) 1.91/1.93 a__U23(tt) -> tt 1.91/1.93 a__U23(X:S) -> U23(X:S) 1.91/1.93 a__U31(tt,V2:S) -> a__U32(a__isNatKind(V2:S)) 1.91/1.93 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 1.91/1.93 a__U32(tt) -> tt 1.91/1.93 a__U32(X:S) -> U32(X:S) 1.91/1.93 a__U41(tt) -> tt 1.91/1.93 a__U41(X:S) -> U41(X:S) 1.91/1.93 a__U51(tt,N:S) -> a__U52(a__isNatKind(N:S),N:S) 1.91/1.93 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.91/1.93 a__U52(tt,N:S) -> mark(N:S) 1.91/1.93 a__U52(X1:S,X2:S) -> U52(X1:S,X2:S) 1.91/1.93 a__U61(tt,M:S,N:S) -> a__U62(a__isNatKind(M:S),M:S,N:S) 1.91/1.93 a__U61(X1:S,X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) 1.91/1.93 a__U62(tt,M:S,N:S) -> a__U63(a__isNat(N:S),M:S,N:S) 1.91/1.93 a__U62(X1:S,X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) 1.91/1.93 a__U63(tt,M:S,N:S) -> a__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.93 a__U63(X1:S,X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) 1.91/1.93 a__U64(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 1.91/1.93 a__U64(X1:S,X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) 1.91/1.93 a__isNat(0) -> tt 1.91/1.93 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.93 a__isNat(s(V1:S)) -> a__U21(a__isNatKind(V1:S),V1:S) 1.91/1.93 a__isNat(X:S) -> isNat(X:S) 1.91/1.93 a__isNatKind(0) -> tt 1.91/1.93 a__isNatKind(plus(V1:S,V2:S)) -> a__U31(a__isNatKind(V1:S),V2:S) 1.91/1.93 a__isNatKind(s(V1:S)) -> a__U41(a__isNatKind(V1:S)) 1.91/1.93 a__isNatKind(X:S) -> isNatKind(X:S) 1.91/1.93 a__plus(N:S,0) -> a__U51(a__isNat(N:S),N:S) 1.91/1.93 a__plus(N:S,s(M:S)) -> a__U61(a__isNat(M:S),M:S,N:S) 1.91/1.93 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 1.91/1.93 mark(0) -> 0 1.91/1.93 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U13(X1:S,X2:S,X3:S)) -> a__U13(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U14(X1:S,X2:S,X3:S)) -> a__U14(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U15(X1:S,X2:S)) -> a__U15(mark(X1:S),X2:S) 1.91/1.93 mark(U16(X:S)) -> a__U16(mark(X:S)) 1.91/1.93 mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) 1.91/1.93 mark(U22(X1:S,X2:S)) -> a__U22(mark(X1:S),X2:S) 1.91/1.93 mark(U23(X:S)) -> a__U23(mark(X:S)) 1.91/1.93 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 1.91/1.93 mark(U32(X:S)) -> a__U32(mark(X:S)) 1.91/1.93 mark(U41(X:S)) -> a__U41(mark(X:S)) 1.91/1.93 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.91/1.93 mark(U52(X1:S,X2:S)) -> a__U52(mark(X1:S),X2:S) 1.91/1.93 mark(U61(X1:S,X2:S,X3:S)) -> a__U61(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U62(X1:S,X2:S,X3:S)) -> a__U62(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U63(X1:S,X2:S,X3:S)) -> a__U63(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U64(X1:S,X2:S,X3:S)) -> a__U64(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(isNat(X:S)) -> a__isNat(X:S) 1.91/1.93 mark(isNatKind(X:S)) -> a__isNatKind(X:S) 1.91/1.93 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 1.91/1.93 mark(s(X:S)) -> s(mark(X:S)) 1.91/1.93 mark(tt) -> tt 1.91/1.93 1.91/1.93 Problem 1.3: 1.91/1.93 1.91/1.93 Reduction Pairs Processor: 1.91/1.93 -> Pairs: 1.91/1.93 A__U52(tt,N:S) -> MARK(N:S) 1.91/1.93 A__U61(tt,M:S,N:S) -> A__U62(a__isNatKind(M:S),M:S,N:S) 1.91/1.93 A__U62(tt,M:S,N:S) -> A__U63(a__isNat(N:S),M:S,N:S) 1.91/1.93 A__U63(tt,M:S,N:S) -> A__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.93 A__U64(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S)) 1.91/1.93 A__U64(tt,M:S,N:S) -> MARK(M:S) 1.91/1.93 A__U64(tt,M:S,N:S) -> MARK(N:S) 1.91/1.93 A__PLUS(N:S,s(M:S)) -> A__U61(a__isNat(M:S),M:S,N:S) 1.91/1.93 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U15(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U16(X:S)) -> MARK(X:S) 1.91/1.93 MARK(U21(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U22(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U23(X:S)) -> MARK(X:S) 1.91/1.93 MARK(U31(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U32(X:S)) -> MARK(X:S) 1.91/1.93 MARK(U41(X:S)) -> MARK(X:S) 1.91/1.93 MARK(U51(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U52(X1:S,X2:S)) -> A__U52(mark(X1:S),X2:S) 1.91/1.93 MARK(U52(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U61(X1:S,X2:S,X3:S)) -> A__U61(mark(X1:S),X2:S,X3:S) 1.91/1.93 MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U62(X1:S,X2:S,X3:S)) -> A__U62(mark(X1:S),X2:S,X3:S) 1.91/1.93 MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U63(X1:S,X2:S,X3:S)) -> A__U63(mark(X1:S),X2:S,X3:S) 1.91/1.93 MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U64(X1:S,X2:S,X3:S)) -> A__U64(mark(X1:S),X2:S,X3:S) 1.91/1.93 MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 1.91/1.93 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.91/1.93 MARK(s(X:S)) -> MARK(X:S) 1.91/1.93 -> Rules: 1.91/1.93 a__U11(tt,V1:S,V2:S) -> a__U12(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.93 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 1.91/1.93 a__U12(tt,V1:S,V2:S) -> a__U13(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.93 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 1.91/1.93 a__U13(tt,V1:S,V2:S) -> a__U14(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.93 a__U13(X1:S,X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) 1.91/1.93 a__U14(tt,V1:S,V2:S) -> a__U15(a__isNat(V1:S),V2:S) 1.91/1.93 a__U14(X1:S,X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) 1.91/1.93 a__U15(tt,V2:S) -> a__U16(a__isNat(V2:S)) 1.91/1.93 a__U15(X1:S,X2:S) -> U15(X1:S,X2:S) 1.91/1.93 a__U16(tt) -> tt 1.91/1.93 a__U16(X:S) -> U16(X:S) 1.91/1.93 a__U21(tt,V1:S) -> a__U22(a__isNatKind(V1:S),V1:S) 1.91/1.93 a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) 1.91/1.93 a__U22(tt,V1:S) -> a__U23(a__isNat(V1:S)) 1.91/1.93 a__U22(X1:S,X2:S) -> U22(X1:S,X2:S) 1.91/1.93 a__U23(tt) -> tt 1.91/1.93 a__U23(X:S) -> U23(X:S) 1.91/1.93 a__U31(tt,V2:S) -> a__U32(a__isNatKind(V2:S)) 1.91/1.93 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 1.91/1.93 a__U32(tt) -> tt 1.91/1.93 a__U32(X:S) -> U32(X:S) 1.91/1.93 a__U41(tt) -> tt 1.91/1.93 a__U41(X:S) -> U41(X:S) 1.91/1.93 a__U51(tt,N:S) -> a__U52(a__isNatKind(N:S),N:S) 1.91/1.93 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.91/1.93 a__U52(tt,N:S) -> mark(N:S) 1.91/1.93 a__U52(X1:S,X2:S) -> U52(X1:S,X2:S) 1.91/1.93 a__U61(tt,M:S,N:S) -> a__U62(a__isNatKind(M:S),M:S,N:S) 1.91/1.93 a__U61(X1:S,X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) 1.91/1.93 a__U62(tt,M:S,N:S) -> a__U63(a__isNat(N:S),M:S,N:S) 1.91/1.93 a__U62(X1:S,X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) 1.91/1.93 a__U63(tt,M:S,N:S) -> a__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.93 a__U63(X1:S,X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) 1.91/1.93 a__U64(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 1.91/1.93 a__U64(X1:S,X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) 1.91/1.93 a__isNat(0) -> tt 1.91/1.93 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.93 a__isNat(s(V1:S)) -> a__U21(a__isNatKind(V1:S),V1:S) 1.91/1.93 a__isNat(X:S) -> isNat(X:S) 1.91/1.93 a__isNatKind(0) -> tt 1.91/1.93 a__isNatKind(plus(V1:S,V2:S)) -> a__U31(a__isNatKind(V1:S),V2:S) 1.91/1.93 a__isNatKind(s(V1:S)) -> a__U41(a__isNatKind(V1:S)) 1.91/1.93 a__isNatKind(X:S) -> isNatKind(X:S) 1.91/1.93 a__plus(N:S,0) -> a__U51(a__isNat(N:S),N:S) 1.91/1.93 a__plus(N:S,s(M:S)) -> a__U61(a__isNat(M:S),M:S,N:S) 1.91/1.93 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 1.91/1.93 mark(0) -> 0 1.91/1.93 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U13(X1:S,X2:S,X3:S)) -> a__U13(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U14(X1:S,X2:S,X3:S)) -> a__U14(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U15(X1:S,X2:S)) -> a__U15(mark(X1:S),X2:S) 1.91/1.93 mark(U16(X:S)) -> a__U16(mark(X:S)) 1.91/1.93 mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) 1.91/1.93 mark(U22(X1:S,X2:S)) -> a__U22(mark(X1:S),X2:S) 1.91/1.93 mark(U23(X:S)) -> a__U23(mark(X:S)) 1.91/1.93 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 1.91/1.93 mark(U32(X:S)) -> a__U32(mark(X:S)) 1.91/1.93 mark(U41(X:S)) -> a__U41(mark(X:S)) 1.91/1.93 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.91/1.93 mark(U52(X1:S,X2:S)) -> a__U52(mark(X1:S),X2:S) 1.91/1.93 mark(U61(X1:S,X2:S,X3:S)) -> a__U61(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U62(X1:S,X2:S,X3:S)) -> a__U62(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U63(X1:S,X2:S,X3:S)) -> a__U63(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U64(X1:S,X2:S,X3:S)) -> a__U64(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(isNat(X:S)) -> a__isNat(X:S) 1.91/1.93 mark(isNatKind(X:S)) -> a__isNatKind(X:S) 1.91/1.93 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 1.91/1.93 mark(s(X:S)) -> s(mark(X:S)) 1.91/1.93 mark(tt) -> tt 1.91/1.93 -> Usable rules: 1.91/1.93 a__U11(tt,V1:S,V2:S) -> a__U12(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.93 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 1.91/1.93 a__U12(tt,V1:S,V2:S) -> a__U13(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.93 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 1.91/1.93 a__U13(tt,V1:S,V2:S) -> a__U14(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.93 a__U13(X1:S,X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) 1.91/1.93 a__U14(tt,V1:S,V2:S) -> a__U15(a__isNat(V1:S),V2:S) 1.91/1.93 a__U14(X1:S,X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) 1.91/1.93 a__U15(tt,V2:S) -> a__U16(a__isNat(V2:S)) 1.91/1.93 a__U15(X1:S,X2:S) -> U15(X1:S,X2:S) 1.91/1.93 a__U16(tt) -> tt 1.91/1.93 a__U16(X:S) -> U16(X:S) 1.91/1.93 a__U21(tt,V1:S) -> a__U22(a__isNatKind(V1:S),V1:S) 1.91/1.93 a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) 1.91/1.93 a__U22(tt,V1:S) -> a__U23(a__isNat(V1:S)) 1.91/1.93 a__U22(X1:S,X2:S) -> U22(X1:S,X2:S) 1.91/1.93 a__U23(tt) -> tt 1.91/1.93 a__U23(X:S) -> U23(X:S) 1.91/1.93 a__U31(tt,V2:S) -> a__U32(a__isNatKind(V2:S)) 1.91/1.93 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 1.91/1.93 a__U32(tt) -> tt 1.91/1.93 a__U32(X:S) -> U32(X:S) 1.91/1.93 a__U41(tt) -> tt 1.91/1.93 a__U41(X:S) -> U41(X:S) 1.91/1.93 a__U51(tt,N:S) -> a__U52(a__isNatKind(N:S),N:S) 1.91/1.93 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.91/1.93 a__U52(tt,N:S) -> mark(N:S) 1.91/1.93 a__U52(X1:S,X2:S) -> U52(X1:S,X2:S) 1.91/1.93 a__U61(tt,M:S,N:S) -> a__U62(a__isNatKind(M:S),M:S,N:S) 1.91/1.93 a__U61(X1:S,X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) 1.91/1.93 a__U62(tt,M:S,N:S) -> a__U63(a__isNat(N:S),M:S,N:S) 1.91/1.93 a__U62(X1:S,X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) 1.91/1.93 a__U63(tt,M:S,N:S) -> a__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.93 a__U63(X1:S,X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) 1.91/1.93 a__U64(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 1.91/1.93 a__U64(X1:S,X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) 1.91/1.93 a__isNat(0) -> tt 1.91/1.93 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.93 a__isNat(s(V1:S)) -> a__U21(a__isNatKind(V1:S),V1:S) 1.91/1.93 a__isNat(X:S) -> isNat(X:S) 1.91/1.93 a__isNatKind(0) -> tt 1.91/1.93 a__isNatKind(plus(V1:S,V2:S)) -> a__U31(a__isNatKind(V1:S),V2:S) 1.91/1.93 a__isNatKind(s(V1:S)) -> a__U41(a__isNatKind(V1:S)) 1.91/1.93 a__isNatKind(X:S) -> isNatKind(X:S) 1.91/1.93 a__plus(N:S,0) -> a__U51(a__isNat(N:S),N:S) 1.91/1.93 a__plus(N:S,s(M:S)) -> a__U61(a__isNat(M:S),M:S,N:S) 1.91/1.93 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 1.91/1.93 mark(0) -> 0 1.91/1.93 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U13(X1:S,X2:S,X3:S)) -> a__U13(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U14(X1:S,X2:S,X3:S)) -> a__U14(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U15(X1:S,X2:S)) -> a__U15(mark(X1:S),X2:S) 1.91/1.93 mark(U16(X:S)) -> a__U16(mark(X:S)) 1.91/1.93 mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) 1.91/1.93 mark(U22(X1:S,X2:S)) -> a__U22(mark(X1:S),X2:S) 1.91/1.93 mark(U23(X:S)) -> a__U23(mark(X:S)) 1.91/1.93 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 1.91/1.93 mark(U32(X:S)) -> a__U32(mark(X:S)) 1.91/1.93 mark(U41(X:S)) -> a__U41(mark(X:S)) 1.91/1.93 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.91/1.93 mark(U52(X1:S,X2:S)) -> a__U52(mark(X1:S),X2:S) 1.91/1.93 mark(U61(X1:S,X2:S,X3:S)) -> a__U61(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U62(X1:S,X2:S,X3:S)) -> a__U62(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U63(X1:S,X2:S,X3:S)) -> a__U63(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U64(X1:S,X2:S,X3:S)) -> a__U64(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(isNat(X:S)) -> a__isNat(X:S) 1.91/1.93 mark(isNatKind(X:S)) -> a__isNatKind(X:S) 1.91/1.93 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 1.91/1.93 mark(s(X:S)) -> s(mark(X:S)) 1.91/1.93 mark(tt) -> tt 1.91/1.93 ->Interpretation type: 1.91/1.93 Linear 1.91/1.93 ->Coefficients: 1.91/1.93 Natural Numbers 1.91/1.93 ->Dimension: 1.91/1.93 1 1.91/1.93 ->Bound: 1.91/1.93 2 1.91/1.93 ->Interpretation: 1.91/1.93 1.91/1.93 [a__U11](X1,X2,X3) = X1 1.91/1.93 [a__U12](X1,X2,X3) = X1 1.91/1.93 [a__U13](X1,X2,X3) = X1 1.91/1.93 [a__U14](X1,X2,X3) = X1 1.91/1.93 [a__U15](X1,X2) = X1 1.91/1.93 [a__U16](X) = X 1.91/1.93 [a__U21](X1,X2) = X1 1.91/1.93 [a__U22](X1,X2) = X1 1.91/1.93 [a__U23](X) = X 1.91/1.93 [a__U31](X1,X2) = X1 1.91/1.93 [a__U32](X) = X 1.91/1.93 [a__U41](X) = X 1.91/1.93 [a__U51](X1,X2) = 2.X1 + 2.X2 + 2 1.91/1.93 [a__U52](X1,X2) = 2.X1 + 2.X2 + 1 1.91/1.93 [a__U61](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 1.91/1.93 [a__U62](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 1 1.91/1.93 [a__U63](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 1 1.91/1.93 [a__U64](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 1.91/1.93 [a__isNat](X) = 2 1.91/1.93 [a__isNatKind](X) = 2 1.91/1.93 [a__plus](X1,X2) = 2.X1 + 2.X2 + 2 1.91/1.93 [mark](X) = X 1.91/1.93 [0] = 2 1.91/1.93 [U11](X1,X2,X3) = X1 1.91/1.93 [U12](X1,X2,X3) = X1 1.91/1.93 [U13](X1,X2,X3) = X1 1.91/1.93 [U14](X1,X2,X3) = X1 1.91/1.93 [U15](X1,X2) = X1 1.91/1.93 [U16](X) = X 1.91/1.93 [U21](X1,X2) = X1 1.91/1.93 [U22](X1,X2) = X1 1.91/1.93 [U23](X) = X 1.91/1.93 [U31](X1,X2) = X1 1.91/1.93 [U32](X) = X 1.91/1.93 [U41](X) = X 1.91/1.93 [U51](X1,X2) = 2.X1 + 2.X2 + 2 1.91/1.93 [U52](X1,X2) = 2.X1 + 2.X2 + 1 1.91/1.93 [U61](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 1.91/1.93 [U62](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 1 1.91/1.93 [U63](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 1 1.91/1.93 [U64](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 1.91/1.93 [fSNonEmpty] = 0 1.91/1.93 [isNat](X) = 2 1.91/1.93 [isNatKind](X) = 2 1.91/1.93 [plus](X1,X2) = 2.X1 + 2.X2 + 2 1.91/1.93 [s](X) = X + 2 1.91/1.93 [tt] = 2 1.91/1.93 [A__U11](X1,X2,X3) = 0 1.91/1.93 [A__U12](X1,X2,X3) = 0 1.91/1.93 [A__U13](X1,X2,X3) = 0 1.91/1.93 [A__U14](X1,X2,X3) = 0 1.91/1.93 [A__U15](X1,X2) = 0 1.91/1.93 [A__U16](X) = 0 1.91/1.93 [A__U21](X1,X2) = 0 1.91/1.93 [A__U22](X1,X2) = 0 1.91/1.93 [A__U23](X) = 0 1.91/1.93 [A__U31](X1,X2) = 0 1.91/1.93 [A__U32](X) = 0 1.91/1.93 [A__U41](X) = 0 1.91/1.93 [A__U51](X1,X2) = 0 1.91/1.93 [A__U52](X1,X2) = X1 + 2.X2 + 2 1.91/1.93 [A__U61](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 1.91/1.93 [A__U62](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 1.91/1.93 [A__U63](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 1.91/1.93 [A__U64](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 1.91/1.93 [A__ISNAT](X) = 0 1.91/1.93 [A__ISNATKIND](X) = 0 1.91/1.93 [A__PLUS](X1,X2) = 2.X1 + 2.X2 + 2 1.91/1.93 [MARK](X) = 2.X + 2 1.91/1.93 1.91/1.93 Problem 1.3: 1.91/1.93 1.91/1.93 SCC Processor: 1.91/1.93 -> Pairs: 1.91/1.93 A__U61(tt,M:S,N:S) -> A__U62(a__isNatKind(M:S),M:S,N:S) 1.91/1.93 A__U62(tt,M:S,N:S) -> A__U63(a__isNat(N:S),M:S,N:S) 1.91/1.93 A__U63(tt,M:S,N:S) -> A__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.93 A__U64(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S)) 1.91/1.93 A__U64(tt,M:S,N:S) -> MARK(M:S) 1.91/1.93 A__U64(tt,M:S,N:S) -> MARK(N:S) 1.91/1.93 A__PLUS(N:S,s(M:S)) -> A__U61(a__isNat(M:S),M:S,N:S) 1.91/1.93 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U15(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U16(X:S)) -> MARK(X:S) 1.91/1.93 MARK(U21(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U22(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U23(X:S)) -> MARK(X:S) 1.91/1.93 MARK(U31(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U32(X:S)) -> MARK(X:S) 1.91/1.93 MARK(U41(X:S)) -> MARK(X:S) 1.91/1.93 MARK(U51(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U52(X1:S,X2:S)) -> A__U52(mark(X1:S),X2:S) 1.91/1.93 MARK(U52(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U61(X1:S,X2:S,X3:S)) -> A__U61(mark(X1:S),X2:S,X3:S) 1.91/1.93 MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U62(X1:S,X2:S,X3:S)) -> A__U62(mark(X1:S),X2:S,X3:S) 1.91/1.93 MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U63(X1:S,X2:S,X3:S)) -> A__U63(mark(X1:S),X2:S,X3:S) 1.91/1.93 MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U64(X1:S,X2:S,X3:S)) -> A__U64(mark(X1:S),X2:S,X3:S) 1.91/1.93 MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 1.91/1.93 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.91/1.93 MARK(s(X:S)) -> MARK(X:S) 1.91/1.93 -> Rules: 1.91/1.93 a__U11(tt,V1:S,V2:S) -> a__U12(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.93 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 1.91/1.93 a__U12(tt,V1:S,V2:S) -> a__U13(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.93 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 1.91/1.93 a__U13(tt,V1:S,V2:S) -> a__U14(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.93 a__U13(X1:S,X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) 1.91/1.93 a__U14(tt,V1:S,V2:S) -> a__U15(a__isNat(V1:S),V2:S) 1.91/1.93 a__U14(X1:S,X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) 1.91/1.93 a__U15(tt,V2:S) -> a__U16(a__isNat(V2:S)) 1.91/1.93 a__U15(X1:S,X2:S) -> U15(X1:S,X2:S) 1.91/1.93 a__U16(tt) -> tt 1.91/1.93 a__U16(X:S) -> U16(X:S) 1.91/1.93 a__U21(tt,V1:S) -> a__U22(a__isNatKind(V1:S),V1:S) 1.91/1.93 a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) 1.91/1.93 a__U22(tt,V1:S) -> a__U23(a__isNat(V1:S)) 1.91/1.93 a__U22(X1:S,X2:S) -> U22(X1:S,X2:S) 1.91/1.93 a__U23(tt) -> tt 1.91/1.93 a__U23(X:S) -> U23(X:S) 1.91/1.93 a__U31(tt,V2:S) -> a__U32(a__isNatKind(V2:S)) 1.91/1.93 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 1.91/1.93 a__U32(tt) -> tt 1.91/1.93 a__U32(X:S) -> U32(X:S) 1.91/1.93 a__U41(tt) -> tt 1.91/1.93 a__U41(X:S) -> U41(X:S) 1.91/1.93 a__U51(tt,N:S) -> a__U52(a__isNatKind(N:S),N:S) 1.91/1.93 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.91/1.93 a__U52(tt,N:S) -> mark(N:S) 1.91/1.93 a__U52(X1:S,X2:S) -> U52(X1:S,X2:S) 1.91/1.93 a__U61(tt,M:S,N:S) -> a__U62(a__isNatKind(M:S),M:S,N:S) 1.91/1.93 a__U61(X1:S,X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) 1.91/1.93 a__U62(tt,M:S,N:S) -> a__U63(a__isNat(N:S),M:S,N:S) 1.91/1.93 a__U62(X1:S,X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) 1.91/1.93 a__U63(tt,M:S,N:S) -> a__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.93 a__U63(X1:S,X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) 1.91/1.93 a__U64(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 1.91/1.93 a__U64(X1:S,X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) 1.91/1.93 a__isNat(0) -> tt 1.91/1.93 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.93 a__isNat(s(V1:S)) -> a__U21(a__isNatKind(V1:S),V1:S) 1.91/1.93 a__isNat(X:S) -> isNat(X:S) 1.91/1.93 a__isNatKind(0) -> tt 1.91/1.93 a__isNatKind(plus(V1:S,V2:S)) -> a__U31(a__isNatKind(V1:S),V2:S) 1.91/1.93 a__isNatKind(s(V1:S)) -> a__U41(a__isNatKind(V1:S)) 1.91/1.93 a__isNatKind(X:S) -> isNatKind(X:S) 1.91/1.93 a__plus(N:S,0) -> a__U51(a__isNat(N:S),N:S) 1.91/1.93 a__plus(N:S,s(M:S)) -> a__U61(a__isNat(M:S),M:S,N:S) 1.91/1.93 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 1.91/1.93 mark(0) -> 0 1.91/1.93 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U13(X1:S,X2:S,X3:S)) -> a__U13(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U14(X1:S,X2:S,X3:S)) -> a__U14(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U15(X1:S,X2:S)) -> a__U15(mark(X1:S),X2:S) 1.91/1.93 mark(U16(X:S)) -> a__U16(mark(X:S)) 1.91/1.93 mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) 1.91/1.93 mark(U22(X1:S,X2:S)) -> a__U22(mark(X1:S),X2:S) 1.91/1.93 mark(U23(X:S)) -> a__U23(mark(X:S)) 1.91/1.93 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 1.91/1.93 mark(U32(X:S)) -> a__U32(mark(X:S)) 1.91/1.93 mark(U41(X:S)) -> a__U41(mark(X:S)) 1.91/1.93 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.91/1.93 mark(U52(X1:S,X2:S)) -> a__U52(mark(X1:S),X2:S) 1.91/1.93 mark(U61(X1:S,X2:S,X3:S)) -> a__U61(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U62(X1:S,X2:S,X3:S)) -> a__U62(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U63(X1:S,X2:S,X3:S)) -> a__U63(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U64(X1:S,X2:S,X3:S)) -> a__U64(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(isNat(X:S)) -> a__isNat(X:S) 1.91/1.93 mark(isNatKind(X:S)) -> a__isNatKind(X:S) 1.91/1.93 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 1.91/1.93 mark(s(X:S)) -> s(mark(X:S)) 1.91/1.93 mark(tt) -> tt 1.91/1.93 ->Strongly Connected Components: 1.91/1.93 ->->Cycle: 1.91/1.93 ->->-> Pairs: 1.91/1.93 A__U61(tt,M:S,N:S) -> A__U62(a__isNatKind(M:S),M:S,N:S) 1.91/1.93 A__U62(tt,M:S,N:S) -> A__U63(a__isNat(N:S),M:S,N:S) 1.91/1.93 A__U63(tt,M:S,N:S) -> A__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.93 A__U64(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S)) 1.91/1.93 A__U64(tt,M:S,N:S) -> MARK(M:S) 1.91/1.93 A__U64(tt,M:S,N:S) -> MARK(N:S) 1.91/1.93 A__PLUS(N:S,s(M:S)) -> A__U61(a__isNat(M:S),M:S,N:S) 1.91/1.93 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U15(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U16(X:S)) -> MARK(X:S) 1.91/1.93 MARK(U21(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U22(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U23(X:S)) -> MARK(X:S) 1.91/1.93 MARK(U31(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U32(X:S)) -> MARK(X:S) 1.91/1.93 MARK(U41(X:S)) -> MARK(X:S) 1.91/1.93 MARK(U51(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U52(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U61(X1:S,X2:S,X3:S)) -> A__U61(mark(X1:S),X2:S,X3:S) 1.91/1.93 MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U62(X1:S,X2:S,X3:S)) -> A__U62(mark(X1:S),X2:S,X3:S) 1.91/1.93 MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U63(X1:S,X2:S,X3:S)) -> A__U63(mark(X1:S),X2:S,X3:S) 1.91/1.93 MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U64(X1:S,X2:S,X3:S)) -> A__U64(mark(X1:S),X2:S,X3:S) 1.91/1.93 MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 1.91/1.93 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.91/1.93 MARK(s(X:S)) -> MARK(X:S) 1.91/1.93 ->->-> Rules: 1.91/1.93 a__U11(tt,V1:S,V2:S) -> a__U12(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.93 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 1.91/1.93 a__U12(tt,V1:S,V2:S) -> a__U13(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.93 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 1.91/1.93 a__U13(tt,V1:S,V2:S) -> a__U14(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.93 a__U13(X1:S,X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) 1.91/1.93 a__U14(tt,V1:S,V2:S) -> a__U15(a__isNat(V1:S),V2:S) 1.91/1.93 a__U14(X1:S,X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) 1.91/1.93 a__U15(tt,V2:S) -> a__U16(a__isNat(V2:S)) 1.91/1.93 a__U15(X1:S,X2:S) -> U15(X1:S,X2:S) 1.91/1.93 a__U16(tt) -> tt 1.91/1.93 a__U16(X:S) -> U16(X:S) 1.91/1.93 a__U21(tt,V1:S) -> a__U22(a__isNatKind(V1:S),V1:S) 1.91/1.93 a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) 1.91/1.93 a__U22(tt,V1:S) -> a__U23(a__isNat(V1:S)) 1.91/1.93 a__U22(X1:S,X2:S) -> U22(X1:S,X2:S) 1.91/1.93 a__U23(tt) -> tt 1.91/1.93 a__U23(X:S) -> U23(X:S) 1.91/1.93 a__U31(tt,V2:S) -> a__U32(a__isNatKind(V2:S)) 1.91/1.93 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 1.91/1.93 a__U32(tt) -> tt 1.91/1.93 a__U32(X:S) -> U32(X:S) 1.91/1.93 a__U41(tt) -> tt 1.91/1.93 a__U41(X:S) -> U41(X:S) 1.91/1.93 a__U51(tt,N:S) -> a__U52(a__isNatKind(N:S),N:S) 1.91/1.93 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.91/1.93 a__U52(tt,N:S) -> mark(N:S) 1.91/1.93 a__U52(X1:S,X2:S) -> U52(X1:S,X2:S) 1.91/1.93 a__U61(tt,M:S,N:S) -> a__U62(a__isNatKind(M:S),M:S,N:S) 1.91/1.93 a__U61(X1:S,X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) 1.91/1.93 a__U62(tt,M:S,N:S) -> a__U63(a__isNat(N:S),M:S,N:S) 1.91/1.93 a__U62(X1:S,X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) 1.91/1.93 a__U63(tt,M:S,N:S) -> a__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.93 a__U63(X1:S,X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) 1.91/1.93 a__U64(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 1.91/1.93 a__U64(X1:S,X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) 1.91/1.93 a__isNat(0) -> tt 1.91/1.93 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.93 a__isNat(s(V1:S)) -> a__U21(a__isNatKind(V1:S),V1:S) 1.91/1.93 a__isNat(X:S) -> isNat(X:S) 1.91/1.93 a__isNatKind(0) -> tt 1.91/1.93 a__isNatKind(plus(V1:S,V2:S)) -> a__U31(a__isNatKind(V1:S),V2:S) 1.91/1.93 a__isNatKind(s(V1:S)) -> a__U41(a__isNatKind(V1:S)) 1.91/1.93 a__isNatKind(X:S) -> isNatKind(X:S) 1.91/1.93 a__plus(N:S,0) -> a__U51(a__isNat(N:S),N:S) 1.91/1.93 a__plus(N:S,s(M:S)) -> a__U61(a__isNat(M:S),M:S,N:S) 1.91/1.93 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 1.91/1.93 mark(0) -> 0 1.91/1.93 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U13(X1:S,X2:S,X3:S)) -> a__U13(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U14(X1:S,X2:S,X3:S)) -> a__U14(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U15(X1:S,X2:S)) -> a__U15(mark(X1:S),X2:S) 1.91/1.93 mark(U16(X:S)) -> a__U16(mark(X:S)) 1.91/1.93 mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) 1.91/1.93 mark(U22(X1:S,X2:S)) -> a__U22(mark(X1:S),X2:S) 1.91/1.93 mark(U23(X:S)) -> a__U23(mark(X:S)) 1.91/1.93 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 1.91/1.93 mark(U32(X:S)) -> a__U32(mark(X:S)) 1.91/1.93 mark(U41(X:S)) -> a__U41(mark(X:S)) 1.91/1.93 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.91/1.93 mark(U52(X1:S,X2:S)) -> a__U52(mark(X1:S),X2:S) 1.91/1.93 mark(U61(X1:S,X2:S,X3:S)) -> a__U61(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U62(X1:S,X2:S,X3:S)) -> a__U62(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U63(X1:S,X2:S,X3:S)) -> a__U63(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U64(X1:S,X2:S,X3:S)) -> a__U64(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(isNat(X:S)) -> a__isNat(X:S) 1.91/1.93 mark(isNatKind(X:S)) -> a__isNatKind(X:S) 1.91/1.93 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 1.91/1.93 mark(s(X:S)) -> s(mark(X:S)) 1.91/1.93 mark(tt) -> tt 1.91/1.93 1.91/1.93 Problem 1.3: 1.91/1.93 1.91/1.93 Reduction Pairs Processor: 1.91/1.93 -> Pairs: 1.91/1.93 A__U61(tt,M:S,N:S) -> A__U62(a__isNatKind(M:S),M:S,N:S) 1.91/1.93 A__U62(tt,M:S,N:S) -> A__U63(a__isNat(N:S),M:S,N:S) 1.91/1.93 A__U63(tt,M:S,N:S) -> A__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.93 A__U64(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S)) 1.91/1.93 A__U64(tt,M:S,N:S) -> MARK(M:S) 1.91/1.93 A__U64(tt,M:S,N:S) -> MARK(N:S) 1.91/1.93 A__PLUS(N:S,s(M:S)) -> A__U61(a__isNat(M:S),M:S,N:S) 1.91/1.93 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U15(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U16(X:S)) -> MARK(X:S) 1.91/1.93 MARK(U21(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U22(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U23(X:S)) -> MARK(X:S) 1.91/1.93 MARK(U31(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U32(X:S)) -> MARK(X:S) 1.91/1.93 MARK(U41(X:S)) -> MARK(X:S) 1.91/1.93 MARK(U51(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U52(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U61(X1:S,X2:S,X3:S)) -> A__U61(mark(X1:S),X2:S,X3:S) 1.91/1.93 MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U62(X1:S,X2:S,X3:S)) -> A__U62(mark(X1:S),X2:S,X3:S) 1.91/1.93 MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U63(X1:S,X2:S,X3:S)) -> A__U63(mark(X1:S),X2:S,X3:S) 1.91/1.93 MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U64(X1:S,X2:S,X3:S)) -> A__U64(mark(X1:S),X2:S,X3:S) 1.91/1.93 MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 1.91/1.93 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.91/1.93 MARK(s(X:S)) -> MARK(X:S) 1.91/1.93 -> Rules: 1.91/1.93 a__U11(tt,V1:S,V2:S) -> a__U12(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.93 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 1.91/1.93 a__U12(tt,V1:S,V2:S) -> a__U13(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.93 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 1.91/1.93 a__U13(tt,V1:S,V2:S) -> a__U14(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.93 a__U13(X1:S,X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) 1.91/1.93 a__U14(tt,V1:S,V2:S) -> a__U15(a__isNat(V1:S),V2:S) 1.91/1.93 a__U14(X1:S,X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) 1.91/1.93 a__U15(tt,V2:S) -> a__U16(a__isNat(V2:S)) 1.91/1.93 a__U15(X1:S,X2:S) -> U15(X1:S,X2:S) 1.91/1.93 a__U16(tt) -> tt 1.91/1.93 a__U16(X:S) -> U16(X:S) 1.91/1.93 a__U21(tt,V1:S) -> a__U22(a__isNatKind(V1:S),V1:S) 1.91/1.93 a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) 1.91/1.93 a__U22(tt,V1:S) -> a__U23(a__isNat(V1:S)) 1.91/1.93 a__U22(X1:S,X2:S) -> U22(X1:S,X2:S) 1.91/1.93 a__U23(tt) -> tt 1.91/1.93 a__U23(X:S) -> U23(X:S) 1.91/1.93 a__U31(tt,V2:S) -> a__U32(a__isNatKind(V2:S)) 1.91/1.93 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 1.91/1.93 a__U32(tt) -> tt 1.91/1.93 a__U32(X:S) -> U32(X:S) 1.91/1.93 a__U41(tt) -> tt 1.91/1.93 a__U41(X:S) -> U41(X:S) 1.91/1.93 a__U51(tt,N:S) -> a__U52(a__isNatKind(N:S),N:S) 1.91/1.93 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.91/1.93 a__U52(tt,N:S) -> mark(N:S) 1.91/1.93 a__U52(X1:S,X2:S) -> U52(X1:S,X2:S) 1.91/1.93 a__U61(tt,M:S,N:S) -> a__U62(a__isNatKind(M:S),M:S,N:S) 1.91/1.93 a__U61(X1:S,X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) 1.91/1.93 a__U62(tt,M:S,N:S) -> a__U63(a__isNat(N:S),M:S,N:S) 1.91/1.93 a__U62(X1:S,X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) 1.91/1.93 a__U63(tt,M:S,N:S) -> a__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.93 a__U63(X1:S,X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) 1.91/1.93 a__U64(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 1.91/1.93 a__U64(X1:S,X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) 1.91/1.93 a__isNat(0) -> tt 1.91/1.93 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.93 a__isNat(s(V1:S)) -> a__U21(a__isNatKind(V1:S),V1:S) 1.91/1.93 a__isNat(X:S) -> isNat(X:S) 1.91/1.93 a__isNatKind(0) -> tt 1.91/1.93 a__isNatKind(plus(V1:S,V2:S)) -> a__U31(a__isNatKind(V1:S),V2:S) 1.91/1.93 a__isNatKind(s(V1:S)) -> a__U41(a__isNatKind(V1:S)) 1.91/1.93 a__isNatKind(X:S) -> isNatKind(X:S) 1.91/1.93 a__plus(N:S,0) -> a__U51(a__isNat(N:S),N:S) 1.91/1.93 a__plus(N:S,s(M:S)) -> a__U61(a__isNat(M:S),M:S,N:S) 1.91/1.93 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 1.91/1.93 mark(0) -> 0 1.91/1.93 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U13(X1:S,X2:S,X3:S)) -> a__U13(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U14(X1:S,X2:S,X3:S)) -> a__U14(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U15(X1:S,X2:S)) -> a__U15(mark(X1:S),X2:S) 1.91/1.93 mark(U16(X:S)) -> a__U16(mark(X:S)) 1.91/1.93 mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) 1.91/1.93 mark(U22(X1:S,X2:S)) -> a__U22(mark(X1:S),X2:S) 1.91/1.93 mark(U23(X:S)) -> a__U23(mark(X:S)) 1.91/1.93 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 1.91/1.93 mark(U32(X:S)) -> a__U32(mark(X:S)) 1.91/1.93 mark(U41(X:S)) -> a__U41(mark(X:S)) 1.91/1.93 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.91/1.93 mark(U52(X1:S,X2:S)) -> a__U52(mark(X1:S),X2:S) 1.91/1.93 mark(U61(X1:S,X2:S,X3:S)) -> a__U61(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U62(X1:S,X2:S,X3:S)) -> a__U62(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U63(X1:S,X2:S,X3:S)) -> a__U63(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U64(X1:S,X2:S,X3:S)) -> a__U64(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(isNat(X:S)) -> a__isNat(X:S) 1.91/1.93 mark(isNatKind(X:S)) -> a__isNatKind(X:S) 1.91/1.93 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 1.91/1.93 mark(s(X:S)) -> s(mark(X:S)) 1.91/1.93 mark(tt) -> tt 1.91/1.93 -> Usable rules: 1.91/1.93 a__U11(tt,V1:S,V2:S) -> a__U12(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.93 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 1.91/1.93 a__U12(tt,V1:S,V2:S) -> a__U13(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.93 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 1.91/1.93 a__U13(tt,V1:S,V2:S) -> a__U14(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.93 a__U13(X1:S,X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) 1.91/1.93 a__U14(tt,V1:S,V2:S) -> a__U15(a__isNat(V1:S),V2:S) 1.91/1.93 a__U14(X1:S,X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) 1.91/1.93 a__U15(tt,V2:S) -> a__U16(a__isNat(V2:S)) 1.91/1.93 a__U15(X1:S,X2:S) -> U15(X1:S,X2:S) 1.91/1.93 a__U16(tt) -> tt 1.91/1.93 a__U16(X:S) -> U16(X:S) 1.91/1.93 a__U21(tt,V1:S) -> a__U22(a__isNatKind(V1:S),V1:S) 1.91/1.93 a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) 1.91/1.93 a__U22(tt,V1:S) -> a__U23(a__isNat(V1:S)) 1.91/1.93 a__U22(X1:S,X2:S) -> U22(X1:S,X2:S) 1.91/1.93 a__U23(tt) -> tt 1.91/1.93 a__U23(X:S) -> U23(X:S) 1.91/1.93 a__U31(tt,V2:S) -> a__U32(a__isNatKind(V2:S)) 1.91/1.93 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 1.91/1.93 a__U32(tt) -> tt 1.91/1.93 a__U32(X:S) -> U32(X:S) 1.91/1.93 a__U41(tt) -> tt 1.91/1.93 a__U41(X:S) -> U41(X:S) 1.91/1.93 a__U51(tt,N:S) -> a__U52(a__isNatKind(N:S),N:S) 1.91/1.93 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.91/1.93 a__U52(tt,N:S) -> mark(N:S) 1.91/1.93 a__U52(X1:S,X2:S) -> U52(X1:S,X2:S) 1.91/1.93 a__U61(tt,M:S,N:S) -> a__U62(a__isNatKind(M:S),M:S,N:S) 1.91/1.93 a__U61(X1:S,X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) 1.91/1.93 a__U62(tt,M:S,N:S) -> a__U63(a__isNat(N:S),M:S,N:S) 1.91/1.93 a__U62(X1:S,X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) 1.91/1.93 a__U63(tt,M:S,N:S) -> a__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.93 a__U63(X1:S,X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) 1.91/1.93 a__U64(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 1.91/1.93 a__U64(X1:S,X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) 1.91/1.93 a__isNat(0) -> tt 1.91/1.93 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.93 a__isNat(s(V1:S)) -> a__U21(a__isNatKind(V1:S),V1:S) 1.91/1.93 a__isNat(X:S) -> isNat(X:S) 1.91/1.93 a__isNatKind(0) -> tt 1.91/1.93 a__isNatKind(plus(V1:S,V2:S)) -> a__U31(a__isNatKind(V1:S),V2:S) 1.91/1.93 a__isNatKind(s(V1:S)) -> a__U41(a__isNatKind(V1:S)) 1.91/1.93 a__isNatKind(X:S) -> isNatKind(X:S) 1.91/1.93 a__plus(N:S,0) -> a__U51(a__isNat(N:S),N:S) 1.91/1.93 a__plus(N:S,s(M:S)) -> a__U61(a__isNat(M:S),M:S,N:S) 1.91/1.93 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 1.91/1.93 mark(0) -> 0 1.91/1.93 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U13(X1:S,X2:S,X3:S)) -> a__U13(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U14(X1:S,X2:S,X3:S)) -> a__U14(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U15(X1:S,X2:S)) -> a__U15(mark(X1:S),X2:S) 1.91/1.93 mark(U16(X:S)) -> a__U16(mark(X:S)) 1.91/1.93 mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) 1.91/1.93 mark(U22(X1:S,X2:S)) -> a__U22(mark(X1:S),X2:S) 1.91/1.93 mark(U23(X:S)) -> a__U23(mark(X:S)) 1.91/1.93 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 1.91/1.93 mark(U32(X:S)) -> a__U32(mark(X:S)) 1.91/1.93 mark(U41(X:S)) -> a__U41(mark(X:S)) 1.91/1.93 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.91/1.93 mark(U52(X1:S,X2:S)) -> a__U52(mark(X1:S),X2:S) 1.91/1.93 mark(U61(X1:S,X2:S,X3:S)) -> a__U61(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U62(X1:S,X2:S,X3:S)) -> a__U62(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U63(X1:S,X2:S,X3:S)) -> a__U63(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U64(X1:S,X2:S,X3:S)) -> a__U64(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(isNat(X:S)) -> a__isNat(X:S) 1.91/1.93 mark(isNatKind(X:S)) -> a__isNatKind(X:S) 1.91/1.93 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 1.91/1.93 mark(s(X:S)) -> s(mark(X:S)) 1.91/1.93 mark(tt) -> tt 1.91/1.93 ->Interpretation type: 1.91/1.93 Linear 1.91/1.93 ->Coefficients: 1.91/1.93 Natural Numbers 1.91/1.93 ->Dimension: 1.91/1.93 1 1.91/1.93 ->Bound: 1.91/1.93 2 1.91/1.93 ->Interpretation: 1.91/1.93 1.91/1.93 [a__U11](X1,X2,X3) = X1 1.91/1.93 [a__U12](X1,X2,X3) = X1 1.91/1.93 [a__U13](X1,X2,X3) = X1 1.91/1.93 [a__U14](X1,X2,X3) = X1 1.91/1.93 [a__U15](X1,X2) = X1 1.91/1.93 [a__U16](X) = X 1.91/1.93 [a__U21](X1,X2) = X1 1.91/1.93 [a__U22](X1,X2) = X1 1.91/1.93 [a__U23](X) = X 1.91/1.93 [a__U31](X1,X2) = X1 1.91/1.93 [a__U32](X) = X 1.91/1.93 [a__U41](X) = X 1.91/1.93 [a__U51](X1,X2) = X1 + X2 + 2 1.91/1.93 [a__U52](X1,X2) = X1 + X2 + 1 1.91/1.93 [a__U61](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 1.91/1.93 [a__U62](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 1.91/1.93 [a__U63](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 1.91/1.93 [a__U64](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 1 1.91/1.93 [a__isNat](X) = 2 1.91/1.93 [a__isNatKind](X) = 2 1.91/1.93 [a__plus](X1,X2) = 2.X1 + 2.X2 + 2 1.91/1.93 [mark](X) = X 1.91/1.93 [0] = 2 1.91/1.93 [U11](X1,X2,X3) = X1 1.91/1.93 [U12](X1,X2,X3) = X1 1.91/1.93 [U13](X1,X2,X3) = X1 1.91/1.93 [U14](X1,X2,X3) = X1 1.91/1.93 [U15](X1,X2) = X1 1.91/1.93 [U16](X) = X 1.91/1.93 [U21](X1,X2) = X1 1.91/1.93 [U22](X1,X2) = X1 1.91/1.93 [U23](X) = X 1.91/1.93 [U31](X1,X2) = X1 1.91/1.93 [U32](X) = X 1.91/1.93 [U41](X) = X 1.91/1.93 [U51](X1,X2) = X1 + X2 + 2 1.91/1.93 [U52](X1,X2) = X1 + X2 + 1 1.91/1.93 [U61](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 1.91/1.93 [U62](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 1.91/1.93 [U63](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 1.91/1.93 [U64](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 1 1.91/1.93 [fSNonEmpty] = 0 1.91/1.93 [isNat](X) = 2 1.91/1.93 [isNatKind](X) = 2 1.91/1.93 [plus](X1,X2) = 2.X1 + 2.X2 + 2 1.91/1.93 [s](X) = X + 2 1.91/1.93 [tt] = 2 1.91/1.93 [A__U11](X1,X2,X3) = 0 1.91/1.93 [A__U12](X1,X2,X3) = 0 1.91/1.93 [A__U13](X1,X2,X3) = 0 1.91/1.93 [A__U14](X1,X2,X3) = 0 1.91/1.93 [A__U15](X1,X2) = 0 1.91/1.93 [A__U16](X) = 0 1.91/1.93 [A__U21](X1,X2) = 0 1.91/1.93 [A__U22](X1,X2) = 0 1.91/1.93 [A__U23](X) = 0 1.91/1.93 [A__U31](X1,X2) = 0 1.91/1.93 [A__U32](X) = 0 1.91/1.93 [A__U41](X) = 0 1.91/1.93 [A__U51](X1,X2) = 0 1.91/1.93 [A__U52](X1,X2) = 0 1.91/1.93 [A__U61](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 1.91/1.93 [A__U62](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 1.91/1.93 [A__U63](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 1.91/1.93 [A__U64](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 1.91/1.93 [A__ISNAT](X) = 0 1.91/1.93 [A__ISNATKIND](X) = 0 1.91/1.93 [A__PLUS](X1,X2) = 2.X1 + 2.X2 + 2 1.91/1.93 [MARK](X) = 2.X + 2 1.91/1.93 1.91/1.93 Problem 1.3: 1.91/1.93 1.91/1.93 SCC Processor: 1.91/1.93 -> Pairs: 1.91/1.93 A__U62(tt,M:S,N:S) -> A__U63(a__isNat(N:S),M:S,N:S) 1.91/1.93 A__U63(tt,M:S,N:S) -> A__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.93 A__U64(tt,M:S,N:S) -> A__PLUS(mark(N:S),mark(M:S)) 1.91/1.93 A__U64(tt,M:S,N:S) -> MARK(M:S) 1.91/1.93 A__U64(tt,M:S,N:S) -> MARK(N:S) 1.91/1.93 A__PLUS(N:S,s(M:S)) -> A__U61(a__isNat(M:S),M:S,N:S) 1.91/1.93 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U15(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U16(X:S)) -> MARK(X:S) 1.91/1.93 MARK(U21(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U22(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U23(X:S)) -> MARK(X:S) 1.91/1.93 MARK(U31(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U32(X:S)) -> MARK(X:S) 1.91/1.93 MARK(U41(X:S)) -> MARK(X:S) 1.91/1.93 MARK(U51(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U52(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U61(X1:S,X2:S,X3:S)) -> A__U61(mark(X1:S),X2:S,X3:S) 1.91/1.93 MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U62(X1:S,X2:S,X3:S)) -> A__U62(mark(X1:S),X2:S,X3:S) 1.91/1.93 MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U63(X1:S,X2:S,X3:S)) -> A__U63(mark(X1:S),X2:S,X3:S) 1.91/1.93 MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U64(X1:S,X2:S,X3:S)) -> A__U64(mark(X1:S),X2:S,X3:S) 1.91/1.93 MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(plus(X1:S,X2:S)) -> A__PLUS(mark(X1:S),mark(X2:S)) 1.91/1.93 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.91/1.93 MARK(s(X:S)) -> MARK(X:S) 1.91/1.93 -> Rules: 1.91/1.93 a__U11(tt,V1:S,V2:S) -> a__U12(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.93 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 1.91/1.93 a__U12(tt,V1:S,V2:S) -> a__U13(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.93 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 1.91/1.93 a__U13(tt,V1:S,V2:S) -> a__U14(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.93 a__U13(X1:S,X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) 1.91/1.93 a__U14(tt,V1:S,V2:S) -> a__U15(a__isNat(V1:S),V2:S) 1.91/1.93 a__U14(X1:S,X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) 1.91/1.93 a__U15(tt,V2:S) -> a__U16(a__isNat(V2:S)) 1.91/1.93 a__U15(X1:S,X2:S) -> U15(X1:S,X2:S) 1.91/1.93 a__U16(tt) -> tt 1.91/1.93 a__U16(X:S) -> U16(X:S) 1.91/1.93 a__U21(tt,V1:S) -> a__U22(a__isNatKind(V1:S),V1:S) 1.91/1.93 a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) 1.91/1.93 a__U22(tt,V1:S) -> a__U23(a__isNat(V1:S)) 1.91/1.93 a__U22(X1:S,X2:S) -> U22(X1:S,X2:S) 1.91/1.93 a__U23(tt) -> tt 1.91/1.93 a__U23(X:S) -> U23(X:S) 1.91/1.93 a__U31(tt,V2:S) -> a__U32(a__isNatKind(V2:S)) 1.91/1.93 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 1.91/1.93 a__U32(tt) -> tt 1.91/1.93 a__U32(X:S) -> U32(X:S) 1.91/1.93 a__U41(tt) -> tt 1.91/1.93 a__U41(X:S) -> U41(X:S) 1.91/1.93 a__U51(tt,N:S) -> a__U52(a__isNatKind(N:S),N:S) 1.91/1.93 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.91/1.93 a__U52(tt,N:S) -> mark(N:S) 1.91/1.93 a__U52(X1:S,X2:S) -> U52(X1:S,X2:S) 1.91/1.93 a__U61(tt,M:S,N:S) -> a__U62(a__isNatKind(M:S),M:S,N:S) 1.91/1.93 a__U61(X1:S,X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) 1.91/1.93 a__U62(tt,M:S,N:S) -> a__U63(a__isNat(N:S),M:S,N:S) 1.91/1.93 a__U62(X1:S,X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) 1.91/1.93 a__U63(tt,M:S,N:S) -> a__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.93 a__U63(X1:S,X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) 1.91/1.93 a__U64(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 1.91/1.93 a__U64(X1:S,X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) 1.91/1.93 a__isNat(0) -> tt 1.91/1.93 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.93 a__isNat(s(V1:S)) -> a__U21(a__isNatKind(V1:S),V1:S) 1.91/1.93 a__isNat(X:S) -> isNat(X:S) 1.91/1.93 a__isNatKind(0) -> tt 1.91/1.93 a__isNatKind(plus(V1:S,V2:S)) -> a__U31(a__isNatKind(V1:S),V2:S) 1.91/1.93 a__isNatKind(s(V1:S)) -> a__U41(a__isNatKind(V1:S)) 1.91/1.93 a__isNatKind(X:S) -> isNatKind(X:S) 1.91/1.93 a__plus(N:S,0) -> a__U51(a__isNat(N:S),N:S) 1.91/1.93 a__plus(N:S,s(M:S)) -> a__U61(a__isNat(M:S),M:S,N:S) 1.91/1.93 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 1.91/1.93 mark(0) -> 0 1.91/1.93 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U13(X1:S,X2:S,X3:S)) -> a__U13(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U14(X1:S,X2:S,X3:S)) -> a__U14(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U15(X1:S,X2:S)) -> a__U15(mark(X1:S),X2:S) 1.91/1.93 mark(U16(X:S)) -> a__U16(mark(X:S)) 1.91/1.93 mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) 1.91/1.93 mark(U22(X1:S,X2:S)) -> a__U22(mark(X1:S),X2:S) 1.91/1.93 mark(U23(X:S)) -> a__U23(mark(X:S)) 1.91/1.93 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 1.91/1.93 mark(U32(X:S)) -> a__U32(mark(X:S)) 1.91/1.93 mark(U41(X:S)) -> a__U41(mark(X:S)) 1.91/1.93 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.91/1.93 mark(U52(X1:S,X2:S)) -> a__U52(mark(X1:S),X2:S) 1.91/1.93 mark(U61(X1:S,X2:S,X3:S)) -> a__U61(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U62(X1:S,X2:S,X3:S)) -> a__U62(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U63(X1:S,X2:S,X3:S)) -> a__U63(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U64(X1:S,X2:S,X3:S)) -> a__U64(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(isNat(X:S)) -> a__isNat(X:S) 1.91/1.93 mark(isNatKind(X:S)) -> a__isNatKind(X:S) 1.91/1.93 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 1.91/1.93 mark(s(X:S)) -> s(mark(X:S)) 1.91/1.93 mark(tt) -> tt 1.91/1.93 ->Strongly Connected Components: 1.91/1.93 ->->Cycle: 1.91/1.93 ->->-> Pairs: 1.91/1.93 A__U62(tt,M:S,N:S) -> A__U63(a__isNat(N:S),M:S,N:S) 1.91/1.93 A__U63(tt,M:S,N:S) -> A__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.93 A__U64(tt,M:S,N:S) -> MARK(M:S) 1.91/1.93 A__U64(tt,M:S,N:S) -> MARK(N:S) 1.91/1.93 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U15(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U16(X:S)) -> MARK(X:S) 1.91/1.93 MARK(U21(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U22(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U23(X:S)) -> MARK(X:S) 1.91/1.93 MARK(U31(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U32(X:S)) -> MARK(X:S) 1.91/1.93 MARK(U41(X:S)) -> MARK(X:S) 1.91/1.93 MARK(U51(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U52(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U62(X1:S,X2:S,X3:S)) -> A__U62(mark(X1:S),X2:S,X3:S) 1.91/1.93 MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U63(X1:S,X2:S,X3:S)) -> A__U63(mark(X1:S),X2:S,X3:S) 1.91/1.93 MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U64(X1:S,X2:S,X3:S)) -> A__U64(mark(X1:S),X2:S,X3:S) 1.91/1.93 MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.91/1.93 MARK(s(X:S)) -> MARK(X:S) 1.91/1.93 ->->-> Rules: 1.91/1.93 a__U11(tt,V1:S,V2:S) -> a__U12(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.93 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 1.91/1.93 a__U12(tt,V1:S,V2:S) -> a__U13(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.93 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 1.91/1.93 a__U13(tt,V1:S,V2:S) -> a__U14(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.93 a__U13(X1:S,X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) 1.91/1.93 a__U14(tt,V1:S,V2:S) -> a__U15(a__isNat(V1:S),V2:S) 1.91/1.93 a__U14(X1:S,X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) 1.91/1.93 a__U15(tt,V2:S) -> a__U16(a__isNat(V2:S)) 1.91/1.93 a__U15(X1:S,X2:S) -> U15(X1:S,X2:S) 1.91/1.93 a__U16(tt) -> tt 1.91/1.93 a__U16(X:S) -> U16(X:S) 1.91/1.93 a__U21(tt,V1:S) -> a__U22(a__isNatKind(V1:S),V1:S) 1.91/1.93 a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) 1.91/1.93 a__U22(tt,V1:S) -> a__U23(a__isNat(V1:S)) 1.91/1.93 a__U22(X1:S,X2:S) -> U22(X1:S,X2:S) 1.91/1.93 a__U23(tt) -> tt 1.91/1.93 a__U23(X:S) -> U23(X:S) 1.91/1.93 a__U31(tt,V2:S) -> a__U32(a__isNatKind(V2:S)) 1.91/1.93 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 1.91/1.93 a__U32(tt) -> tt 1.91/1.93 a__U32(X:S) -> U32(X:S) 1.91/1.93 a__U41(tt) -> tt 1.91/1.93 a__U41(X:S) -> U41(X:S) 1.91/1.93 a__U51(tt,N:S) -> a__U52(a__isNatKind(N:S),N:S) 1.91/1.93 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.91/1.93 a__U52(tt,N:S) -> mark(N:S) 1.91/1.93 a__U52(X1:S,X2:S) -> U52(X1:S,X2:S) 1.91/1.93 a__U61(tt,M:S,N:S) -> a__U62(a__isNatKind(M:S),M:S,N:S) 1.91/1.93 a__U61(X1:S,X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) 1.91/1.93 a__U62(tt,M:S,N:S) -> a__U63(a__isNat(N:S),M:S,N:S) 1.91/1.93 a__U62(X1:S,X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) 1.91/1.93 a__U63(tt,M:S,N:S) -> a__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.93 a__U63(X1:S,X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) 1.91/1.93 a__U64(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 1.91/1.93 a__U64(X1:S,X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) 1.91/1.93 a__isNat(0) -> tt 1.91/1.93 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.93 a__isNat(s(V1:S)) -> a__U21(a__isNatKind(V1:S),V1:S) 1.91/1.93 a__isNat(X:S) -> isNat(X:S) 1.91/1.93 a__isNatKind(0) -> tt 1.91/1.93 a__isNatKind(plus(V1:S,V2:S)) -> a__U31(a__isNatKind(V1:S),V2:S) 1.91/1.93 a__isNatKind(s(V1:S)) -> a__U41(a__isNatKind(V1:S)) 1.91/1.93 a__isNatKind(X:S) -> isNatKind(X:S) 1.91/1.93 a__plus(N:S,0) -> a__U51(a__isNat(N:S),N:S) 1.91/1.93 a__plus(N:S,s(M:S)) -> a__U61(a__isNat(M:S),M:S,N:S) 1.91/1.93 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 1.91/1.93 mark(0) -> 0 1.91/1.93 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U13(X1:S,X2:S,X3:S)) -> a__U13(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U14(X1:S,X2:S,X3:S)) -> a__U14(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U15(X1:S,X2:S)) -> a__U15(mark(X1:S),X2:S) 1.91/1.93 mark(U16(X:S)) -> a__U16(mark(X:S)) 1.91/1.93 mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) 1.91/1.93 mark(U22(X1:S,X2:S)) -> a__U22(mark(X1:S),X2:S) 1.91/1.93 mark(U23(X:S)) -> a__U23(mark(X:S)) 1.91/1.93 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 1.91/1.93 mark(U32(X:S)) -> a__U32(mark(X:S)) 1.91/1.93 mark(U41(X:S)) -> a__U41(mark(X:S)) 1.91/1.93 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.91/1.93 mark(U52(X1:S,X2:S)) -> a__U52(mark(X1:S),X2:S) 1.91/1.93 mark(U61(X1:S,X2:S,X3:S)) -> a__U61(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U62(X1:S,X2:S,X3:S)) -> a__U62(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U63(X1:S,X2:S,X3:S)) -> a__U63(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U64(X1:S,X2:S,X3:S)) -> a__U64(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(isNat(X:S)) -> a__isNat(X:S) 1.91/1.93 mark(isNatKind(X:S)) -> a__isNatKind(X:S) 1.91/1.93 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 1.91/1.93 mark(s(X:S)) -> s(mark(X:S)) 1.91/1.93 mark(tt) -> tt 1.91/1.93 1.91/1.93 Problem 1.3: 1.91/1.93 1.91/1.93 Reduction Pairs Processor: 1.91/1.93 -> Pairs: 1.91/1.93 A__U62(tt,M:S,N:S) -> A__U63(a__isNat(N:S),M:S,N:S) 1.91/1.93 A__U63(tt,M:S,N:S) -> A__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.93 A__U64(tt,M:S,N:S) -> MARK(M:S) 1.91/1.93 A__U64(tt,M:S,N:S) -> MARK(N:S) 1.91/1.93 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U15(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U16(X:S)) -> MARK(X:S) 1.91/1.93 MARK(U21(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U22(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U23(X:S)) -> MARK(X:S) 1.91/1.93 MARK(U31(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U32(X:S)) -> MARK(X:S) 1.91/1.93 MARK(U41(X:S)) -> MARK(X:S) 1.91/1.93 MARK(U51(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U52(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U62(X1:S,X2:S,X3:S)) -> A__U62(mark(X1:S),X2:S,X3:S) 1.91/1.93 MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U63(X1:S,X2:S,X3:S)) -> A__U63(mark(X1:S),X2:S,X3:S) 1.91/1.93 MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U64(X1:S,X2:S,X3:S)) -> A__U64(mark(X1:S),X2:S,X3:S) 1.91/1.93 MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.91/1.93 MARK(s(X:S)) -> MARK(X:S) 1.91/1.93 -> Rules: 1.91/1.93 a__U11(tt,V1:S,V2:S) -> a__U12(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.93 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 1.91/1.93 a__U12(tt,V1:S,V2:S) -> a__U13(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.93 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 1.91/1.93 a__U13(tt,V1:S,V2:S) -> a__U14(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.93 a__U13(X1:S,X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) 1.91/1.93 a__U14(tt,V1:S,V2:S) -> a__U15(a__isNat(V1:S),V2:S) 1.91/1.93 a__U14(X1:S,X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) 1.91/1.93 a__U15(tt,V2:S) -> a__U16(a__isNat(V2:S)) 1.91/1.93 a__U15(X1:S,X2:S) -> U15(X1:S,X2:S) 1.91/1.93 a__U16(tt) -> tt 1.91/1.93 a__U16(X:S) -> U16(X:S) 1.91/1.93 a__U21(tt,V1:S) -> a__U22(a__isNatKind(V1:S),V1:S) 1.91/1.93 a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) 1.91/1.93 a__U22(tt,V1:S) -> a__U23(a__isNat(V1:S)) 1.91/1.93 a__U22(X1:S,X2:S) -> U22(X1:S,X2:S) 1.91/1.93 a__U23(tt) -> tt 1.91/1.93 a__U23(X:S) -> U23(X:S) 1.91/1.93 a__U31(tt,V2:S) -> a__U32(a__isNatKind(V2:S)) 1.91/1.93 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 1.91/1.93 a__U32(tt) -> tt 1.91/1.93 a__U32(X:S) -> U32(X:S) 1.91/1.93 a__U41(tt) -> tt 1.91/1.93 a__U41(X:S) -> U41(X:S) 1.91/1.93 a__U51(tt,N:S) -> a__U52(a__isNatKind(N:S),N:S) 1.91/1.93 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.91/1.93 a__U52(tt,N:S) -> mark(N:S) 1.91/1.93 a__U52(X1:S,X2:S) -> U52(X1:S,X2:S) 1.91/1.93 a__U61(tt,M:S,N:S) -> a__U62(a__isNatKind(M:S),M:S,N:S) 1.91/1.93 a__U61(X1:S,X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) 1.91/1.93 a__U62(tt,M:S,N:S) -> a__U63(a__isNat(N:S),M:S,N:S) 1.91/1.93 a__U62(X1:S,X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) 1.91/1.93 a__U63(tt,M:S,N:S) -> a__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.93 a__U63(X1:S,X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) 1.91/1.93 a__U64(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 1.91/1.93 a__U64(X1:S,X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) 1.91/1.93 a__isNat(0) -> tt 1.91/1.93 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.93 a__isNat(s(V1:S)) -> a__U21(a__isNatKind(V1:S),V1:S) 1.91/1.93 a__isNat(X:S) -> isNat(X:S) 1.91/1.93 a__isNatKind(0) -> tt 1.91/1.93 a__isNatKind(plus(V1:S,V2:S)) -> a__U31(a__isNatKind(V1:S),V2:S) 1.91/1.93 a__isNatKind(s(V1:S)) -> a__U41(a__isNatKind(V1:S)) 1.91/1.93 a__isNatKind(X:S) -> isNatKind(X:S) 1.91/1.93 a__plus(N:S,0) -> a__U51(a__isNat(N:S),N:S) 1.91/1.93 a__plus(N:S,s(M:S)) -> a__U61(a__isNat(M:S),M:S,N:S) 1.91/1.93 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 1.91/1.93 mark(0) -> 0 1.91/1.93 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U13(X1:S,X2:S,X3:S)) -> a__U13(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U14(X1:S,X2:S,X3:S)) -> a__U14(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U15(X1:S,X2:S)) -> a__U15(mark(X1:S),X2:S) 1.91/1.93 mark(U16(X:S)) -> a__U16(mark(X:S)) 1.91/1.93 mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) 1.91/1.93 mark(U22(X1:S,X2:S)) -> a__U22(mark(X1:S),X2:S) 1.91/1.93 mark(U23(X:S)) -> a__U23(mark(X:S)) 1.91/1.93 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 1.91/1.93 mark(U32(X:S)) -> a__U32(mark(X:S)) 1.91/1.93 mark(U41(X:S)) -> a__U41(mark(X:S)) 1.91/1.93 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.91/1.93 mark(U52(X1:S,X2:S)) -> a__U52(mark(X1:S),X2:S) 1.91/1.93 mark(U61(X1:S,X2:S,X3:S)) -> a__U61(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U62(X1:S,X2:S,X3:S)) -> a__U62(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U63(X1:S,X2:S,X3:S)) -> a__U63(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U64(X1:S,X2:S,X3:S)) -> a__U64(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(isNat(X:S)) -> a__isNat(X:S) 1.91/1.93 mark(isNatKind(X:S)) -> a__isNatKind(X:S) 1.91/1.93 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 1.91/1.93 mark(s(X:S)) -> s(mark(X:S)) 1.91/1.93 mark(tt) -> tt 1.91/1.93 -> Usable rules: 1.91/1.93 a__U11(tt,V1:S,V2:S) -> a__U12(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.93 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 1.91/1.93 a__U12(tt,V1:S,V2:S) -> a__U13(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.93 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 1.91/1.93 a__U13(tt,V1:S,V2:S) -> a__U14(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.93 a__U13(X1:S,X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) 1.91/1.93 a__U14(tt,V1:S,V2:S) -> a__U15(a__isNat(V1:S),V2:S) 1.91/1.93 a__U14(X1:S,X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) 1.91/1.93 a__U15(tt,V2:S) -> a__U16(a__isNat(V2:S)) 1.91/1.93 a__U15(X1:S,X2:S) -> U15(X1:S,X2:S) 1.91/1.93 a__U16(tt) -> tt 1.91/1.93 a__U16(X:S) -> U16(X:S) 1.91/1.93 a__U21(tt,V1:S) -> a__U22(a__isNatKind(V1:S),V1:S) 1.91/1.93 a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) 1.91/1.93 a__U22(tt,V1:S) -> a__U23(a__isNat(V1:S)) 1.91/1.93 a__U22(X1:S,X2:S) -> U22(X1:S,X2:S) 1.91/1.93 a__U23(tt) -> tt 1.91/1.93 a__U23(X:S) -> U23(X:S) 1.91/1.93 a__U31(tt,V2:S) -> a__U32(a__isNatKind(V2:S)) 1.91/1.93 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 1.91/1.93 a__U32(tt) -> tt 1.91/1.93 a__U32(X:S) -> U32(X:S) 1.91/1.93 a__U41(tt) -> tt 1.91/1.93 a__U41(X:S) -> U41(X:S) 1.91/1.93 a__U51(tt,N:S) -> a__U52(a__isNatKind(N:S),N:S) 1.91/1.93 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.91/1.93 a__U52(tt,N:S) -> mark(N:S) 1.91/1.93 a__U52(X1:S,X2:S) -> U52(X1:S,X2:S) 1.91/1.93 a__U61(tt,M:S,N:S) -> a__U62(a__isNatKind(M:S),M:S,N:S) 1.91/1.93 a__U61(X1:S,X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) 1.91/1.93 a__U62(tt,M:S,N:S) -> a__U63(a__isNat(N:S),M:S,N:S) 1.91/1.93 a__U62(X1:S,X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) 1.91/1.93 a__U63(tt,M:S,N:S) -> a__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.93 a__U63(X1:S,X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) 1.91/1.93 a__U64(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 1.91/1.93 a__U64(X1:S,X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) 1.91/1.93 a__isNat(0) -> tt 1.91/1.93 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.93 a__isNat(s(V1:S)) -> a__U21(a__isNatKind(V1:S),V1:S) 1.91/1.93 a__isNat(X:S) -> isNat(X:S) 1.91/1.93 a__isNatKind(0) -> tt 1.91/1.93 a__isNatKind(plus(V1:S,V2:S)) -> a__U31(a__isNatKind(V1:S),V2:S) 1.91/1.93 a__isNatKind(s(V1:S)) -> a__U41(a__isNatKind(V1:S)) 1.91/1.93 a__isNatKind(X:S) -> isNatKind(X:S) 1.91/1.93 a__plus(N:S,0) -> a__U51(a__isNat(N:S),N:S) 1.91/1.93 a__plus(N:S,s(M:S)) -> a__U61(a__isNat(M:S),M:S,N:S) 1.91/1.93 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 1.91/1.93 mark(0) -> 0 1.91/1.93 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U13(X1:S,X2:S,X3:S)) -> a__U13(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U14(X1:S,X2:S,X3:S)) -> a__U14(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U15(X1:S,X2:S)) -> a__U15(mark(X1:S),X2:S) 1.91/1.93 mark(U16(X:S)) -> a__U16(mark(X:S)) 1.91/1.93 mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) 1.91/1.93 mark(U22(X1:S,X2:S)) -> a__U22(mark(X1:S),X2:S) 1.91/1.93 mark(U23(X:S)) -> a__U23(mark(X:S)) 1.91/1.93 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 1.91/1.93 mark(U32(X:S)) -> a__U32(mark(X:S)) 1.91/1.93 mark(U41(X:S)) -> a__U41(mark(X:S)) 1.91/1.93 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.91/1.93 mark(U52(X1:S,X2:S)) -> a__U52(mark(X1:S),X2:S) 1.91/1.93 mark(U61(X1:S,X2:S,X3:S)) -> a__U61(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U62(X1:S,X2:S,X3:S)) -> a__U62(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U63(X1:S,X2:S,X3:S)) -> a__U63(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U64(X1:S,X2:S,X3:S)) -> a__U64(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(isNat(X:S)) -> a__isNat(X:S) 1.91/1.93 mark(isNatKind(X:S)) -> a__isNatKind(X:S) 1.91/1.93 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 1.91/1.93 mark(s(X:S)) -> s(mark(X:S)) 1.91/1.93 mark(tt) -> tt 1.91/1.93 ->Interpretation type: 1.91/1.93 Linear 1.91/1.93 ->Coefficients: 1.91/1.93 Natural Numbers 1.91/1.93 ->Dimension: 1.91/1.93 1 1.91/1.93 ->Bound: 1.91/1.93 2 1.91/1.93 ->Interpretation: 1.91/1.93 1.91/1.93 [a__U11](X1,X2,X3) = X1 1.91/1.93 [a__U12](X1,X2,X3) = X1 1.91/1.93 [a__U13](X1,X2,X3) = X1 1.91/1.93 [a__U14](X1,X2,X3) = X1 1.91/1.93 [a__U15](X1,X2) = X1 1.91/1.93 [a__U16](X) = X 1.91/1.93 [a__U21](X1,X2) = X1 1.91/1.93 [a__U22](X1,X2) = X1 1.91/1.93 [a__U23](X) = X 1.91/1.93 [a__U31](X1,X2) = X1 1.91/1.93 [a__U32](X) = X 1.91/1.93 [a__U41](X) = X 1.91/1.93 [a__U51](X1,X2) = 2.X1 + 2.X2 1.91/1.93 [a__U52](X1,X2) = 2.X1 + 2.X2 1.91/1.93 [a__U61](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 1.91/1.93 [a__U62](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 1.91/1.93 [a__U63](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 1.91/1.93 [a__U64](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 1.91/1.93 [a__isNat](X) = 2 1.91/1.93 [a__isNatKind](X) = 2 1.91/1.93 [a__plus](X1,X2) = 2.X1 + 2.X2 + 2 1.91/1.93 [mark](X) = X 1.91/1.93 [0] = 2 1.91/1.93 [U11](X1,X2,X3) = X1 1.91/1.93 [U12](X1,X2,X3) = X1 1.91/1.93 [U13](X1,X2,X3) = X1 1.91/1.93 [U14](X1,X2,X3) = X1 1.91/1.93 [U15](X1,X2) = X1 1.91/1.93 [U16](X) = X 1.91/1.93 [U21](X1,X2) = X1 1.91/1.93 [U22](X1,X2) = X1 1.91/1.93 [U23](X) = X 1.91/1.93 [U31](X1,X2) = X1 1.91/1.93 [U32](X) = X 1.91/1.93 [U41](X) = X 1.91/1.93 [U51](X1,X2) = 2.X1 + 2.X2 1.91/1.93 [U52](X1,X2) = 2.X1 + 2.X2 1.91/1.93 [U61](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 1.91/1.93 [U62](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 1.91/1.93 [U63](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 1.91/1.93 [U64](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 1.91/1.93 [fSNonEmpty] = 0 1.91/1.93 [isNat](X) = 2 1.91/1.93 [isNatKind](X) = 2 1.91/1.93 [plus](X1,X2) = 2.X1 + 2.X2 + 2 1.91/1.93 [s](X) = X + 2 1.91/1.93 [tt] = 2 1.91/1.93 [A__U11](X1,X2,X3) = 0 1.91/1.93 [A__U12](X1,X2,X3) = 0 1.91/1.93 [A__U13](X1,X2,X3) = 0 1.91/1.93 [A__U14](X1,X2,X3) = 0 1.91/1.93 [A__U15](X1,X2) = 0 1.91/1.93 [A__U16](X) = 0 1.91/1.93 [A__U21](X1,X2) = 0 1.91/1.93 [A__U22](X1,X2) = 0 1.91/1.93 [A__U23](X) = 0 1.91/1.93 [A__U31](X1,X2) = 0 1.91/1.93 [A__U32](X) = 0 1.91/1.93 [A__U41](X) = 0 1.91/1.93 [A__U51](X1,X2) = 0 1.91/1.93 [A__U52](X1,X2) = 0 1.91/1.93 [A__U61](X1,X2,X3) = 0 1.91/1.93 [A__U62](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 1.91/1.93 [A__U63](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 1.91/1.93 [A__U64](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 1.91/1.93 [A__ISNAT](X) = 0 1.91/1.93 [A__ISNATKIND](X) = 0 1.91/1.93 [A__PLUS](X1,X2) = 0 1.91/1.93 [MARK](X) = 2.X + 2 1.91/1.93 1.91/1.93 Problem 1.3: 1.91/1.93 1.91/1.93 SCC Processor: 1.91/1.93 -> Pairs: 1.91/1.93 A__U63(tt,M:S,N:S) -> A__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.93 A__U64(tt,M:S,N:S) -> MARK(M:S) 1.91/1.93 A__U64(tt,M:S,N:S) -> MARK(N:S) 1.91/1.93 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U15(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U16(X:S)) -> MARK(X:S) 1.91/1.93 MARK(U21(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U22(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U23(X:S)) -> MARK(X:S) 1.91/1.93 MARK(U31(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U32(X:S)) -> MARK(X:S) 1.91/1.93 MARK(U41(X:S)) -> MARK(X:S) 1.91/1.93 MARK(U51(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U52(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U62(X1:S,X2:S,X3:S)) -> A__U62(mark(X1:S),X2:S,X3:S) 1.91/1.93 MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U63(X1:S,X2:S,X3:S)) -> A__U63(mark(X1:S),X2:S,X3:S) 1.91/1.93 MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U64(X1:S,X2:S,X3:S)) -> A__U64(mark(X1:S),X2:S,X3:S) 1.91/1.93 MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.91/1.93 MARK(s(X:S)) -> MARK(X:S) 1.91/1.93 -> Rules: 1.91/1.93 a__U11(tt,V1:S,V2:S) -> a__U12(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.93 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 1.91/1.93 a__U12(tt,V1:S,V2:S) -> a__U13(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.93 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 1.91/1.93 a__U13(tt,V1:S,V2:S) -> a__U14(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.93 a__U13(X1:S,X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) 1.91/1.93 a__U14(tt,V1:S,V2:S) -> a__U15(a__isNat(V1:S),V2:S) 1.91/1.93 a__U14(X1:S,X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) 1.91/1.93 a__U15(tt,V2:S) -> a__U16(a__isNat(V2:S)) 1.91/1.93 a__U15(X1:S,X2:S) -> U15(X1:S,X2:S) 1.91/1.93 a__U16(tt) -> tt 1.91/1.93 a__U16(X:S) -> U16(X:S) 1.91/1.93 a__U21(tt,V1:S) -> a__U22(a__isNatKind(V1:S),V1:S) 1.91/1.93 a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) 1.91/1.93 a__U22(tt,V1:S) -> a__U23(a__isNat(V1:S)) 1.91/1.93 a__U22(X1:S,X2:S) -> U22(X1:S,X2:S) 1.91/1.93 a__U23(tt) -> tt 1.91/1.93 a__U23(X:S) -> U23(X:S) 1.91/1.93 a__U31(tt,V2:S) -> a__U32(a__isNatKind(V2:S)) 1.91/1.93 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 1.91/1.93 a__U32(tt) -> tt 1.91/1.93 a__U32(X:S) -> U32(X:S) 1.91/1.93 a__U41(tt) -> tt 1.91/1.93 a__U41(X:S) -> U41(X:S) 1.91/1.93 a__U51(tt,N:S) -> a__U52(a__isNatKind(N:S),N:S) 1.91/1.93 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.91/1.93 a__U52(tt,N:S) -> mark(N:S) 1.91/1.93 a__U52(X1:S,X2:S) -> U52(X1:S,X2:S) 1.91/1.93 a__U61(tt,M:S,N:S) -> a__U62(a__isNatKind(M:S),M:S,N:S) 1.91/1.93 a__U61(X1:S,X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) 1.91/1.93 a__U62(tt,M:S,N:S) -> a__U63(a__isNat(N:S),M:S,N:S) 1.91/1.93 a__U62(X1:S,X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) 1.91/1.93 a__U63(tt,M:S,N:S) -> a__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.93 a__U63(X1:S,X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) 1.91/1.93 a__U64(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 1.91/1.93 a__U64(X1:S,X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) 1.91/1.93 a__isNat(0) -> tt 1.91/1.93 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.93 a__isNat(s(V1:S)) -> a__U21(a__isNatKind(V1:S),V1:S) 1.91/1.93 a__isNat(X:S) -> isNat(X:S) 1.91/1.93 a__isNatKind(0) -> tt 1.91/1.93 a__isNatKind(plus(V1:S,V2:S)) -> a__U31(a__isNatKind(V1:S),V2:S) 1.91/1.93 a__isNatKind(s(V1:S)) -> a__U41(a__isNatKind(V1:S)) 1.91/1.93 a__isNatKind(X:S) -> isNatKind(X:S) 1.91/1.93 a__plus(N:S,0) -> a__U51(a__isNat(N:S),N:S) 1.91/1.93 a__plus(N:S,s(M:S)) -> a__U61(a__isNat(M:S),M:S,N:S) 1.91/1.93 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 1.91/1.93 mark(0) -> 0 1.91/1.93 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U13(X1:S,X2:S,X3:S)) -> a__U13(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U14(X1:S,X2:S,X3:S)) -> a__U14(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U15(X1:S,X2:S)) -> a__U15(mark(X1:S),X2:S) 1.91/1.93 mark(U16(X:S)) -> a__U16(mark(X:S)) 1.91/1.93 mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) 1.91/1.93 mark(U22(X1:S,X2:S)) -> a__U22(mark(X1:S),X2:S) 1.91/1.93 mark(U23(X:S)) -> a__U23(mark(X:S)) 1.91/1.93 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 1.91/1.93 mark(U32(X:S)) -> a__U32(mark(X:S)) 1.91/1.93 mark(U41(X:S)) -> a__U41(mark(X:S)) 1.91/1.93 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.91/1.93 mark(U52(X1:S,X2:S)) -> a__U52(mark(X1:S),X2:S) 1.91/1.93 mark(U61(X1:S,X2:S,X3:S)) -> a__U61(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U62(X1:S,X2:S,X3:S)) -> a__U62(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U63(X1:S,X2:S,X3:S)) -> a__U63(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U64(X1:S,X2:S,X3:S)) -> a__U64(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(isNat(X:S)) -> a__isNat(X:S) 1.91/1.93 mark(isNatKind(X:S)) -> a__isNatKind(X:S) 1.91/1.93 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 1.91/1.93 mark(s(X:S)) -> s(mark(X:S)) 1.91/1.93 mark(tt) -> tt 1.91/1.93 ->Strongly Connected Components: 1.91/1.93 ->->Cycle: 1.91/1.93 ->->-> Pairs: 1.91/1.93 A__U63(tt,M:S,N:S) -> A__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.93 A__U64(tt,M:S,N:S) -> MARK(M:S) 1.91/1.93 A__U64(tt,M:S,N:S) -> MARK(N:S) 1.91/1.93 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U15(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U16(X:S)) -> MARK(X:S) 1.91/1.93 MARK(U21(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U22(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U23(X:S)) -> MARK(X:S) 1.91/1.93 MARK(U31(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U32(X:S)) -> MARK(X:S) 1.91/1.93 MARK(U41(X:S)) -> MARK(X:S) 1.91/1.93 MARK(U51(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U52(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U63(X1:S,X2:S,X3:S)) -> A__U63(mark(X1:S),X2:S,X3:S) 1.91/1.93 MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U64(X1:S,X2:S,X3:S)) -> A__U64(mark(X1:S),X2:S,X3:S) 1.91/1.93 MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.93 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.91/1.93 MARK(s(X:S)) -> MARK(X:S) 1.91/1.93 ->->-> Rules: 1.91/1.93 a__U11(tt,V1:S,V2:S) -> a__U12(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.93 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 1.91/1.93 a__U12(tt,V1:S,V2:S) -> a__U13(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.93 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 1.91/1.93 a__U13(tt,V1:S,V2:S) -> a__U14(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.93 a__U13(X1:S,X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) 1.91/1.93 a__U14(tt,V1:S,V2:S) -> a__U15(a__isNat(V1:S),V2:S) 1.91/1.93 a__U14(X1:S,X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) 1.91/1.93 a__U15(tt,V2:S) -> a__U16(a__isNat(V2:S)) 1.91/1.93 a__U15(X1:S,X2:S) -> U15(X1:S,X2:S) 1.91/1.93 a__U16(tt) -> tt 1.91/1.93 a__U16(X:S) -> U16(X:S) 1.91/1.93 a__U21(tt,V1:S) -> a__U22(a__isNatKind(V1:S),V1:S) 1.91/1.93 a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) 1.91/1.93 a__U22(tt,V1:S) -> a__U23(a__isNat(V1:S)) 1.91/1.93 a__U22(X1:S,X2:S) -> U22(X1:S,X2:S) 1.91/1.93 a__U23(tt) -> tt 1.91/1.93 a__U23(X:S) -> U23(X:S) 1.91/1.93 a__U31(tt,V2:S) -> a__U32(a__isNatKind(V2:S)) 1.91/1.93 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 1.91/1.93 a__U32(tt) -> tt 1.91/1.93 a__U32(X:S) -> U32(X:S) 1.91/1.93 a__U41(tt) -> tt 1.91/1.93 a__U41(X:S) -> U41(X:S) 1.91/1.93 a__U51(tt,N:S) -> a__U52(a__isNatKind(N:S),N:S) 1.91/1.93 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.91/1.93 a__U52(tt,N:S) -> mark(N:S) 1.91/1.93 a__U52(X1:S,X2:S) -> U52(X1:S,X2:S) 1.91/1.93 a__U61(tt,M:S,N:S) -> a__U62(a__isNatKind(M:S),M:S,N:S) 1.91/1.93 a__U61(X1:S,X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) 1.91/1.93 a__U62(tt,M:S,N:S) -> a__U63(a__isNat(N:S),M:S,N:S) 1.91/1.93 a__U62(X1:S,X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) 1.91/1.93 a__U63(tt,M:S,N:S) -> a__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.93 a__U63(X1:S,X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) 1.91/1.93 a__U64(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 1.91/1.93 a__U64(X1:S,X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) 1.91/1.93 a__isNat(0) -> tt 1.91/1.93 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.93 a__isNat(s(V1:S)) -> a__U21(a__isNatKind(V1:S),V1:S) 1.91/1.93 a__isNat(X:S) -> isNat(X:S) 1.91/1.93 a__isNatKind(0) -> tt 1.91/1.93 a__isNatKind(plus(V1:S,V2:S)) -> a__U31(a__isNatKind(V1:S),V2:S) 1.91/1.93 a__isNatKind(s(V1:S)) -> a__U41(a__isNatKind(V1:S)) 1.91/1.93 a__isNatKind(X:S) -> isNatKind(X:S) 1.91/1.93 a__plus(N:S,0) -> a__U51(a__isNat(N:S),N:S) 1.91/1.93 a__plus(N:S,s(M:S)) -> a__U61(a__isNat(M:S),M:S,N:S) 1.91/1.93 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 1.91/1.93 mark(0) -> 0 1.91/1.93 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U13(X1:S,X2:S,X3:S)) -> a__U13(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U14(X1:S,X2:S,X3:S)) -> a__U14(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U15(X1:S,X2:S)) -> a__U15(mark(X1:S),X2:S) 1.91/1.93 mark(U16(X:S)) -> a__U16(mark(X:S)) 1.91/1.93 mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) 1.91/1.93 mark(U22(X1:S,X2:S)) -> a__U22(mark(X1:S),X2:S) 1.91/1.93 mark(U23(X:S)) -> a__U23(mark(X:S)) 1.91/1.93 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 1.91/1.93 mark(U32(X:S)) -> a__U32(mark(X:S)) 1.91/1.93 mark(U41(X:S)) -> a__U41(mark(X:S)) 1.91/1.93 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.91/1.93 mark(U52(X1:S,X2:S)) -> a__U52(mark(X1:S),X2:S) 1.91/1.93 mark(U61(X1:S,X2:S,X3:S)) -> a__U61(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U62(X1:S,X2:S,X3:S)) -> a__U62(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U63(X1:S,X2:S,X3:S)) -> a__U63(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(U64(X1:S,X2:S,X3:S)) -> a__U64(mark(X1:S),X2:S,X3:S) 1.91/1.93 mark(isNat(X:S)) -> a__isNat(X:S) 1.91/1.93 mark(isNatKind(X:S)) -> a__isNatKind(X:S) 1.91/1.93 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 1.91/1.93 mark(s(X:S)) -> s(mark(X:S)) 1.91/1.93 mark(tt) -> tt 1.91/1.93 1.91/1.93 Problem 1.3: 1.91/1.93 1.91/1.93 Reduction Pairs Processor: 1.91/1.93 -> Pairs: 1.91/1.93 A__U63(tt,M:S,N:S) -> A__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.93 A__U64(tt,M:S,N:S) -> MARK(M:S) 1.91/1.93 A__U64(tt,M:S,N:S) -> MARK(N:S) 1.91/1.93 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.93 MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(U15(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.94 MARK(U16(X:S)) -> MARK(X:S) 1.91/1.94 MARK(U21(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.94 MARK(U22(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.94 MARK(U23(X:S)) -> MARK(X:S) 1.91/1.94 MARK(U31(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.94 MARK(U32(X:S)) -> MARK(X:S) 1.91/1.94 MARK(U41(X:S)) -> MARK(X:S) 1.91/1.94 MARK(U51(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.94 MARK(U52(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.94 MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(U63(X1:S,X2:S,X3:S)) -> A__U63(mark(X1:S),X2:S,X3:S) 1.91/1.94 MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(U64(X1:S,X2:S,X3:S)) -> A__U64(mark(X1:S),X2:S,X3:S) 1.91/1.94 MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.94 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.91/1.94 MARK(s(X:S)) -> MARK(X:S) 1.91/1.94 -> Rules: 1.91/1.94 a__U11(tt,V1:S,V2:S) -> a__U12(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.94 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 1.91/1.94 a__U12(tt,V1:S,V2:S) -> a__U13(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.94 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 1.91/1.94 a__U13(tt,V1:S,V2:S) -> a__U14(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.94 a__U13(X1:S,X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) 1.91/1.94 a__U14(tt,V1:S,V2:S) -> a__U15(a__isNat(V1:S),V2:S) 1.91/1.94 a__U14(X1:S,X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) 1.91/1.94 a__U15(tt,V2:S) -> a__U16(a__isNat(V2:S)) 1.91/1.94 a__U15(X1:S,X2:S) -> U15(X1:S,X2:S) 1.91/1.94 a__U16(tt) -> tt 1.91/1.94 a__U16(X:S) -> U16(X:S) 1.91/1.94 a__U21(tt,V1:S) -> a__U22(a__isNatKind(V1:S),V1:S) 1.91/1.94 a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) 1.91/1.94 a__U22(tt,V1:S) -> a__U23(a__isNat(V1:S)) 1.91/1.94 a__U22(X1:S,X2:S) -> U22(X1:S,X2:S) 1.91/1.94 a__U23(tt) -> tt 1.91/1.94 a__U23(X:S) -> U23(X:S) 1.91/1.94 a__U31(tt,V2:S) -> a__U32(a__isNatKind(V2:S)) 1.91/1.94 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 1.91/1.94 a__U32(tt) -> tt 1.91/1.94 a__U32(X:S) -> U32(X:S) 1.91/1.94 a__U41(tt) -> tt 1.91/1.94 a__U41(X:S) -> U41(X:S) 1.91/1.94 a__U51(tt,N:S) -> a__U52(a__isNatKind(N:S),N:S) 1.91/1.94 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.91/1.94 a__U52(tt,N:S) -> mark(N:S) 1.91/1.94 a__U52(X1:S,X2:S) -> U52(X1:S,X2:S) 1.91/1.94 a__U61(tt,M:S,N:S) -> a__U62(a__isNatKind(M:S),M:S,N:S) 1.91/1.94 a__U61(X1:S,X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) 1.91/1.94 a__U62(tt,M:S,N:S) -> a__U63(a__isNat(N:S),M:S,N:S) 1.91/1.94 a__U62(X1:S,X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) 1.91/1.94 a__U63(tt,M:S,N:S) -> a__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.94 a__U63(X1:S,X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) 1.91/1.94 a__U64(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 1.91/1.94 a__U64(X1:S,X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) 1.91/1.94 a__isNat(0) -> tt 1.91/1.94 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.94 a__isNat(s(V1:S)) -> a__U21(a__isNatKind(V1:S),V1:S) 1.91/1.94 a__isNat(X:S) -> isNat(X:S) 1.91/1.94 a__isNatKind(0) -> tt 1.91/1.94 a__isNatKind(plus(V1:S,V2:S)) -> a__U31(a__isNatKind(V1:S),V2:S) 1.91/1.94 a__isNatKind(s(V1:S)) -> a__U41(a__isNatKind(V1:S)) 1.91/1.94 a__isNatKind(X:S) -> isNatKind(X:S) 1.91/1.94 a__plus(N:S,0) -> a__U51(a__isNat(N:S),N:S) 1.91/1.94 a__plus(N:S,s(M:S)) -> a__U61(a__isNat(M:S),M:S,N:S) 1.91/1.94 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 1.91/1.94 mark(0) -> 0 1.91/1.94 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U13(X1:S,X2:S,X3:S)) -> a__U13(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U14(X1:S,X2:S,X3:S)) -> a__U14(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U15(X1:S,X2:S)) -> a__U15(mark(X1:S),X2:S) 1.91/1.94 mark(U16(X:S)) -> a__U16(mark(X:S)) 1.91/1.94 mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) 1.91/1.94 mark(U22(X1:S,X2:S)) -> a__U22(mark(X1:S),X2:S) 1.91/1.94 mark(U23(X:S)) -> a__U23(mark(X:S)) 1.91/1.94 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 1.91/1.94 mark(U32(X:S)) -> a__U32(mark(X:S)) 1.91/1.94 mark(U41(X:S)) -> a__U41(mark(X:S)) 1.91/1.94 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.91/1.94 mark(U52(X1:S,X2:S)) -> a__U52(mark(X1:S),X2:S) 1.91/1.94 mark(U61(X1:S,X2:S,X3:S)) -> a__U61(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U62(X1:S,X2:S,X3:S)) -> a__U62(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U63(X1:S,X2:S,X3:S)) -> a__U63(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U64(X1:S,X2:S,X3:S)) -> a__U64(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(isNat(X:S)) -> a__isNat(X:S) 1.91/1.94 mark(isNatKind(X:S)) -> a__isNatKind(X:S) 1.91/1.94 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 1.91/1.94 mark(s(X:S)) -> s(mark(X:S)) 1.91/1.94 mark(tt) -> tt 1.91/1.94 -> Usable rules: 1.91/1.94 a__U11(tt,V1:S,V2:S) -> a__U12(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.94 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 1.91/1.94 a__U12(tt,V1:S,V2:S) -> a__U13(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.94 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 1.91/1.94 a__U13(tt,V1:S,V2:S) -> a__U14(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.94 a__U13(X1:S,X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) 1.91/1.94 a__U14(tt,V1:S,V2:S) -> a__U15(a__isNat(V1:S),V2:S) 1.91/1.94 a__U14(X1:S,X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) 1.91/1.94 a__U15(tt,V2:S) -> a__U16(a__isNat(V2:S)) 1.91/1.94 a__U15(X1:S,X2:S) -> U15(X1:S,X2:S) 1.91/1.94 a__U16(tt) -> tt 1.91/1.94 a__U16(X:S) -> U16(X:S) 1.91/1.94 a__U21(tt,V1:S) -> a__U22(a__isNatKind(V1:S),V1:S) 1.91/1.94 a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) 1.91/1.94 a__U22(tt,V1:S) -> a__U23(a__isNat(V1:S)) 1.91/1.94 a__U22(X1:S,X2:S) -> U22(X1:S,X2:S) 1.91/1.94 a__U23(tt) -> tt 1.91/1.94 a__U23(X:S) -> U23(X:S) 1.91/1.94 a__U31(tt,V2:S) -> a__U32(a__isNatKind(V2:S)) 1.91/1.94 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 1.91/1.94 a__U32(tt) -> tt 1.91/1.94 a__U32(X:S) -> U32(X:S) 1.91/1.94 a__U41(tt) -> tt 1.91/1.94 a__U41(X:S) -> U41(X:S) 1.91/1.94 a__U51(tt,N:S) -> a__U52(a__isNatKind(N:S),N:S) 1.91/1.94 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.91/1.94 a__U52(tt,N:S) -> mark(N:S) 1.91/1.94 a__U52(X1:S,X2:S) -> U52(X1:S,X2:S) 1.91/1.94 a__U61(tt,M:S,N:S) -> a__U62(a__isNatKind(M:S),M:S,N:S) 1.91/1.94 a__U61(X1:S,X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) 1.91/1.94 a__U62(tt,M:S,N:S) -> a__U63(a__isNat(N:S),M:S,N:S) 1.91/1.94 a__U62(X1:S,X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) 1.91/1.94 a__U63(tt,M:S,N:S) -> a__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.94 a__U63(X1:S,X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) 1.91/1.94 a__U64(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 1.91/1.94 a__U64(X1:S,X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) 1.91/1.94 a__isNat(0) -> tt 1.91/1.94 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.94 a__isNat(s(V1:S)) -> a__U21(a__isNatKind(V1:S),V1:S) 1.91/1.94 a__isNat(X:S) -> isNat(X:S) 1.91/1.94 a__isNatKind(0) -> tt 1.91/1.94 a__isNatKind(plus(V1:S,V2:S)) -> a__U31(a__isNatKind(V1:S),V2:S) 1.91/1.94 a__isNatKind(s(V1:S)) -> a__U41(a__isNatKind(V1:S)) 1.91/1.94 a__isNatKind(X:S) -> isNatKind(X:S) 1.91/1.94 a__plus(N:S,0) -> a__U51(a__isNat(N:S),N:S) 1.91/1.94 a__plus(N:S,s(M:S)) -> a__U61(a__isNat(M:S),M:S,N:S) 1.91/1.94 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 1.91/1.94 mark(0) -> 0 1.91/1.94 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U13(X1:S,X2:S,X3:S)) -> a__U13(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U14(X1:S,X2:S,X3:S)) -> a__U14(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U15(X1:S,X2:S)) -> a__U15(mark(X1:S),X2:S) 1.91/1.94 mark(U16(X:S)) -> a__U16(mark(X:S)) 1.91/1.94 mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) 1.91/1.94 mark(U22(X1:S,X2:S)) -> a__U22(mark(X1:S),X2:S) 1.91/1.94 mark(U23(X:S)) -> a__U23(mark(X:S)) 1.91/1.94 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 1.91/1.94 mark(U32(X:S)) -> a__U32(mark(X:S)) 1.91/1.94 mark(U41(X:S)) -> a__U41(mark(X:S)) 1.91/1.94 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.91/1.94 mark(U52(X1:S,X2:S)) -> a__U52(mark(X1:S),X2:S) 1.91/1.94 mark(U61(X1:S,X2:S,X3:S)) -> a__U61(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U62(X1:S,X2:S,X3:S)) -> a__U62(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U63(X1:S,X2:S,X3:S)) -> a__U63(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U64(X1:S,X2:S,X3:S)) -> a__U64(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(isNat(X:S)) -> a__isNat(X:S) 1.91/1.94 mark(isNatKind(X:S)) -> a__isNatKind(X:S) 1.91/1.94 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 1.91/1.94 mark(s(X:S)) -> s(mark(X:S)) 1.91/1.94 mark(tt) -> tt 1.91/1.94 ->Interpretation type: 1.91/1.94 Linear 1.91/1.94 ->Coefficients: 1.91/1.94 Natural Numbers 1.91/1.94 ->Dimension: 1.91/1.94 1 1.91/1.94 ->Bound: 1.91/1.94 2 1.91/1.94 ->Interpretation: 1.91/1.94 1.91/1.94 [a__U11](X1,X2,X3) = X1 1.91/1.94 [a__U12](X1,X2,X3) = X1 1.91/1.94 [a__U13](X1,X2,X3) = X1 1.91/1.94 [a__U14](X1,X2,X3) = X1 1.91/1.94 [a__U15](X1,X2) = X1 1.91/1.94 [a__U16](X) = X 1.91/1.94 [a__U21](X1,X2) = X1 1.91/1.94 [a__U22](X1,X2) = X1 1.91/1.94 [a__U23](X) = X 1.91/1.94 [a__U31](X1,X2) = X1 1.91/1.94 [a__U32](X) = X 1.91/1.94 [a__U41](X) = X 1.91/1.94 [a__U51](X1,X2) = 2.X1 + 2.X2 + 2 1.91/1.94 [a__U52](X1,X2) = 2.X1 + 2.X2 + 2 1.91/1.94 [a__U61](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 1.91/1.94 [a__U62](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 1.91/1.94 [a__U63](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 1.91/1.94 [a__U64](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 1.91/1.94 [a__isNat](X) = 2 1.91/1.94 [a__isNatKind](X) = 2 1.91/1.94 [a__plus](X1,X2) = 2.X1 + 2.X2 + 2 1.91/1.94 [mark](X) = X 1.91/1.94 [0] = 2 1.91/1.94 [U11](X1,X2,X3) = X1 1.91/1.94 [U12](X1,X2,X3) = X1 1.91/1.94 [U13](X1,X2,X3) = X1 1.91/1.94 [U14](X1,X2,X3) = X1 1.91/1.94 [U15](X1,X2) = X1 1.91/1.94 [U16](X) = X 1.91/1.94 [U21](X1,X2) = X1 1.91/1.94 [U22](X1,X2) = X1 1.91/1.94 [U23](X) = X 1.91/1.94 [U31](X1,X2) = X1 1.91/1.94 [U32](X) = X 1.91/1.94 [U41](X) = X 1.91/1.94 [U51](X1,X2) = 2.X1 + 2.X2 + 2 1.91/1.94 [U52](X1,X2) = 2.X1 + 2.X2 + 2 1.91/1.94 [U61](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 1.91/1.94 [U62](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 1.91/1.94 [U63](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 1.91/1.94 [U64](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 1.91/1.94 [fSNonEmpty] = 0 1.91/1.94 [isNat](X) = 2 1.91/1.94 [isNatKind](X) = 2 1.91/1.94 [plus](X1,X2) = 2.X1 + 2.X2 + 2 1.91/1.94 [s](X) = X + 2 1.91/1.94 [tt] = 2 1.91/1.94 [A__U11](X1,X2,X3) = 0 1.91/1.94 [A__U12](X1,X2,X3) = 0 1.91/1.94 [A__U13](X1,X2,X3) = 0 1.91/1.94 [A__U14](X1,X2,X3) = 0 1.91/1.94 [A__U15](X1,X2) = 0 1.91/1.94 [A__U16](X) = 0 1.91/1.94 [A__U21](X1,X2) = 0 1.91/1.94 [A__U22](X1,X2) = 0 1.91/1.94 [A__U23](X) = 0 1.91/1.94 [A__U31](X1,X2) = 0 1.91/1.94 [A__U32](X) = 0 1.91/1.94 [A__U41](X) = 0 1.91/1.94 [A__U51](X1,X2) = 0 1.91/1.94 [A__U52](X1,X2) = 0 1.91/1.94 [A__U61](X1,X2,X3) = 0 1.91/1.94 [A__U62](X1,X2,X3) = 0 1.91/1.94 [A__U63](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 1.91/1.94 [A__U64](X1,X2,X3) = 2.X2 + 2.X3 + 2 1.91/1.94 [A__ISNAT](X) = 0 1.91/1.94 [A__ISNATKIND](X) = 0 1.91/1.94 [A__PLUS](X1,X2) = 0 1.91/1.94 [MARK](X) = 2.X + 2 1.91/1.94 1.91/1.94 Problem 1.3: 1.91/1.94 1.91/1.94 SCC Processor: 1.91/1.94 -> Pairs: 1.91/1.94 A__U64(tt,M:S,N:S) -> MARK(M:S) 1.91/1.94 A__U64(tt,M:S,N:S) -> MARK(N:S) 1.91/1.94 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(U15(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.94 MARK(U16(X:S)) -> MARK(X:S) 1.91/1.94 MARK(U21(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.94 MARK(U22(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.94 MARK(U23(X:S)) -> MARK(X:S) 1.91/1.94 MARK(U31(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.94 MARK(U32(X:S)) -> MARK(X:S) 1.91/1.94 MARK(U41(X:S)) -> MARK(X:S) 1.91/1.94 MARK(U51(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.94 MARK(U52(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.94 MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(U63(X1:S,X2:S,X3:S)) -> A__U63(mark(X1:S),X2:S,X3:S) 1.91/1.94 MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(U64(X1:S,X2:S,X3:S)) -> A__U64(mark(X1:S),X2:S,X3:S) 1.91/1.94 MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.94 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.91/1.94 MARK(s(X:S)) -> MARK(X:S) 1.91/1.94 -> Rules: 1.91/1.94 a__U11(tt,V1:S,V2:S) -> a__U12(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.94 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 1.91/1.94 a__U12(tt,V1:S,V2:S) -> a__U13(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.94 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 1.91/1.94 a__U13(tt,V1:S,V2:S) -> a__U14(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.94 a__U13(X1:S,X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) 1.91/1.94 a__U14(tt,V1:S,V2:S) -> a__U15(a__isNat(V1:S),V2:S) 1.91/1.94 a__U14(X1:S,X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) 1.91/1.94 a__U15(tt,V2:S) -> a__U16(a__isNat(V2:S)) 1.91/1.94 a__U15(X1:S,X2:S) -> U15(X1:S,X2:S) 1.91/1.94 a__U16(tt) -> tt 1.91/1.94 a__U16(X:S) -> U16(X:S) 1.91/1.94 a__U21(tt,V1:S) -> a__U22(a__isNatKind(V1:S),V1:S) 1.91/1.94 a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) 1.91/1.94 a__U22(tt,V1:S) -> a__U23(a__isNat(V1:S)) 1.91/1.94 a__U22(X1:S,X2:S) -> U22(X1:S,X2:S) 1.91/1.94 a__U23(tt) -> tt 1.91/1.94 a__U23(X:S) -> U23(X:S) 1.91/1.94 a__U31(tt,V2:S) -> a__U32(a__isNatKind(V2:S)) 1.91/1.94 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 1.91/1.94 a__U32(tt) -> tt 1.91/1.94 a__U32(X:S) -> U32(X:S) 1.91/1.94 a__U41(tt) -> tt 1.91/1.94 a__U41(X:S) -> U41(X:S) 1.91/1.94 a__U51(tt,N:S) -> a__U52(a__isNatKind(N:S),N:S) 1.91/1.94 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.91/1.94 a__U52(tt,N:S) -> mark(N:S) 1.91/1.94 a__U52(X1:S,X2:S) -> U52(X1:S,X2:S) 1.91/1.94 a__U61(tt,M:S,N:S) -> a__U62(a__isNatKind(M:S),M:S,N:S) 1.91/1.94 a__U61(X1:S,X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) 1.91/1.94 a__U62(tt,M:S,N:S) -> a__U63(a__isNat(N:S),M:S,N:S) 1.91/1.94 a__U62(X1:S,X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) 1.91/1.94 a__U63(tt,M:S,N:S) -> a__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.94 a__U63(X1:S,X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) 1.91/1.94 a__U64(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 1.91/1.94 a__U64(X1:S,X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) 1.91/1.94 a__isNat(0) -> tt 1.91/1.94 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.94 a__isNat(s(V1:S)) -> a__U21(a__isNatKind(V1:S),V1:S) 1.91/1.94 a__isNat(X:S) -> isNat(X:S) 1.91/1.94 a__isNatKind(0) -> tt 1.91/1.94 a__isNatKind(plus(V1:S,V2:S)) -> a__U31(a__isNatKind(V1:S),V2:S) 1.91/1.94 a__isNatKind(s(V1:S)) -> a__U41(a__isNatKind(V1:S)) 1.91/1.94 a__isNatKind(X:S) -> isNatKind(X:S) 1.91/1.94 a__plus(N:S,0) -> a__U51(a__isNat(N:S),N:S) 1.91/1.94 a__plus(N:S,s(M:S)) -> a__U61(a__isNat(M:S),M:S,N:S) 1.91/1.94 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 1.91/1.94 mark(0) -> 0 1.91/1.94 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U13(X1:S,X2:S,X3:S)) -> a__U13(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U14(X1:S,X2:S,X3:S)) -> a__U14(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U15(X1:S,X2:S)) -> a__U15(mark(X1:S),X2:S) 1.91/1.94 mark(U16(X:S)) -> a__U16(mark(X:S)) 1.91/1.94 mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) 1.91/1.94 mark(U22(X1:S,X2:S)) -> a__U22(mark(X1:S),X2:S) 1.91/1.94 mark(U23(X:S)) -> a__U23(mark(X:S)) 1.91/1.94 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 1.91/1.94 mark(U32(X:S)) -> a__U32(mark(X:S)) 1.91/1.94 mark(U41(X:S)) -> a__U41(mark(X:S)) 1.91/1.94 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.91/1.94 mark(U52(X1:S,X2:S)) -> a__U52(mark(X1:S),X2:S) 1.91/1.94 mark(U61(X1:S,X2:S,X3:S)) -> a__U61(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U62(X1:S,X2:S,X3:S)) -> a__U62(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U63(X1:S,X2:S,X3:S)) -> a__U63(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U64(X1:S,X2:S,X3:S)) -> a__U64(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(isNat(X:S)) -> a__isNat(X:S) 1.91/1.94 mark(isNatKind(X:S)) -> a__isNatKind(X:S) 1.91/1.94 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 1.91/1.94 mark(s(X:S)) -> s(mark(X:S)) 1.91/1.94 mark(tt) -> tt 1.91/1.94 ->Strongly Connected Components: 1.91/1.94 ->->Cycle: 1.91/1.94 ->->-> Pairs: 1.91/1.94 A__U64(tt,M:S,N:S) -> MARK(M:S) 1.91/1.94 A__U64(tt,M:S,N:S) -> MARK(N:S) 1.91/1.94 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(U15(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.94 MARK(U16(X:S)) -> MARK(X:S) 1.91/1.94 MARK(U21(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.94 MARK(U22(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.94 MARK(U23(X:S)) -> MARK(X:S) 1.91/1.94 MARK(U31(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.94 MARK(U32(X:S)) -> MARK(X:S) 1.91/1.94 MARK(U41(X:S)) -> MARK(X:S) 1.91/1.94 MARK(U51(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.94 MARK(U52(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.94 MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(U64(X1:S,X2:S,X3:S)) -> A__U64(mark(X1:S),X2:S,X3:S) 1.91/1.94 MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.94 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.91/1.94 MARK(s(X:S)) -> MARK(X:S) 1.91/1.94 ->->-> Rules: 1.91/1.94 a__U11(tt,V1:S,V2:S) -> a__U12(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.94 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 1.91/1.94 a__U12(tt,V1:S,V2:S) -> a__U13(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.94 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 1.91/1.94 a__U13(tt,V1:S,V2:S) -> a__U14(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.94 a__U13(X1:S,X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) 1.91/1.94 a__U14(tt,V1:S,V2:S) -> a__U15(a__isNat(V1:S),V2:S) 1.91/1.94 a__U14(X1:S,X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) 1.91/1.94 a__U15(tt,V2:S) -> a__U16(a__isNat(V2:S)) 1.91/1.94 a__U15(X1:S,X2:S) -> U15(X1:S,X2:S) 1.91/1.94 a__U16(tt) -> tt 1.91/1.94 a__U16(X:S) -> U16(X:S) 1.91/1.94 a__U21(tt,V1:S) -> a__U22(a__isNatKind(V1:S),V1:S) 1.91/1.94 a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) 1.91/1.94 a__U22(tt,V1:S) -> a__U23(a__isNat(V1:S)) 1.91/1.94 a__U22(X1:S,X2:S) -> U22(X1:S,X2:S) 1.91/1.94 a__U23(tt) -> tt 1.91/1.94 a__U23(X:S) -> U23(X:S) 1.91/1.94 a__U31(tt,V2:S) -> a__U32(a__isNatKind(V2:S)) 1.91/1.94 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 1.91/1.94 a__U32(tt) -> tt 1.91/1.94 a__U32(X:S) -> U32(X:S) 1.91/1.94 a__U41(tt) -> tt 1.91/1.94 a__U41(X:S) -> U41(X:S) 1.91/1.94 a__U51(tt,N:S) -> a__U52(a__isNatKind(N:S),N:S) 1.91/1.94 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.91/1.94 a__U52(tt,N:S) -> mark(N:S) 1.91/1.94 a__U52(X1:S,X2:S) -> U52(X1:S,X2:S) 1.91/1.94 a__U61(tt,M:S,N:S) -> a__U62(a__isNatKind(M:S),M:S,N:S) 1.91/1.94 a__U61(X1:S,X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) 1.91/1.94 a__U62(tt,M:S,N:S) -> a__U63(a__isNat(N:S),M:S,N:S) 1.91/1.94 a__U62(X1:S,X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) 1.91/1.94 a__U63(tt,M:S,N:S) -> a__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.94 a__U63(X1:S,X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) 1.91/1.94 a__U64(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 1.91/1.94 a__U64(X1:S,X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) 1.91/1.94 a__isNat(0) -> tt 1.91/1.94 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.94 a__isNat(s(V1:S)) -> a__U21(a__isNatKind(V1:S),V1:S) 1.91/1.94 a__isNat(X:S) -> isNat(X:S) 1.91/1.94 a__isNatKind(0) -> tt 1.91/1.94 a__isNatKind(plus(V1:S,V2:S)) -> a__U31(a__isNatKind(V1:S),V2:S) 1.91/1.94 a__isNatKind(s(V1:S)) -> a__U41(a__isNatKind(V1:S)) 1.91/1.94 a__isNatKind(X:S) -> isNatKind(X:S) 1.91/1.94 a__plus(N:S,0) -> a__U51(a__isNat(N:S),N:S) 1.91/1.94 a__plus(N:S,s(M:S)) -> a__U61(a__isNat(M:S),M:S,N:S) 1.91/1.94 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 1.91/1.94 mark(0) -> 0 1.91/1.94 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U13(X1:S,X2:S,X3:S)) -> a__U13(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U14(X1:S,X2:S,X3:S)) -> a__U14(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U15(X1:S,X2:S)) -> a__U15(mark(X1:S),X2:S) 1.91/1.94 mark(U16(X:S)) -> a__U16(mark(X:S)) 1.91/1.94 mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) 1.91/1.94 mark(U22(X1:S,X2:S)) -> a__U22(mark(X1:S),X2:S) 1.91/1.94 mark(U23(X:S)) -> a__U23(mark(X:S)) 1.91/1.94 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 1.91/1.94 mark(U32(X:S)) -> a__U32(mark(X:S)) 1.91/1.94 mark(U41(X:S)) -> a__U41(mark(X:S)) 1.91/1.94 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.91/1.94 mark(U52(X1:S,X2:S)) -> a__U52(mark(X1:S),X2:S) 1.91/1.94 mark(U61(X1:S,X2:S,X3:S)) -> a__U61(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U62(X1:S,X2:S,X3:S)) -> a__U62(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U63(X1:S,X2:S,X3:S)) -> a__U63(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U64(X1:S,X2:S,X3:S)) -> a__U64(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(isNat(X:S)) -> a__isNat(X:S) 1.91/1.94 mark(isNatKind(X:S)) -> a__isNatKind(X:S) 1.91/1.94 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 1.91/1.94 mark(s(X:S)) -> s(mark(X:S)) 1.91/1.94 mark(tt) -> tt 1.91/1.94 1.91/1.94 Problem 1.3: 1.91/1.94 1.91/1.94 Reduction Pairs Processor: 1.91/1.94 -> Pairs: 1.91/1.94 A__U64(tt,M:S,N:S) -> MARK(M:S) 1.91/1.94 A__U64(tt,M:S,N:S) -> MARK(N:S) 1.91/1.94 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(U15(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.94 MARK(U16(X:S)) -> MARK(X:S) 1.91/1.94 MARK(U21(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.94 MARK(U22(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.94 MARK(U23(X:S)) -> MARK(X:S) 1.91/1.94 MARK(U31(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.94 MARK(U32(X:S)) -> MARK(X:S) 1.91/1.94 MARK(U41(X:S)) -> MARK(X:S) 1.91/1.94 MARK(U51(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.94 MARK(U52(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.94 MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(U64(X1:S,X2:S,X3:S)) -> A__U64(mark(X1:S),X2:S,X3:S) 1.91/1.94 MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.94 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.91/1.94 MARK(s(X:S)) -> MARK(X:S) 1.91/1.94 -> Rules: 1.91/1.94 a__U11(tt,V1:S,V2:S) -> a__U12(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.94 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 1.91/1.94 a__U12(tt,V1:S,V2:S) -> a__U13(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.94 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 1.91/1.94 a__U13(tt,V1:S,V2:S) -> a__U14(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.94 a__U13(X1:S,X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) 1.91/1.94 a__U14(tt,V1:S,V2:S) -> a__U15(a__isNat(V1:S),V2:S) 1.91/1.94 a__U14(X1:S,X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) 1.91/1.94 a__U15(tt,V2:S) -> a__U16(a__isNat(V2:S)) 1.91/1.94 a__U15(X1:S,X2:S) -> U15(X1:S,X2:S) 1.91/1.94 a__U16(tt) -> tt 1.91/1.94 a__U16(X:S) -> U16(X:S) 1.91/1.94 a__U21(tt,V1:S) -> a__U22(a__isNatKind(V1:S),V1:S) 1.91/1.94 a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) 1.91/1.94 a__U22(tt,V1:S) -> a__U23(a__isNat(V1:S)) 1.91/1.94 a__U22(X1:S,X2:S) -> U22(X1:S,X2:S) 1.91/1.94 a__U23(tt) -> tt 1.91/1.94 a__U23(X:S) -> U23(X:S) 1.91/1.94 a__U31(tt,V2:S) -> a__U32(a__isNatKind(V2:S)) 1.91/1.94 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 1.91/1.94 a__U32(tt) -> tt 1.91/1.94 a__U32(X:S) -> U32(X:S) 1.91/1.94 a__U41(tt) -> tt 1.91/1.94 a__U41(X:S) -> U41(X:S) 1.91/1.94 a__U51(tt,N:S) -> a__U52(a__isNatKind(N:S),N:S) 1.91/1.94 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.91/1.94 a__U52(tt,N:S) -> mark(N:S) 1.91/1.94 a__U52(X1:S,X2:S) -> U52(X1:S,X2:S) 1.91/1.94 a__U61(tt,M:S,N:S) -> a__U62(a__isNatKind(M:S),M:S,N:S) 1.91/1.94 a__U61(X1:S,X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) 1.91/1.94 a__U62(tt,M:S,N:S) -> a__U63(a__isNat(N:S),M:S,N:S) 1.91/1.94 a__U62(X1:S,X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) 1.91/1.94 a__U63(tt,M:S,N:S) -> a__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.94 a__U63(X1:S,X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) 1.91/1.94 a__U64(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 1.91/1.94 a__U64(X1:S,X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) 1.91/1.94 a__isNat(0) -> tt 1.91/1.94 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.94 a__isNat(s(V1:S)) -> a__U21(a__isNatKind(V1:S),V1:S) 1.91/1.94 a__isNat(X:S) -> isNat(X:S) 1.91/1.94 a__isNatKind(0) -> tt 1.91/1.94 a__isNatKind(plus(V1:S,V2:S)) -> a__U31(a__isNatKind(V1:S),V2:S) 1.91/1.94 a__isNatKind(s(V1:S)) -> a__U41(a__isNatKind(V1:S)) 1.91/1.94 a__isNatKind(X:S) -> isNatKind(X:S) 1.91/1.94 a__plus(N:S,0) -> a__U51(a__isNat(N:S),N:S) 1.91/1.94 a__plus(N:S,s(M:S)) -> a__U61(a__isNat(M:S),M:S,N:S) 1.91/1.94 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 1.91/1.94 mark(0) -> 0 1.91/1.94 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U13(X1:S,X2:S,X3:S)) -> a__U13(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U14(X1:S,X2:S,X3:S)) -> a__U14(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U15(X1:S,X2:S)) -> a__U15(mark(X1:S),X2:S) 1.91/1.94 mark(U16(X:S)) -> a__U16(mark(X:S)) 1.91/1.94 mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) 1.91/1.94 mark(U22(X1:S,X2:S)) -> a__U22(mark(X1:S),X2:S) 1.91/1.94 mark(U23(X:S)) -> a__U23(mark(X:S)) 1.91/1.94 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 1.91/1.94 mark(U32(X:S)) -> a__U32(mark(X:S)) 1.91/1.94 mark(U41(X:S)) -> a__U41(mark(X:S)) 1.91/1.94 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.91/1.94 mark(U52(X1:S,X2:S)) -> a__U52(mark(X1:S),X2:S) 1.91/1.94 mark(U61(X1:S,X2:S,X3:S)) -> a__U61(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U62(X1:S,X2:S,X3:S)) -> a__U62(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U63(X1:S,X2:S,X3:S)) -> a__U63(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U64(X1:S,X2:S,X3:S)) -> a__U64(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(isNat(X:S)) -> a__isNat(X:S) 1.91/1.94 mark(isNatKind(X:S)) -> a__isNatKind(X:S) 1.91/1.94 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 1.91/1.94 mark(s(X:S)) -> s(mark(X:S)) 1.91/1.94 mark(tt) -> tt 1.91/1.94 -> Usable rules: 1.91/1.94 a__U11(tt,V1:S,V2:S) -> a__U12(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.94 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 1.91/1.94 a__U12(tt,V1:S,V2:S) -> a__U13(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.94 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 1.91/1.94 a__U13(tt,V1:S,V2:S) -> a__U14(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.94 a__U13(X1:S,X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) 1.91/1.94 a__U14(tt,V1:S,V2:S) -> a__U15(a__isNat(V1:S),V2:S) 1.91/1.94 a__U14(X1:S,X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) 1.91/1.94 a__U15(tt,V2:S) -> a__U16(a__isNat(V2:S)) 1.91/1.94 a__U15(X1:S,X2:S) -> U15(X1:S,X2:S) 1.91/1.94 a__U16(tt) -> tt 1.91/1.94 a__U16(X:S) -> U16(X:S) 1.91/1.94 a__U21(tt,V1:S) -> a__U22(a__isNatKind(V1:S),V1:S) 1.91/1.94 a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) 1.91/1.94 a__U22(tt,V1:S) -> a__U23(a__isNat(V1:S)) 1.91/1.94 a__U22(X1:S,X2:S) -> U22(X1:S,X2:S) 1.91/1.94 a__U23(tt) -> tt 1.91/1.94 a__U23(X:S) -> U23(X:S) 1.91/1.94 a__U31(tt,V2:S) -> a__U32(a__isNatKind(V2:S)) 1.91/1.94 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 1.91/1.94 a__U32(tt) -> tt 1.91/1.94 a__U32(X:S) -> U32(X:S) 1.91/1.94 a__U41(tt) -> tt 1.91/1.94 a__U41(X:S) -> U41(X:S) 1.91/1.94 a__U51(tt,N:S) -> a__U52(a__isNatKind(N:S),N:S) 1.91/1.94 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.91/1.94 a__U52(tt,N:S) -> mark(N:S) 1.91/1.94 a__U52(X1:S,X2:S) -> U52(X1:S,X2:S) 1.91/1.94 a__U61(tt,M:S,N:S) -> a__U62(a__isNatKind(M:S),M:S,N:S) 1.91/1.94 a__U61(X1:S,X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) 1.91/1.94 a__U62(tt,M:S,N:S) -> a__U63(a__isNat(N:S),M:S,N:S) 1.91/1.94 a__U62(X1:S,X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) 1.91/1.94 a__U63(tt,M:S,N:S) -> a__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.94 a__U63(X1:S,X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) 1.91/1.94 a__U64(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 1.91/1.94 a__U64(X1:S,X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) 1.91/1.94 a__isNat(0) -> tt 1.91/1.94 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.94 a__isNat(s(V1:S)) -> a__U21(a__isNatKind(V1:S),V1:S) 1.91/1.94 a__isNat(X:S) -> isNat(X:S) 1.91/1.94 a__isNatKind(0) -> tt 1.91/1.94 a__isNatKind(plus(V1:S,V2:S)) -> a__U31(a__isNatKind(V1:S),V2:S) 1.91/1.94 a__isNatKind(s(V1:S)) -> a__U41(a__isNatKind(V1:S)) 1.91/1.94 a__isNatKind(X:S) -> isNatKind(X:S) 1.91/1.94 a__plus(N:S,0) -> a__U51(a__isNat(N:S),N:S) 1.91/1.94 a__plus(N:S,s(M:S)) -> a__U61(a__isNat(M:S),M:S,N:S) 1.91/1.94 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 1.91/1.94 mark(0) -> 0 1.91/1.94 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U13(X1:S,X2:S,X3:S)) -> a__U13(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U14(X1:S,X2:S,X3:S)) -> a__U14(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U15(X1:S,X2:S)) -> a__U15(mark(X1:S),X2:S) 1.91/1.94 mark(U16(X:S)) -> a__U16(mark(X:S)) 1.91/1.94 mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) 1.91/1.94 mark(U22(X1:S,X2:S)) -> a__U22(mark(X1:S),X2:S) 1.91/1.94 mark(U23(X:S)) -> a__U23(mark(X:S)) 1.91/1.94 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 1.91/1.94 mark(U32(X:S)) -> a__U32(mark(X:S)) 1.91/1.94 mark(U41(X:S)) -> a__U41(mark(X:S)) 1.91/1.94 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.91/1.94 mark(U52(X1:S,X2:S)) -> a__U52(mark(X1:S),X2:S) 1.91/1.94 mark(U61(X1:S,X2:S,X3:S)) -> a__U61(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U62(X1:S,X2:S,X3:S)) -> a__U62(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U63(X1:S,X2:S,X3:S)) -> a__U63(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U64(X1:S,X2:S,X3:S)) -> a__U64(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(isNat(X:S)) -> a__isNat(X:S) 1.91/1.94 mark(isNatKind(X:S)) -> a__isNatKind(X:S) 1.91/1.94 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 1.91/1.94 mark(s(X:S)) -> s(mark(X:S)) 1.91/1.94 mark(tt) -> tt 1.91/1.94 ->Interpretation type: 1.91/1.94 Linear 1.91/1.94 ->Coefficients: 1.91/1.94 Natural Numbers 1.91/1.94 ->Dimension: 1.91/1.94 1 1.91/1.94 ->Bound: 1.91/1.94 2 1.91/1.94 ->Interpretation: 1.91/1.94 1.91/1.94 [a__U11](X1,X2,X3) = X1 1.91/1.94 [a__U12](X1,X2,X3) = X1 1.91/1.94 [a__U13](X1,X2,X3) = X1 1.91/1.94 [a__U14](X1,X2,X3) = X1 1.91/1.94 [a__U15](X1,X2) = X1 1.91/1.94 [a__U16](X) = X 1.91/1.94 [a__U21](X1,X2) = X1 1.91/1.94 [a__U22](X1,X2) = X1 1.91/1.94 [a__U23](X) = X 1.91/1.94 [a__U31](X1,X2) = X1 1.91/1.94 [a__U32](X) = X 1.91/1.94 [a__U41](X) = X 1.91/1.94 [a__U51](X1,X2) = 2.X1 + X2 + 2 1.91/1.94 [a__U52](X1,X2) = 2.X1 + X2 + 2 1.91/1.94 [a__U61](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 1.91/1.94 [a__U62](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 1.91/1.94 [a__U63](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 1.91/1.94 [a__U64](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 1.91/1.94 [a__isNat](X) = 2 1.91/1.94 [a__isNatKind](X) = 2 1.91/1.94 [a__plus](X1,X2) = 2.X1 + 2.X2 + 2 1.91/1.94 [mark](X) = X 1.91/1.94 [0] = 2 1.91/1.94 [U11](X1,X2,X3) = X1 1.91/1.94 [U12](X1,X2,X3) = X1 1.91/1.94 [U13](X1,X2,X3) = X1 1.91/1.94 [U14](X1,X2,X3) = X1 1.91/1.94 [U15](X1,X2) = X1 1.91/1.94 [U16](X) = X 1.91/1.94 [U21](X1,X2) = X1 1.91/1.94 [U22](X1,X2) = X1 1.91/1.94 [U23](X) = X 1.91/1.94 [U31](X1,X2) = X1 1.91/1.94 [U32](X) = X 1.91/1.94 [U41](X) = X 1.91/1.94 [U51](X1,X2) = 2.X1 + X2 + 2 1.91/1.94 [U52](X1,X2) = 2.X1 + X2 + 2 1.91/1.94 [U61](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 1.91/1.94 [U62](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 1.91/1.94 [U63](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 1.91/1.94 [U64](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 1.91/1.94 [fSNonEmpty] = 0 1.91/1.94 [isNat](X) = 2 1.91/1.94 [isNatKind](X) = 2 1.91/1.94 [plus](X1,X2) = 2.X1 + 2.X2 + 2 1.91/1.94 [s](X) = X + 2 1.91/1.94 [tt] = 2 1.91/1.94 [A__U11](X1,X2,X3) = 0 1.91/1.94 [A__U12](X1,X2,X3) = 0 1.91/1.94 [A__U13](X1,X2,X3) = 0 1.91/1.94 [A__U14](X1,X2,X3) = 0 1.91/1.94 [A__U15](X1,X2) = 0 1.91/1.94 [A__U16](X) = 0 1.91/1.94 [A__U21](X1,X2) = 0 1.91/1.94 [A__U22](X1,X2) = 0 1.91/1.94 [A__U23](X) = 0 1.91/1.94 [A__U31](X1,X2) = 0 1.91/1.94 [A__U32](X) = 0 1.91/1.94 [A__U41](X) = 0 1.91/1.94 [A__U51](X1,X2) = 0 1.91/1.94 [A__U52](X1,X2) = 0 1.91/1.94 [A__U61](X1,X2,X3) = 0 1.91/1.94 [A__U62](X1,X2,X3) = 0 1.91/1.94 [A__U63](X1,X2,X3) = 0 1.91/1.94 [A__U64](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 1.91/1.94 [A__ISNAT](X) = 0 1.91/1.94 [A__ISNATKIND](X) = 0 1.91/1.94 [A__PLUS](X1,X2) = 0 1.91/1.94 [MARK](X) = 2.X 1.91/1.94 1.91/1.94 Problem 1.3: 1.91/1.94 1.91/1.94 SCC Processor: 1.91/1.94 -> Pairs: 1.91/1.94 A__U64(tt,M:S,N:S) -> MARK(N:S) 1.91/1.94 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(U15(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.94 MARK(U16(X:S)) -> MARK(X:S) 1.91/1.94 MARK(U21(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.94 MARK(U22(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.94 MARK(U23(X:S)) -> MARK(X:S) 1.91/1.94 MARK(U31(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.94 MARK(U32(X:S)) -> MARK(X:S) 1.91/1.94 MARK(U41(X:S)) -> MARK(X:S) 1.91/1.94 MARK(U51(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.94 MARK(U52(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.94 MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(U64(X1:S,X2:S,X3:S)) -> A__U64(mark(X1:S),X2:S,X3:S) 1.91/1.94 MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.94 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.91/1.94 MARK(s(X:S)) -> MARK(X:S) 1.91/1.94 -> Rules: 1.91/1.94 a__U11(tt,V1:S,V2:S) -> a__U12(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.94 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 1.91/1.94 a__U12(tt,V1:S,V2:S) -> a__U13(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.94 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 1.91/1.94 a__U13(tt,V1:S,V2:S) -> a__U14(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.94 a__U13(X1:S,X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) 1.91/1.94 a__U14(tt,V1:S,V2:S) -> a__U15(a__isNat(V1:S),V2:S) 1.91/1.94 a__U14(X1:S,X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) 1.91/1.94 a__U15(tt,V2:S) -> a__U16(a__isNat(V2:S)) 1.91/1.94 a__U15(X1:S,X2:S) -> U15(X1:S,X2:S) 1.91/1.94 a__U16(tt) -> tt 1.91/1.94 a__U16(X:S) -> U16(X:S) 1.91/1.94 a__U21(tt,V1:S) -> a__U22(a__isNatKind(V1:S),V1:S) 1.91/1.94 a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) 1.91/1.94 a__U22(tt,V1:S) -> a__U23(a__isNat(V1:S)) 1.91/1.94 a__U22(X1:S,X2:S) -> U22(X1:S,X2:S) 1.91/1.94 a__U23(tt) -> tt 1.91/1.94 a__U23(X:S) -> U23(X:S) 1.91/1.94 a__U31(tt,V2:S) -> a__U32(a__isNatKind(V2:S)) 1.91/1.94 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 1.91/1.94 a__U32(tt) -> tt 1.91/1.94 a__U32(X:S) -> U32(X:S) 1.91/1.94 a__U41(tt) -> tt 1.91/1.94 a__U41(X:S) -> U41(X:S) 1.91/1.94 a__U51(tt,N:S) -> a__U52(a__isNatKind(N:S),N:S) 1.91/1.94 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.91/1.94 a__U52(tt,N:S) -> mark(N:S) 1.91/1.94 a__U52(X1:S,X2:S) -> U52(X1:S,X2:S) 1.91/1.94 a__U61(tt,M:S,N:S) -> a__U62(a__isNatKind(M:S),M:S,N:S) 1.91/1.94 a__U61(X1:S,X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) 1.91/1.94 a__U62(tt,M:S,N:S) -> a__U63(a__isNat(N:S),M:S,N:S) 1.91/1.94 a__U62(X1:S,X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) 1.91/1.94 a__U63(tt,M:S,N:S) -> a__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.94 a__U63(X1:S,X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) 1.91/1.94 a__U64(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 1.91/1.94 a__U64(X1:S,X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) 1.91/1.94 a__isNat(0) -> tt 1.91/1.94 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.94 a__isNat(s(V1:S)) -> a__U21(a__isNatKind(V1:S),V1:S) 1.91/1.94 a__isNat(X:S) -> isNat(X:S) 1.91/1.94 a__isNatKind(0) -> tt 1.91/1.94 a__isNatKind(plus(V1:S,V2:S)) -> a__U31(a__isNatKind(V1:S),V2:S) 1.91/1.94 a__isNatKind(s(V1:S)) -> a__U41(a__isNatKind(V1:S)) 1.91/1.94 a__isNatKind(X:S) -> isNatKind(X:S) 1.91/1.94 a__plus(N:S,0) -> a__U51(a__isNat(N:S),N:S) 1.91/1.94 a__plus(N:S,s(M:S)) -> a__U61(a__isNat(M:S),M:S,N:S) 1.91/1.94 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 1.91/1.94 mark(0) -> 0 1.91/1.94 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U13(X1:S,X2:S,X3:S)) -> a__U13(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U14(X1:S,X2:S,X3:S)) -> a__U14(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U15(X1:S,X2:S)) -> a__U15(mark(X1:S),X2:S) 1.91/1.94 mark(U16(X:S)) -> a__U16(mark(X:S)) 1.91/1.94 mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) 1.91/1.94 mark(U22(X1:S,X2:S)) -> a__U22(mark(X1:S),X2:S) 1.91/1.94 mark(U23(X:S)) -> a__U23(mark(X:S)) 1.91/1.94 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 1.91/1.94 mark(U32(X:S)) -> a__U32(mark(X:S)) 1.91/1.94 mark(U41(X:S)) -> a__U41(mark(X:S)) 1.91/1.94 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.91/1.94 mark(U52(X1:S,X2:S)) -> a__U52(mark(X1:S),X2:S) 1.91/1.94 mark(U61(X1:S,X2:S,X3:S)) -> a__U61(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U62(X1:S,X2:S,X3:S)) -> a__U62(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U63(X1:S,X2:S,X3:S)) -> a__U63(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U64(X1:S,X2:S,X3:S)) -> a__U64(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(isNat(X:S)) -> a__isNat(X:S) 1.91/1.94 mark(isNatKind(X:S)) -> a__isNatKind(X:S) 1.91/1.94 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 1.91/1.94 mark(s(X:S)) -> s(mark(X:S)) 1.91/1.94 mark(tt) -> tt 1.91/1.94 ->Strongly Connected Components: 1.91/1.94 ->->Cycle: 1.91/1.94 ->->-> Pairs: 1.91/1.94 A__U64(tt,M:S,N:S) -> MARK(N:S) 1.91/1.94 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(U15(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.94 MARK(U16(X:S)) -> MARK(X:S) 1.91/1.94 MARK(U21(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.94 MARK(U22(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.94 MARK(U23(X:S)) -> MARK(X:S) 1.91/1.94 MARK(U31(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.94 MARK(U32(X:S)) -> MARK(X:S) 1.91/1.94 MARK(U41(X:S)) -> MARK(X:S) 1.91/1.94 MARK(U51(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.94 MARK(U52(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.94 MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(U64(X1:S,X2:S,X3:S)) -> A__U64(mark(X1:S),X2:S,X3:S) 1.91/1.94 MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.94 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.91/1.94 MARK(s(X:S)) -> MARK(X:S) 1.91/1.94 ->->-> Rules: 1.91/1.94 a__U11(tt,V1:S,V2:S) -> a__U12(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.94 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 1.91/1.94 a__U12(tt,V1:S,V2:S) -> a__U13(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.94 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 1.91/1.94 a__U13(tt,V1:S,V2:S) -> a__U14(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.94 a__U13(X1:S,X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) 1.91/1.94 a__U14(tt,V1:S,V2:S) -> a__U15(a__isNat(V1:S),V2:S) 1.91/1.94 a__U14(X1:S,X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) 1.91/1.94 a__U15(tt,V2:S) -> a__U16(a__isNat(V2:S)) 1.91/1.94 a__U15(X1:S,X2:S) -> U15(X1:S,X2:S) 1.91/1.94 a__U16(tt) -> tt 1.91/1.94 a__U16(X:S) -> U16(X:S) 1.91/1.94 a__U21(tt,V1:S) -> a__U22(a__isNatKind(V1:S),V1:S) 1.91/1.94 a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) 1.91/1.94 a__U22(tt,V1:S) -> a__U23(a__isNat(V1:S)) 1.91/1.94 a__U22(X1:S,X2:S) -> U22(X1:S,X2:S) 1.91/1.94 a__U23(tt) -> tt 1.91/1.94 a__U23(X:S) -> U23(X:S) 1.91/1.94 a__U31(tt,V2:S) -> a__U32(a__isNatKind(V2:S)) 1.91/1.94 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 1.91/1.94 a__U32(tt) -> tt 1.91/1.94 a__U32(X:S) -> U32(X:S) 1.91/1.94 a__U41(tt) -> tt 1.91/1.94 a__U41(X:S) -> U41(X:S) 1.91/1.94 a__U51(tt,N:S) -> a__U52(a__isNatKind(N:S),N:S) 1.91/1.94 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.91/1.94 a__U52(tt,N:S) -> mark(N:S) 1.91/1.94 a__U52(X1:S,X2:S) -> U52(X1:S,X2:S) 1.91/1.94 a__U61(tt,M:S,N:S) -> a__U62(a__isNatKind(M:S),M:S,N:S) 1.91/1.94 a__U61(X1:S,X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) 1.91/1.94 a__U62(tt,M:S,N:S) -> a__U63(a__isNat(N:S),M:S,N:S) 1.91/1.94 a__U62(X1:S,X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) 1.91/1.94 a__U63(tt,M:S,N:S) -> a__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.94 a__U63(X1:S,X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) 1.91/1.94 a__U64(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 1.91/1.94 a__U64(X1:S,X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) 1.91/1.94 a__isNat(0) -> tt 1.91/1.94 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.94 a__isNat(s(V1:S)) -> a__U21(a__isNatKind(V1:S),V1:S) 1.91/1.94 a__isNat(X:S) -> isNat(X:S) 1.91/1.94 a__isNatKind(0) -> tt 1.91/1.94 a__isNatKind(plus(V1:S,V2:S)) -> a__U31(a__isNatKind(V1:S),V2:S) 1.91/1.94 a__isNatKind(s(V1:S)) -> a__U41(a__isNatKind(V1:S)) 1.91/1.94 a__isNatKind(X:S) -> isNatKind(X:S) 1.91/1.94 a__plus(N:S,0) -> a__U51(a__isNat(N:S),N:S) 1.91/1.94 a__plus(N:S,s(M:S)) -> a__U61(a__isNat(M:S),M:S,N:S) 1.91/1.94 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 1.91/1.94 mark(0) -> 0 1.91/1.94 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U13(X1:S,X2:S,X3:S)) -> a__U13(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U14(X1:S,X2:S,X3:S)) -> a__U14(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U15(X1:S,X2:S)) -> a__U15(mark(X1:S),X2:S) 1.91/1.94 mark(U16(X:S)) -> a__U16(mark(X:S)) 1.91/1.94 mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) 1.91/1.94 mark(U22(X1:S,X2:S)) -> a__U22(mark(X1:S),X2:S) 1.91/1.94 mark(U23(X:S)) -> a__U23(mark(X:S)) 1.91/1.94 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 1.91/1.94 mark(U32(X:S)) -> a__U32(mark(X:S)) 1.91/1.94 mark(U41(X:S)) -> a__U41(mark(X:S)) 1.91/1.94 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.91/1.94 mark(U52(X1:S,X2:S)) -> a__U52(mark(X1:S),X2:S) 1.91/1.94 mark(U61(X1:S,X2:S,X3:S)) -> a__U61(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U62(X1:S,X2:S,X3:S)) -> a__U62(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U63(X1:S,X2:S,X3:S)) -> a__U63(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U64(X1:S,X2:S,X3:S)) -> a__U64(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(isNat(X:S)) -> a__isNat(X:S) 1.91/1.94 mark(isNatKind(X:S)) -> a__isNatKind(X:S) 1.91/1.94 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 1.91/1.94 mark(s(X:S)) -> s(mark(X:S)) 1.91/1.94 mark(tt) -> tt 1.91/1.94 1.91/1.94 Problem 1.3: 1.91/1.94 1.91/1.94 Subterm Processor: 1.91/1.94 -> Pairs: 1.91/1.94 A__U64(tt,M:S,N:S) -> MARK(N:S) 1.91/1.94 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(U12(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(U13(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(U14(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(U15(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.94 MARK(U16(X:S)) -> MARK(X:S) 1.91/1.94 MARK(U21(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.94 MARK(U22(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.94 MARK(U23(X:S)) -> MARK(X:S) 1.91/1.94 MARK(U31(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.94 MARK(U32(X:S)) -> MARK(X:S) 1.91/1.94 MARK(U41(X:S)) -> MARK(X:S) 1.91/1.94 MARK(U51(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.94 MARK(U52(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.94 MARK(U61(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(U62(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(U63(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(U64(X1:S,X2:S,X3:S)) -> A__U64(mark(X1:S),X2:S,X3:S) 1.91/1.94 MARK(U64(X1:S,X2:S,X3:S)) -> MARK(X1:S) 1.91/1.94 MARK(plus(X1:S,X2:S)) -> MARK(X1:S) 1.91/1.94 MARK(plus(X1:S,X2:S)) -> MARK(X2:S) 1.91/1.94 MARK(s(X:S)) -> MARK(X:S) 1.91/1.94 -> Rules: 1.91/1.94 a__U11(tt,V1:S,V2:S) -> a__U12(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.94 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 1.91/1.94 a__U12(tt,V1:S,V2:S) -> a__U13(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.94 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 1.91/1.94 a__U13(tt,V1:S,V2:S) -> a__U14(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.94 a__U13(X1:S,X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) 1.91/1.94 a__U14(tt,V1:S,V2:S) -> a__U15(a__isNat(V1:S),V2:S) 1.91/1.94 a__U14(X1:S,X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) 1.91/1.94 a__U15(tt,V2:S) -> a__U16(a__isNat(V2:S)) 1.91/1.94 a__U15(X1:S,X2:S) -> U15(X1:S,X2:S) 1.91/1.94 a__U16(tt) -> tt 1.91/1.94 a__U16(X:S) -> U16(X:S) 1.91/1.94 a__U21(tt,V1:S) -> a__U22(a__isNatKind(V1:S),V1:S) 1.91/1.94 a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) 1.91/1.94 a__U22(tt,V1:S) -> a__U23(a__isNat(V1:S)) 1.91/1.94 a__U22(X1:S,X2:S) -> U22(X1:S,X2:S) 1.91/1.94 a__U23(tt) -> tt 1.91/1.94 a__U23(X:S) -> U23(X:S) 1.91/1.94 a__U31(tt,V2:S) -> a__U32(a__isNatKind(V2:S)) 1.91/1.94 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 1.91/1.94 a__U32(tt) -> tt 1.91/1.94 a__U32(X:S) -> U32(X:S) 1.91/1.94 a__U41(tt) -> tt 1.91/1.94 a__U41(X:S) -> U41(X:S) 1.91/1.94 a__U51(tt,N:S) -> a__U52(a__isNatKind(N:S),N:S) 1.91/1.94 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.91/1.94 a__U52(tt,N:S) -> mark(N:S) 1.91/1.94 a__U52(X1:S,X2:S) -> U52(X1:S,X2:S) 1.91/1.94 a__U61(tt,M:S,N:S) -> a__U62(a__isNatKind(M:S),M:S,N:S) 1.91/1.94 a__U61(X1:S,X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) 1.91/1.94 a__U62(tt,M:S,N:S) -> a__U63(a__isNat(N:S),M:S,N:S) 1.91/1.94 a__U62(X1:S,X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) 1.91/1.94 a__U63(tt,M:S,N:S) -> a__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.94 a__U63(X1:S,X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) 1.91/1.94 a__U64(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 1.91/1.94 a__U64(X1:S,X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) 1.91/1.94 a__isNat(0) -> tt 1.91/1.94 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.94 a__isNat(s(V1:S)) -> a__U21(a__isNatKind(V1:S),V1:S) 1.91/1.94 a__isNat(X:S) -> isNat(X:S) 1.91/1.94 a__isNatKind(0) -> tt 1.91/1.94 a__isNatKind(plus(V1:S,V2:S)) -> a__U31(a__isNatKind(V1:S),V2:S) 1.91/1.94 a__isNatKind(s(V1:S)) -> a__U41(a__isNatKind(V1:S)) 1.91/1.94 a__isNatKind(X:S) -> isNatKind(X:S) 1.91/1.94 a__plus(N:S,0) -> a__U51(a__isNat(N:S),N:S) 1.91/1.94 a__plus(N:S,s(M:S)) -> a__U61(a__isNat(M:S),M:S,N:S) 1.91/1.94 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 1.91/1.94 mark(0) -> 0 1.91/1.94 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U13(X1:S,X2:S,X3:S)) -> a__U13(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U14(X1:S,X2:S,X3:S)) -> a__U14(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U15(X1:S,X2:S)) -> a__U15(mark(X1:S),X2:S) 1.91/1.94 mark(U16(X:S)) -> a__U16(mark(X:S)) 1.91/1.94 mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) 1.91/1.94 mark(U22(X1:S,X2:S)) -> a__U22(mark(X1:S),X2:S) 1.91/1.94 mark(U23(X:S)) -> a__U23(mark(X:S)) 1.91/1.94 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 1.91/1.94 mark(U32(X:S)) -> a__U32(mark(X:S)) 1.91/1.94 mark(U41(X:S)) -> a__U41(mark(X:S)) 1.91/1.94 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.91/1.94 mark(U52(X1:S,X2:S)) -> a__U52(mark(X1:S),X2:S) 1.91/1.94 mark(U61(X1:S,X2:S,X3:S)) -> a__U61(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U62(X1:S,X2:S,X3:S)) -> a__U62(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U63(X1:S,X2:S,X3:S)) -> a__U63(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U64(X1:S,X2:S,X3:S)) -> a__U64(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(isNat(X:S)) -> a__isNat(X:S) 1.91/1.94 mark(isNatKind(X:S)) -> a__isNatKind(X:S) 1.91/1.94 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 1.91/1.94 mark(s(X:S)) -> s(mark(X:S)) 1.91/1.94 mark(tt) -> tt 1.91/1.94 ->Projection: 1.91/1.94 pi(A__U64) = 3 1.91/1.94 pi(MARK) = 1 1.91/1.94 1.91/1.94 Problem 1.3: 1.91/1.94 1.91/1.94 SCC Processor: 1.91/1.94 -> Pairs: 1.91/1.94 A__U64(tt,M:S,N:S) -> MARK(N:S) 1.91/1.94 -> Rules: 1.91/1.94 a__U11(tt,V1:S,V2:S) -> a__U12(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.94 a__U11(X1:S,X2:S,X3:S) -> U11(X1:S,X2:S,X3:S) 1.91/1.94 a__U12(tt,V1:S,V2:S) -> a__U13(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.94 a__U12(X1:S,X2:S,X3:S) -> U12(X1:S,X2:S,X3:S) 1.91/1.94 a__U13(tt,V1:S,V2:S) -> a__U14(a__isNatKind(V2:S),V1:S,V2:S) 1.91/1.94 a__U13(X1:S,X2:S,X3:S) -> U13(X1:S,X2:S,X3:S) 1.91/1.94 a__U14(tt,V1:S,V2:S) -> a__U15(a__isNat(V1:S),V2:S) 1.91/1.94 a__U14(X1:S,X2:S,X3:S) -> U14(X1:S,X2:S,X3:S) 1.91/1.94 a__U15(tt,V2:S) -> a__U16(a__isNat(V2:S)) 1.91/1.94 a__U15(X1:S,X2:S) -> U15(X1:S,X2:S) 1.91/1.94 a__U16(tt) -> tt 1.91/1.94 a__U16(X:S) -> U16(X:S) 1.91/1.94 a__U21(tt,V1:S) -> a__U22(a__isNatKind(V1:S),V1:S) 1.91/1.94 a__U21(X1:S,X2:S) -> U21(X1:S,X2:S) 1.91/1.94 a__U22(tt,V1:S) -> a__U23(a__isNat(V1:S)) 1.91/1.94 a__U22(X1:S,X2:S) -> U22(X1:S,X2:S) 1.91/1.94 a__U23(tt) -> tt 1.91/1.94 a__U23(X:S) -> U23(X:S) 1.91/1.94 a__U31(tt,V2:S) -> a__U32(a__isNatKind(V2:S)) 1.91/1.94 a__U31(X1:S,X2:S) -> U31(X1:S,X2:S) 1.91/1.94 a__U32(tt) -> tt 1.91/1.94 a__U32(X:S) -> U32(X:S) 1.91/1.94 a__U41(tt) -> tt 1.91/1.94 a__U41(X:S) -> U41(X:S) 1.91/1.94 a__U51(tt,N:S) -> a__U52(a__isNatKind(N:S),N:S) 1.91/1.94 a__U51(X1:S,X2:S) -> U51(X1:S,X2:S) 1.91/1.94 a__U52(tt,N:S) -> mark(N:S) 1.91/1.94 a__U52(X1:S,X2:S) -> U52(X1:S,X2:S) 1.91/1.94 a__U61(tt,M:S,N:S) -> a__U62(a__isNatKind(M:S),M:S,N:S) 1.91/1.94 a__U61(X1:S,X2:S,X3:S) -> U61(X1:S,X2:S,X3:S) 1.91/1.94 a__U62(tt,M:S,N:S) -> a__U63(a__isNat(N:S),M:S,N:S) 1.91/1.94 a__U62(X1:S,X2:S,X3:S) -> U62(X1:S,X2:S,X3:S) 1.91/1.94 a__U63(tt,M:S,N:S) -> a__U64(a__isNatKind(N:S),M:S,N:S) 1.91/1.94 a__U63(X1:S,X2:S,X3:S) -> U63(X1:S,X2:S,X3:S) 1.91/1.94 a__U64(tt,M:S,N:S) -> s(a__plus(mark(N:S),mark(M:S))) 1.91/1.94 a__U64(X1:S,X2:S,X3:S) -> U64(X1:S,X2:S,X3:S) 1.91/1.94 a__isNat(0) -> tt 1.91/1.94 a__isNat(plus(V1:S,V2:S)) -> a__U11(a__isNatKind(V1:S),V1:S,V2:S) 1.91/1.94 a__isNat(s(V1:S)) -> a__U21(a__isNatKind(V1:S),V1:S) 1.91/1.94 a__isNat(X:S) -> isNat(X:S) 1.91/1.94 a__isNatKind(0) -> tt 1.91/1.94 a__isNatKind(plus(V1:S,V2:S)) -> a__U31(a__isNatKind(V1:S),V2:S) 1.91/1.94 a__isNatKind(s(V1:S)) -> a__U41(a__isNatKind(V1:S)) 1.91/1.94 a__isNatKind(X:S) -> isNatKind(X:S) 1.91/1.94 a__plus(N:S,0) -> a__U51(a__isNat(N:S),N:S) 1.91/1.94 a__plus(N:S,s(M:S)) -> a__U61(a__isNat(M:S),M:S,N:S) 1.91/1.94 a__plus(X1:S,X2:S) -> plus(X1:S,X2:S) 1.91/1.94 mark(0) -> 0 1.91/1.94 mark(U11(X1:S,X2:S,X3:S)) -> a__U11(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U12(X1:S,X2:S,X3:S)) -> a__U12(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U13(X1:S,X2:S,X3:S)) -> a__U13(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U14(X1:S,X2:S,X3:S)) -> a__U14(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U15(X1:S,X2:S)) -> a__U15(mark(X1:S),X2:S) 1.91/1.94 mark(U16(X:S)) -> a__U16(mark(X:S)) 1.91/1.94 mark(U21(X1:S,X2:S)) -> a__U21(mark(X1:S),X2:S) 1.91/1.94 mark(U22(X1:S,X2:S)) -> a__U22(mark(X1:S),X2:S) 1.91/1.94 mark(U23(X:S)) -> a__U23(mark(X:S)) 1.91/1.94 mark(U31(X1:S,X2:S)) -> a__U31(mark(X1:S),X2:S) 1.91/1.94 mark(U32(X:S)) -> a__U32(mark(X:S)) 1.91/1.94 mark(U41(X:S)) -> a__U41(mark(X:S)) 1.91/1.94 mark(U51(X1:S,X2:S)) -> a__U51(mark(X1:S),X2:S) 1.91/1.94 mark(U52(X1:S,X2:S)) -> a__U52(mark(X1:S),X2:S) 1.91/1.94 mark(U61(X1:S,X2:S,X3:S)) -> a__U61(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U62(X1:S,X2:S,X3:S)) -> a__U62(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U63(X1:S,X2:S,X3:S)) -> a__U63(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(U64(X1:S,X2:S,X3:S)) -> a__U64(mark(X1:S),X2:S,X3:S) 1.91/1.94 mark(isNat(X:S)) -> a__isNat(X:S) 1.91/1.94 mark(isNatKind(X:S)) -> a__isNatKind(X:S) 1.91/1.94 mark(plus(X1:S,X2:S)) -> a__plus(mark(X1:S),mark(X2:S)) 1.91/1.94 mark(s(X:S)) -> s(mark(X:S)) 1.91/1.94 mark(tt) -> tt 1.91/1.94 ->Strongly Connected Components: 1.91/1.94 There is no strongly connected component 1.91/1.94 1.91/1.94 The problem is finite. 1.91/1.94 EOF