/export/starexec/sandbox2/solver/bin/starexec_run_ttt2-1.17+nonreach /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES Problem: U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X Proof: DP Processor: DPs: U11#(tt(),V1,V2) -> activate#(V2) U11#(tt(),V1,V2) -> activate#(V1) U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) U12#(tt(),V1,V2) -> activate#(V1) U12#(tt(),V1,V2) -> activate#(V2) U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) U13#(tt(),V1,V2) -> activate#(V1) U13#(tt(),V1,V2) -> activate#(V2) U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) U14#(tt(),V1,V2) -> activate#(V2) U14#(tt(),V1,V2) -> activate#(V1) U14#(tt(),V1,V2) -> isNat#(activate(V1)) U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)) U15#(tt(),V2) -> activate#(V2) U15#(tt(),V2) -> isNat#(activate(V2)) U15#(tt(),V2) -> U16#(isNat(activate(V2))) U21#(tt(),V1) -> activate#(V1) U21#(tt(),V1) -> isNatKind#(activate(V1)) U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)) U22#(tt(),V1) -> activate#(V1) U22#(tt(),V1) -> isNat#(activate(V1)) U22#(tt(),V1) -> U23#(isNat(activate(V1))) U31#(tt(),V2) -> activate#(V2) U31#(tt(),V2) -> isNatKind#(activate(V2)) U31#(tt(),V2) -> U32#(isNatKind(activate(V2))) U51#(tt(),N) -> activate#(N) U51#(tt(),N) -> isNatKind#(activate(N)) U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N)) U52#(tt(),N) -> activate#(N) U61#(tt(),M,N) -> activate#(N) U61#(tt(),M,N) -> activate#(M) U61#(tt(),M,N) -> isNatKind#(activate(M)) U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N)) U62#(tt(),M,N) -> activate#(M) U62#(tt(),M,N) -> activate#(N) U62#(tt(),M,N) -> isNat#(activate(N)) U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)) U63#(tt(),M,N) -> activate#(M) U63#(tt(),M,N) -> activate#(N) U63#(tt(),M,N) -> isNatKind#(activate(N)) U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) U64#(tt(),M,N) -> activate#(M) U64#(tt(),M,N) -> activate#(N) U64#(tt(),M,N) -> plus#(activate(N),activate(M)) U64#(tt(),M,N) -> s#(plus(activate(N),activate(M))) isNat#(n__plus(V1,V2)) -> activate#(V2) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) isNatKind#(n__plus(V1,V2)) -> activate#(V2) isNatKind#(n__plus(V1,V2)) -> activate#(V1) isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)) isNatKind#(n__s(V1)) -> activate#(V1) isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) isNatKind#(n__s(V1)) -> U41#(isNatKind(activate(V1))) plus#(N,0()) -> isNat#(N) plus#(N,0()) -> U51#(isNat(N),N) plus#(N,s(M)) -> isNat#(M) plus#(N,s(M)) -> U61#(isNat(M),M,N) activate#(n__0()) -> 0#() activate#(n__plus(X1,X2)) -> activate#(X2) activate#(n__plus(X1,X2)) -> activate#(X1) activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) activate#(n__s(X)) -> activate#(X) activate#(n__s(X)) -> s#(activate(X)) TRS: U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X TDG Processor: DPs: U11#(tt(),V1,V2) -> activate#(V2) U11#(tt(),V1,V2) -> activate#(V1) U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) U12#(tt(),V1,V2) -> activate#(V1) U12#(tt(),V1,V2) -> activate#(V2) U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) U13#(tt(),V1,V2) -> activate#(V1) U13#(tt(),V1,V2) -> activate#(V2) U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) U14#(tt(),V1,V2) -> activate#(V2) U14#(tt(),V1,V2) -> activate#(V1) U14#(tt(),V1,V2) -> isNat#(activate(V1)) U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)) U15#(tt(),V2) -> activate#(V2) U15#(tt(),V2) -> isNat#(activate(V2)) U15#(tt(),V2) -> U16#(isNat(activate(V2))) U21#(tt(),V1) -> activate#(V1) U21#(tt(),V1) -> isNatKind#(activate(V1)) U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)) U22#(tt(),V1) -> activate#(V1) U22#(tt(),V1) -> isNat#(activate(V1)) U22#(tt(),V1) -> U23#(isNat(activate(V1))) U31#(tt(),V2) -> activate#(V2) U31#(tt(),V2) -> isNatKind#(activate(V2)) U31#(tt(),V2) -> U32#(isNatKind(activate(V2))) U51#(tt(),N) -> activate#(N) U51#(tt(),N) -> isNatKind#(activate(N)) U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N)) U52#(tt(),N) -> activate#(N) U61#(tt(),M,N) -> activate#(N) U61#(tt(),M,N) -> activate#(M) U61#(tt(),M,N) -> isNatKind#(activate(M)) U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N)) U62#(tt(),M,N) -> activate#(M) U62#(tt(),M,N) -> activate#(N) U62#(tt(),M,N) -> isNat#(activate(N)) U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)) U63#(tt(),M,N) -> activate#(M) U63#(tt(),M,N) -> activate#(N) U63#(tt(),M,N) -> isNatKind#(activate(N)) U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) U64#(tt(),M,N) -> activate#(M) U64#(tt(),M,N) -> activate#(N) U64#(tt(),M,N) -> plus#(activate(N),activate(M)) U64#(tt(),M,N) -> s#(plus(activate(N),activate(M))) isNat#(n__plus(V1,V2)) -> activate#(V2) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) isNatKind#(n__plus(V1,V2)) -> activate#(V2) isNatKind#(n__plus(V1,V2)) -> activate#(V1) isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)) isNatKind#(n__s(V1)) -> activate#(V1) isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) isNatKind#(n__s(V1)) -> U41#(isNatKind(activate(V1))) plus#(N,0()) -> isNat#(N) plus#(N,0()) -> U51#(isNat(N),N) plus#(N,s(M)) -> isNat#(M) plus#(N,s(M)) -> U61#(isNat(M),M,N) activate#(n__0()) -> 0#() activate#(n__plus(X1,X2)) -> activate#(X2) activate#(n__plus(X1,X2)) -> activate#(X1) activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) activate#(n__s(X)) -> activate#(X) activate#(n__s(X)) -> s#(activate(X)) TRS: U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X graph: plus#(N,0()) -> U51#(isNat(N),N) -> U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N)) plus#(N,0()) -> U51#(isNat(N),N) -> U51#(tt(),N) -> isNatKind#(activate(N)) plus#(N,0()) -> U51#(isNat(N),N) -> U51#(tt(),N) -> activate#(N) plus#(N,0()) -> isNat#(N) -> isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) plus#(N,0()) -> isNat#(N) -> isNat#(n__s(V1)) -> isNatKind#(activate(V1)) plus#(N,0()) -> isNat#(N) -> isNat#(n__s(V1)) -> activate#(V1) plus#(N,0()) -> isNat#(N) -> isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) plus#(N,0()) -> isNat#(N) -> isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) plus#(N,0()) -> isNat#(N) -> isNat#(n__plus(V1,V2)) -> activate#(V1) plus#(N,0()) -> isNat#(N) -> isNat#(n__plus(V1,V2)) -> activate#(V2) plus#(N,s(M)) -> U61#(isNat(M),M,N) -> U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N)) plus#(N,s(M)) -> U61#(isNat(M),M,N) -> U61#(tt(),M,N) -> isNatKind#(activate(M)) plus#(N,s(M)) -> U61#(isNat(M),M,N) -> U61#(tt(),M,N) -> activate#(M) plus#(N,s(M)) -> U61#(isNat(M),M,N) -> U61#(tt(),M,N) -> activate#(N) plus#(N,s(M)) -> isNat#(M) -> isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) plus#(N,s(M)) -> isNat#(M) -> isNat#(n__s(V1)) -> isNatKind#(activate(V1)) plus#(N,s(M)) -> isNat#(M) -> isNat#(n__s(V1)) -> activate#(V1) plus#(N,s(M)) -> isNat#(M) -> isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) plus#(N,s(M)) -> isNat#(M) -> isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) plus#(N,s(M)) -> isNat#(M) -> isNat#(n__plus(V1,V2)) -> activate#(V1) plus#(N,s(M)) -> isNat#(M) -> isNat#(n__plus(V1,V2)) -> activate#(V2) U64#(tt(),M,N) -> plus#(activate(N),activate(M)) -> plus#(N,s(M)) -> U61#(isNat(M),M,N) U64#(tt(),M,N) -> plus#(activate(N),activate(M)) -> plus#(N,s(M)) -> isNat#(M) U64#(tt(),M,N) -> plus#(activate(N),activate(M)) -> plus#(N,0()) -> U51#(isNat(N),N) U64#(tt(),M,N) -> plus#(activate(N),activate(M)) -> plus#(N,0()) -> isNat#(N) U64#(tt(),M,N) -> activate#(M) -> activate#(n__s(X)) -> s#(activate(X)) U64#(tt(),M,N) -> activate#(M) -> activate#(n__s(X)) -> activate#(X) U64#(tt(),M,N) -> activate#(M) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) U64#(tt(),M,N) -> activate#(M) -> activate#(n__plus(X1,X2)) -> activate#(X1) U64#(tt(),M,N) -> activate#(M) -> activate#(n__plus(X1,X2)) -> activate#(X2) U64#(tt(),M,N) -> activate#(M) -> activate#(n__0()) -> 0#() U64#(tt(),M,N) -> activate#(N) -> activate#(n__s(X)) -> s#(activate(X)) U64#(tt(),M,N) -> activate#(N) -> activate#(n__s(X)) -> activate#(X) U64#(tt(),M,N) -> activate#(N) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) U64#(tt(),M,N) -> activate#(N) -> activate#(n__plus(X1,X2)) -> activate#(X1) U64#(tt(),M,N) -> activate#(N) -> activate#(n__plus(X1,X2)) -> activate#(X2) U64#(tt(),M,N) -> activate#(N) -> activate#(n__0()) -> 0#() U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) -> U64#(tt(),M,N) -> s#(plus(activate(N),activate(M))) U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) -> U64#(tt(),M,N) -> plus#(activate(N),activate(M)) U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) -> U64#(tt(),M,N) -> activate#(N) U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) -> U64#(tt(),M,N) -> activate#(M) U63#(tt(),M,N) -> isNatKind#(activate(N)) -> isNatKind#(n__s(V1)) -> U41#(isNatKind(activate(V1))) U63#(tt(),M,N) -> isNatKind#(activate(N)) -> isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) U63#(tt(),M,N) -> isNatKind#(activate(N)) -> isNatKind#(n__s(V1)) -> activate#(V1) U63#(tt(),M,N) -> isNatKind#(activate(N)) -> isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)) U63#(tt(),M,N) -> isNatKind#(activate(N)) -> isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) U63#(tt(),M,N) -> isNatKind#(activate(N)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V1) U63#(tt(),M,N) -> isNatKind#(activate(N)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V2) U63#(tt(),M,N) -> activate#(M) -> activate#(n__s(X)) -> s#(activate(X)) U63#(tt(),M,N) -> activate#(M) -> activate#(n__s(X)) -> activate#(X) U63#(tt(),M,N) -> activate#(M) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) U63#(tt(),M,N) -> activate#(M) -> activate#(n__plus(X1,X2)) -> activate#(X1) U63#(tt(),M,N) -> activate#(M) -> activate#(n__plus(X1,X2)) -> activate#(X2) U63#(tt(),M,N) -> activate#(M) -> activate#(n__0()) -> 0#() U63#(tt(),M,N) -> activate#(N) -> activate#(n__s(X)) -> s#(activate(X)) U63#(tt(),M,N) -> activate#(N) -> activate#(n__s(X)) -> activate#(X) U63#(tt(),M,N) -> activate#(N) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) U63#(tt(),M,N) -> activate#(N) -> activate#(n__plus(X1,X2)) -> activate#(X1) U63#(tt(),M,N) -> activate#(N) -> activate#(n__plus(X1,X2)) -> activate#(X2) U63#(tt(),M,N) -> activate#(N) -> activate#(n__0()) -> 0#() U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)) -> U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)) -> U63#(tt(),M,N) -> isNatKind#(activate(N)) U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)) -> U63#(tt(),M,N) -> activate#(N) U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)) -> U63#(tt(),M,N) -> activate#(M) U62#(tt(),M,N) -> isNat#(activate(N)) -> isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) U62#(tt(),M,N) -> isNat#(activate(N)) -> isNat#(n__s(V1)) -> isNatKind#(activate(V1)) U62#(tt(),M,N) -> isNat#(activate(N)) -> isNat#(n__s(V1)) -> activate#(V1) U62#(tt(),M,N) -> isNat#(activate(N)) -> isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) U62#(tt(),M,N) -> isNat#(activate(N)) -> isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) U62#(tt(),M,N) -> isNat#(activate(N)) -> isNat#(n__plus(V1,V2)) -> activate#(V1) U62#(tt(),M,N) -> isNat#(activate(N)) -> isNat#(n__plus(V1,V2)) -> activate#(V2) U62#(tt(),M,N) -> activate#(M) -> activate#(n__s(X)) -> s#(activate(X)) U62#(tt(),M,N) -> activate#(M) -> activate#(n__s(X)) -> activate#(X) U62#(tt(),M,N) -> activate#(M) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) U62#(tt(),M,N) -> activate#(M) -> activate#(n__plus(X1,X2)) -> activate#(X1) U62#(tt(),M,N) -> activate#(M) -> activate#(n__plus(X1,X2)) -> activate#(X2) U62#(tt(),M,N) -> activate#(M) -> activate#(n__0()) -> 0#() U62#(tt(),M,N) -> activate#(N) -> activate#(n__s(X)) -> s#(activate(X)) U62#(tt(),M,N) -> activate#(N) -> activate#(n__s(X)) -> activate#(X) U62#(tt(),M,N) -> activate#(N) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) U62#(tt(),M,N) -> activate#(N) -> activate#(n__plus(X1,X2)) -> activate#(X1) U62#(tt(),M,N) -> activate#(N) -> activate#(n__plus(X1,X2)) -> activate#(X2) U62#(tt(),M,N) -> activate#(N) -> activate#(n__0()) -> 0#() U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N)) -> U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)) U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N)) -> U62#(tt(),M,N) -> isNat#(activate(N)) U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N)) -> U62#(tt(),M,N) -> activate#(N) U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N)) -> U62#(tt(),M,N) -> activate#(M) U61#(tt(),M,N) -> isNatKind#(activate(M)) -> isNatKind#(n__s(V1)) -> U41#(isNatKind(activate(V1))) U61#(tt(),M,N) -> isNatKind#(activate(M)) -> isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) U61#(tt(),M,N) -> isNatKind#(activate(M)) -> isNatKind#(n__s(V1)) -> activate#(V1) U61#(tt(),M,N) -> isNatKind#(activate(M)) -> isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)) U61#(tt(),M,N) -> isNatKind#(activate(M)) -> isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) U61#(tt(),M,N) -> isNatKind#(activate(M)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V1) U61#(tt(),M,N) -> isNatKind#(activate(M)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V2) U61#(tt(),M,N) -> activate#(M) -> activate#(n__s(X)) -> s#(activate(X)) U61#(tt(),M,N) -> activate#(M) -> activate#(n__s(X)) -> activate#(X) U61#(tt(),M,N) -> activate#(M) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) U61#(tt(),M,N) -> activate#(M) -> activate#(n__plus(X1,X2)) -> activate#(X1) U61#(tt(),M,N) -> activate#(M) -> activate#(n__plus(X1,X2)) -> activate#(X2) U61#(tt(),M,N) -> activate#(M) -> activate#(n__0()) -> 0#() U61#(tt(),M,N) -> activate#(N) -> activate#(n__s(X)) -> s#(activate(X)) U61#(tt(),M,N) -> activate#(N) -> activate#(n__s(X)) -> activate#(X) U61#(tt(),M,N) -> activate#(N) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) U61#(tt(),M,N) -> activate#(N) -> activate#(n__plus(X1,X2)) -> activate#(X1) U61#(tt(),M,N) -> activate#(N) -> activate#(n__plus(X1,X2)) -> activate#(X2) U61#(tt(),M,N) -> activate#(N) -> activate#(n__0()) -> 0#() U52#(tt(),N) -> activate#(N) -> activate#(n__s(X)) -> s#(activate(X)) U52#(tt(),N) -> activate#(N) -> activate#(n__s(X)) -> activate#(X) U52#(tt(),N) -> activate#(N) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) U52#(tt(),N) -> activate#(N) -> activate#(n__plus(X1,X2)) -> activate#(X1) U52#(tt(),N) -> activate#(N) -> activate#(n__plus(X1,X2)) -> activate#(X2) U52#(tt(),N) -> activate#(N) -> activate#(n__0()) -> 0#() U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N)) -> U52#(tt(),N) -> activate#(N) U51#(tt(),N) -> isNatKind#(activate(N)) -> isNatKind#(n__s(V1)) -> U41#(isNatKind(activate(V1))) U51#(tt(),N) -> isNatKind#(activate(N)) -> isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) U51#(tt(),N) -> isNatKind#(activate(N)) -> isNatKind#(n__s(V1)) -> activate#(V1) U51#(tt(),N) -> isNatKind#(activate(N)) -> isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)) U51#(tt(),N) -> isNatKind#(activate(N)) -> isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) U51#(tt(),N) -> isNatKind#(activate(N)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V1) U51#(tt(),N) -> isNatKind#(activate(N)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V2) U51#(tt(),N) -> activate#(N) -> activate#(n__s(X)) -> s#(activate(X)) U51#(tt(),N) -> activate#(N) -> activate#(n__s(X)) -> activate#(X) U51#(tt(),N) -> activate#(N) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) U51#(tt(),N) -> activate#(N) -> activate#(n__plus(X1,X2)) -> activate#(X1) U51#(tt(),N) -> activate#(N) -> activate#(n__plus(X1,X2)) -> activate#(X2) U51#(tt(),N) -> activate#(N) -> activate#(n__0()) -> 0#() U31#(tt(),V2) -> isNatKind#(activate(V2)) -> isNatKind#(n__s(V1)) -> U41#(isNatKind(activate(V1))) U31#(tt(),V2) -> isNatKind#(activate(V2)) -> isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) U31#(tt(),V2) -> isNatKind#(activate(V2)) -> isNatKind#(n__s(V1)) -> activate#(V1) U31#(tt(),V2) -> isNatKind#(activate(V2)) -> isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)) U31#(tt(),V2) -> isNatKind#(activate(V2)) -> isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) U31#(tt(),V2) -> isNatKind#(activate(V2)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V1) U31#(tt(),V2) -> isNatKind#(activate(V2)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V2) U31#(tt(),V2) -> activate#(V2) -> activate#(n__s(X)) -> s#(activate(X)) U31#(tt(),V2) -> activate#(V2) -> activate#(n__s(X)) -> activate#(X) U31#(tt(),V2) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) U31#(tt(),V2) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> activate#(X1) U31#(tt(),V2) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> activate#(X2) U31#(tt(),V2) -> activate#(V2) -> activate#(n__0()) -> 0#() U22#(tt(),V1) -> isNat#(activate(V1)) -> isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) U22#(tt(),V1) -> isNat#(activate(V1)) -> isNat#(n__s(V1)) -> isNatKind#(activate(V1)) U22#(tt(),V1) -> isNat#(activate(V1)) -> isNat#(n__s(V1)) -> activate#(V1) U22#(tt(),V1) -> isNat#(activate(V1)) -> isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) U22#(tt(),V1) -> isNat#(activate(V1)) -> isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) U22#(tt(),V1) -> isNat#(activate(V1)) -> isNat#(n__plus(V1,V2)) -> activate#(V1) U22#(tt(),V1) -> isNat#(activate(V1)) -> isNat#(n__plus(V1,V2)) -> activate#(V2) U22#(tt(),V1) -> activate#(V1) -> activate#(n__s(X)) -> s#(activate(X)) U22#(tt(),V1) -> activate#(V1) -> activate#(n__s(X)) -> activate#(X) U22#(tt(),V1) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) U22#(tt(),V1) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> activate#(X1) U22#(tt(),V1) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> activate#(X2) U22#(tt(),V1) -> activate#(V1) -> activate#(n__0()) -> 0#() U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)) -> U22#(tt(),V1) -> U23#(isNat(activate(V1))) U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)) -> U22#(tt(),V1) -> isNat#(activate(V1)) U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)) -> U22#(tt(),V1) -> activate#(V1) U21#(tt(),V1) -> isNatKind#(activate(V1)) -> isNatKind#(n__s(V1)) -> U41#(isNatKind(activate(V1))) U21#(tt(),V1) -> isNatKind#(activate(V1)) -> isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) U21#(tt(),V1) -> isNatKind#(activate(V1)) -> isNatKind#(n__s(V1)) -> activate#(V1) U21#(tt(),V1) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)) U21#(tt(),V1) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) U21#(tt(),V1) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V1) U21#(tt(),V1) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V2) U21#(tt(),V1) -> activate#(V1) -> activate#(n__s(X)) -> s#(activate(X)) U21#(tt(),V1) -> activate#(V1) -> activate#(n__s(X)) -> activate#(X) U21#(tt(),V1) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) U21#(tt(),V1) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> activate#(X1) U21#(tt(),V1) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> activate#(X2) U21#(tt(),V1) -> activate#(V1) -> activate#(n__0()) -> 0#() U15#(tt(),V2) -> isNat#(activate(V2)) -> isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) U15#(tt(),V2) -> isNat#(activate(V2)) -> isNat#(n__s(V1)) -> isNatKind#(activate(V1)) U15#(tt(),V2) -> isNat#(activate(V2)) -> isNat#(n__s(V1)) -> activate#(V1) U15#(tt(),V2) -> isNat#(activate(V2)) -> isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) U15#(tt(),V2) -> isNat#(activate(V2)) -> isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) U15#(tt(),V2) -> isNat#(activate(V2)) -> isNat#(n__plus(V1,V2)) -> activate#(V1) U15#(tt(),V2) -> isNat#(activate(V2)) -> isNat#(n__plus(V1,V2)) -> activate#(V2) U15#(tt(),V2) -> activate#(V2) -> activate#(n__s(X)) -> s#(activate(X)) U15#(tt(),V2) -> activate#(V2) -> activate#(n__s(X)) -> activate#(X) U15#(tt(),V2) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) U15#(tt(),V2) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> activate#(X1) U15#(tt(),V2) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> activate#(X2) U15#(tt(),V2) -> activate#(V2) -> activate#(n__0()) -> 0#() isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) -> U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)) isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) -> U21#(tt(),V1) -> isNatKind#(activate(V1)) isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) -> U21#(tt(),V1) -> activate#(V1) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) -> isNatKind#(n__s(V1)) -> U41#(isNatKind(activate(V1))) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) -> isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) -> isNatKind#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V2) isNat#(n__s(V1)) -> activate#(V1) -> activate#(n__s(X)) -> s#(activate(X)) isNat#(n__s(V1)) -> activate#(V1) -> activate#(n__s(X)) -> activate#(X) isNat#(n__s(V1)) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) isNat#(n__s(V1)) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> activate#(X1) isNat#(n__s(V1)) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> activate#(X2) isNat#(n__s(V1)) -> activate#(V1) -> activate#(n__0()) -> 0#() isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) -> isNatKind#(n__s(V1)) -> U41#(isNatKind(activate(V1))) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) -> isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) -> isNatKind#(n__s(V1)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V2) isNat#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__s(X)) -> s#(activate(X)) isNat#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__s(X)) -> activate#(X) isNat#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) isNat#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> activate#(X1) isNat#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> activate#(X2) isNat#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__0()) -> 0#() isNat#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__s(X)) -> s#(activate(X)) isNat#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__s(X)) -> activate#(X) isNat#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) isNat#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> activate#(X1) isNat#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> activate#(X2) isNat#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__0()) -> 0#() isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) -> U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) -> U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) -> U11#(tt(),V1,V2) -> activate#(V1) isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) -> U11#(tt(),V1,V2) -> activate#(V2) U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)) -> U15#(tt(),V2) -> U16#(isNat(activate(V2))) U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)) -> U15#(tt(),V2) -> isNat#(activate(V2)) U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)) -> U15#(tt(),V2) -> activate#(V2) U14#(tt(),V1,V2) -> isNat#(activate(V1)) -> isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) U14#(tt(),V1,V2) -> isNat#(activate(V1)) -> isNat#(n__s(V1)) -> isNatKind#(activate(V1)) U14#(tt(),V1,V2) -> isNat#(activate(V1)) -> isNat#(n__s(V1)) -> activate#(V1) U14#(tt(),V1,V2) -> isNat#(activate(V1)) -> isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) U14#(tt(),V1,V2) -> isNat#(activate(V1)) -> isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) U14#(tt(),V1,V2) -> isNat#(activate(V1)) -> isNat#(n__plus(V1,V2)) -> activate#(V1) U14#(tt(),V1,V2) -> isNat#(activate(V1)) -> isNat#(n__plus(V1,V2)) -> activate#(V2) U14#(tt(),V1,V2) -> activate#(V2) -> activate#(n__s(X)) -> s#(activate(X)) U14#(tt(),V1,V2) -> activate#(V2) -> activate#(n__s(X)) -> activate#(X) U14#(tt(),V1,V2) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) U14#(tt(),V1,V2) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> activate#(X1) U14#(tt(),V1,V2) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> activate#(X2) U14#(tt(),V1,V2) -> activate#(V2) -> activate#(n__0()) -> 0#() U14#(tt(),V1,V2) -> activate#(V1) -> activate#(n__s(X)) -> s#(activate(X)) U14#(tt(),V1,V2) -> activate#(V1) -> activate#(n__s(X)) -> activate#(X) U14#(tt(),V1,V2) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) U14#(tt(),V1,V2) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> activate#(X1) U14#(tt(),V1,V2) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> activate#(X2) U14#(tt(),V1,V2) -> activate#(V1) -> activate#(n__0()) -> 0#() U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) -> U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)) U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) -> U14#(tt(),V1,V2) -> isNat#(activate(V1)) U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) -> U14#(tt(),V1,V2) -> activate#(V1) U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) -> U14#(tt(),V1,V2) -> activate#(V2) U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) -> isNatKind#(n__s(V1)) -> U41#(isNatKind(activate(V1))) U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) -> isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) -> isNatKind#(n__s(V1)) -> activate#(V1) U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) -> isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)) U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) -> isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V1) U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V2) U13#(tt(),V1,V2) -> activate#(V2) -> activate#(n__s(X)) -> s#(activate(X)) U13#(tt(),V1,V2) -> activate#(V2) -> activate#(n__s(X)) -> activate#(X) U13#(tt(),V1,V2) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) U13#(tt(),V1,V2) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> activate#(X1) U13#(tt(),V1,V2) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> activate#(X2) U13#(tt(),V1,V2) -> activate#(V2) -> activate#(n__0()) -> 0#() U13#(tt(),V1,V2) -> activate#(V1) -> activate#(n__s(X)) -> s#(activate(X)) U13#(tt(),V1,V2) -> activate#(V1) -> activate#(n__s(X)) -> activate#(X) U13#(tt(),V1,V2) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) U13#(tt(),V1,V2) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> activate#(X1) U13#(tt(),V1,V2) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> activate#(X2) U13#(tt(),V1,V2) -> activate#(V1) -> activate#(n__0()) -> 0#() U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) -> U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) -> U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) -> U13#(tt(),V1,V2) -> activate#(V2) U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) -> U13#(tt(),V1,V2) -> activate#(V1) U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) -> isNatKind#(n__s(V1)) -> U41#(isNatKind(activate(V1))) U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) -> isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) -> isNatKind#(n__s(V1)) -> activate#(V1) U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) -> isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)) U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) -> isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V1) U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V2) U12#(tt(),V1,V2) -> activate#(V2) -> activate#(n__s(X)) -> s#(activate(X)) U12#(tt(),V1,V2) -> activate#(V2) -> activate#(n__s(X)) -> activate#(X) U12#(tt(),V1,V2) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) U12#(tt(),V1,V2) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> activate#(X1) U12#(tt(),V1,V2) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> activate#(X2) U12#(tt(),V1,V2) -> activate#(V2) -> activate#(n__0()) -> 0#() U12#(tt(),V1,V2) -> activate#(V1) -> activate#(n__s(X)) -> s#(activate(X)) U12#(tt(),V1,V2) -> activate#(V1) -> activate#(n__s(X)) -> activate#(X) U12#(tt(),V1,V2) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) U12#(tt(),V1,V2) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> activate#(X1) U12#(tt(),V1,V2) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> activate#(X2) U12#(tt(),V1,V2) -> activate#(V1) -> activate#(n__0()) -> 0#() isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) -> isNatKind#(n__s(V1)) -> U41#(isNatKind(activate(V1))) isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) -> isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) -> isNatKind#(n__s(V1)) -> activate#(V1) isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)) isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V1) isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V2) isNatKind#(n__s(V1)) -> activate#(V1) -> activate#(n__s(X)) -> s#(activate(X)) isNatKind#(n__s(V1)) -> activate#(V1) -> activate#(n__s(X)) -> activate#(X) isNatKind#(n__s(V1)) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) isNatKind#(n__s(V1)) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> activate#(X1) isNatKind#(n__s(V1)) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> activate#(X2) isNatKind#(n__s(V1)) -> activate#(V1) -> activate#(n__0()) -> 0#() isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)) -> U31#(tt(),V2) -> U32#(isNatKind(activate(V2))) isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)) -> U31#(tt(),V2) -> isNatKind#(activate(V2)) isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)) -> U31#(tt(),V2) -> activate#(V2) isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) -> isNatKind#(n__s(V1)) -> U41#(isNatKind(activate(V1))) isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) -> isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) -> isNatKind#(n__s(V1)) -> activate#(V1) isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)) isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V1) isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V2) isNatKind#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__s(X)) -> s#(activate(X)) isNatKind#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__s(X)) -> activate#(X) isNatKind#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) isNatKind#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> activate#(X1) isNatKind#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> activate#(X2) isNatKind#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__0()) -> 0#() isNatKind#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__s(X)) -> s#(activate(X)) isNatKind#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__s(X)) -> activate#(X) isNatKind#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) isNatKind#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> activate#(X1) isNatKind#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> activate#(X2) isNatKind#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__0()) -> 0#() activate#(n__s(X)) -> activate#(X) -> activate#(n__s(X)) -> s#(activate(X)) activate#(n__s(X)) -> activate#(X) -> activate#(n__s(X)) -> activate#(X) activate#(n__s(X)) -> activate#(X) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) activate#(n__s(X)) -> activate#(X) -> activate#(n__plus(X1,X2)) -> activate#(X1) activate#(n__s(X)) -> activate#(X) -> activate#(n__plus(X1,X2)) -> activate#(X2) activate#(n__s(X)) -> activate#(X) -> activate#(n__0()) -> 0#() activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) -> plus#(N,s(M)) -> U61#(isNat(M),M,N) activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) -> plus#(N,s(M)) -> isNat#(M) activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) -> plus#(N,0()) -> U51#(isNat(N),N) activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) -> plus#(N,0()) -> isNat#(N) activate#(n__plus(X1,X2)) -> activate#(X2) -> activate#(n__s(X)) -> s#(activate(X)) activate#(n__plus(X1,X2)) -> activate#(X2) -> activate#(n__s(X)) -> activate#(X) activate#(n__plus(X1,X2)) -> activate#(X2) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) activate#(n__plus(X1,X2)) -> activate#(X2) -> activate#(n__plus(X1,X2)) -> activate#(X1) activate#(n__plus(X1,X2)) -> activate#(X2) -> activate#(n__plus(X1,X2)) -> activate#(X2) activate#(n__plus(X1,X2)) -> activate#(X2) -> activate#(n__0()) -> 0#() activate#(n__plus(X1,X2)) -> activate#(X1) -> activate#(n__s(X)) -> s#(activate(X)) activate#(n__plus(X1,X2)) -> activate#(X1) -> activate#(n__s(X)) -> activate#(X) activate#(n__plus(X1,X2)) -> activate#(X1) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) activate#(n__plus(X1,X2)) -> activate#(X1) -> activate#(n__plus(X1,X2)) -> activate#(X1) activate#(n__plus(X1,X2)) -> activate#(X1) -> activate#(n__plus(X1,X2)) -> activate#(X2) activate#(n__plus(X1,X2)) -> activate#(X1) -> activate#(n__0()) -> 0#() U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) -> U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) -> U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) -> U12#(tt(),V1,V2) -> activate#(V2) U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) -> U12#(tt(),V1,V2) -> activate#(V1) U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) -> isNatKind#(n__s(V1)) -> U41#(isNatKind(activate(V1))) U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) -> isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) -> isNatKind#(n__s(V1)) -> activate#(V1) U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)) U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V1) U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V2) U11#(tt(),V1,V2) -> activate#(V2) -> activate#(n__s(X)) -> s#(activate(X)) U11#(tt(),V1,V2) -> activate#(V2) -> activate#(n__s(X)) -> activate#(X) U11#(tt(),V1,V2) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) U11#(tt(),V1,V2) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> activate#(X1) U11#(tt(),V1,V2) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> activate#(X2) U11#(tt(),V1,V2) -> activate#(V2) -> activate#(n__0()) -> 0#() U11#(tt(),V1,V2) -> activate#(V1) -> activate#(n__s(X)) -> s#(activate(X)) U11#(tt(),V1,V2) -> activate#(V1) -> activate#(n__s(X)) -> activate#(X) U11#(tt(),V1,V2) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) U11#(tt(),V1,V2) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> activate#(X1) U11#(tt(),V1,V2) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> activate#(X2) U11#(tt(),V1,V2) -> activate#(V1) -> activate#(n__0()) -> 0#() SCC Processor: #sccs: 1 #rules: 65 #arcs: 368/5184 DPs: plus#(N,0()) -> U51#(isNat(N),N) U51#(tt(),N) -> activate#(N) activate#(n__plus(X1,X2)) -> activate#(X2) activate#(n__plus(X1,X2)) -> activate#(X1) activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) plus#(N,0()) -> isNat#(N) isNat#(n__plus(V1,V2)) -> activate#(V2) activate#(n__s(X)) -> activate#(X) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) -> activate#(V2) isNatKind#(n__plus(V1,V2)) -> activate#(V1) isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)) U31#(tt(),V2) -> activate#(V2) U31#(tt(),V2) -> isNatKind#(activate(V2)) isNatKind#(n__s(V1)) -> activate#(V1) isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) U11#(tt(),V1,V2) -> activate#(V2) U11#(tt(),V1,V2) -> activate#(V1) U11#(tt(),V1,V2) -> isNatKind#(activate(V1)) U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) U12#(tt(),V1,V2) -> activate#(V1) U12#(tt(),V1,V2) -> activate#(V2) U12#(tt(),V1,V2) -> isNatKind#(activate(V2)) U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) U13#(tt(),V1,V2) -> activate#(V1) U13#(tt(),V1,V2) -> activate#(V2) U13#(tt(),V1,V2) -> isNatKind#(activate(V2)) U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) U14#(tt(),V1,V2) -> activate#(V2) U14#(tt(),V1,V2) -> activate#(V1) U14#(tt(),V1,V2) -> isNat#(activate(V1)) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) -> activate#(V1) U21#(tt(),V1) -> isNatKind#(activate(V1)) U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)) U22#(tt(),V1) -> activate#(V1) U22#(tt(),V1) -> isNat#(activate(V1)) U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)) U15#(tt(),V2) -> activate#(V2) U15#(tt(),V2) -> isNat#(activate(V2)) plus#(N,s(M)) -> isNat#(M) plus#(N,s(M)) -> U61#(isNat(M),M,N) U61#(tt(),M,N) -> activate#(N) U61#(tt(),M,N) -> activate#(M) U61#(tt(),M,N) -> isNatKind#(activate(M)) U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N)) U62#(tt(),M,N) -> activate#(M) U62#(tt(),M,N) -> activate#(N) U62#(tt(),M,N) -> isNat#(activate(N)) U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)) U63#(tt(),M,N) -> activate#(M) U63#(tt(),M,N) -> activate#(N) U63#(tt(),M,N) -> isNatKind#(activate(N)) U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) U64#(tt(),M,N) -> activate#(M) U64#(tt(),M,N) -> activate#(N) U64#(tt(),M,N) -> plus#(activate(N),activate(M)) U51#(tt(),N) -> isNatKind#(activate(N)) U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N)) U52#(tt(),N) -> activate#(N) TRS: U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X Matrix Interpretation Processor: dim=1 usable rules: U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X interpretation: [plus#](x0, x1) = 4x0 + 2x1, [U64#](x0, x1, x2) = 2x1 + 4x2, [U63#](x0, x1, x2) = 2x1 + 4x2, [U62#](x0, x1, x2) = 2x1 + 4x2, [U61#](x0, x1, x2) = 2x1 + 4x2, [U52#](x0, x1) = 2x1, [U51#](x0, x1) = x0 + 2x1, [U31#](x0, x1) = 2x1 + 4, [U22#](x0, x1) = 2x1, [U21#](x0, x1) = 2x1, [U15#](x0, x1) = 2x1 + 1, [isNat#](x0) = 2x0, [U14#](x0, x1, x2) = 4x1 + 2x2 + 1, [U13#](x0, x1, x2) = 4x1 + 2x2 + 1, [U12#](x0, x1, x2) = 4x1 + 2x2 + 1, [isNatKind#](x0) = 2x0, [activate#](x0) = 2x0, [U11#](x0, x1, x2) = 6x1 + 2x2 + 1, [0] = 0, [n__s](x0) = x0, [n__plus](x0, x1) = 3x0 + x1 + 2, [n__0] = 0, [s](x0) = x0, [plus](x0, x1) = 3x0 + x1 + 2, [U64](x0, x1, x2) = x1 + 3x2 + 2, [U63](x0, x1, x2) = x1 + 3x2 + 2, [U62](x0, x1, x2) = x1 + 3x2 + 2, [U61](x0, x1, x2) = x1 + 3x2 + 2, [U52](x0, x1) = 2x1, [U51](x0, x1) = 3x1, [U41](x0) = 5, [U32](x0) = 0, [U31](x0, x1) = 0, [U23](x0) = 0, [U22](x0, x1) = 2x1, [U21](x0, x1) = 2x1, [U16](x0) = 4, [U15](x0, x1) = 4, [isNat](x0) = 2x0, [U14](x0, x1, x2) = 4, [U13](x0, x1, x2) = 4, [U12](x0, x1, x2) = 4x1 + 4, [isNatKind](x0) = 0, [activate](x0) = x0, [U11](x0, x1, x2) = 6x1 + x2 + 4, [tt] = 0 orientation: plus#(N,0()) = 4N >= 4N = U51#(isNat(N),N) U51#(tt(),N) = 2N >= 2N = activate#(N) activate#(n__plus(X1,X2)) = 6X1 + 2X2 + 4 >= 2X2 = activate#(X2) activate#(n__plus(X1,X2)) = 6X1 + 2X2 + 4 >= 2X1 = activate#(X1) activate#(n__plus(X1,X2)) = 6X1 + 2X2 + 4 >= 4X1 + 2X2 = plus#(activate(X1),activate(X2)) plus#(N,0()) = 4N >= 2N = isNat#(N) isNat#(n__plus(V1,V2)) = 6V1 + 2V2 + 4 >= 2V2 = activate#(V2) activate#(n__s(X)) = 2X >= 2X = activate#(X) isNat#(n__plus(V1,V2)) = 6V1 + 2V2 + 4 >= 2V1 = activate#(V1) isNat#(n__plus(V1,V2)) = 6V1 + 2V2 + 4 >= 2V1 = isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) = 6V1 + 2V2 + 4 >= 2V2 = activate#(V2) isNatKind#(n__plus(V1,V2)) = 6V1 + 2V2 + 4 >= 2V1 = activate#(V1) isNatKind#(n__plus(V1,V2)) = 6V1 + 2V2 + 4 >= 2V1 = isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) = 6V1 + 2V2 + 4 >= 2V2 + 4 = U31#(isNatKind(activate(V1)),activate(V2)) U31#(tt(),V2) = 2V2 + 4 >= 2V2 = activate#(V2) U31#(tt(),V2) = 2V2 + 4 >= 2V2 = isNatKind#(activate(V2)) isNatKind#(n__s(V1)) = 2V1 >= 2V1 = activate#(V1) isNatKind#(n__s(V1)) = 2V1 >= 2V1 = isNatKind#(activate(V1)) isNat#(n__plus(V1,V2)) = 6V1 + 2V2 + 4 >= 6V1 + 2V2 + 1 = U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) U11#(tt(),V1,V2) = 6V1 + 2V2 + 1 >= 2V2 = activate#(V2) U11#(tt(),V1,V2) = 6V1 + 2V2 + 1 >= 2V1 = activate#(V1) U11#(tt(),V1,V2) = 6V1 + 2V2 + 1 >= 2V1 = isNatKind#(activate(V1)) U11#(tt(),V1,V2) = 6V1 + 2V2 + 1 >= 4V1 + 2V2 + 1 = U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) U12#(tt(),V1,V2) = 4V1 + 2V2 + 1 >= 2V1 = activate#(V1) U12#(tt(),V1,V2) = 4V1 + 2V2 + 1 >= 2V2 = activate#(V2) U12#(tt(),V1,V2) = 4V1 + 2V2 + 1 >= 2V2 = isNatKind#(activate(V2)) U12#(tt(),V1,V2) = 4V1 + 2V2 + 1 >= 4V1 + 2V2 + 1 = U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) U13#(tt(),V1,V2) = 4V1 + 2V2 + 1 >= 2V1 = activate#(V1) U13#(tt(),V1,V2) = 4V1 + 2V2 + 1 >= 2V2 = activate#(V2) U13#(tt(),V1,V2) = 4V1 + 2V2 + 1 >= 2V2 = isNatKind#(activate(V2)) U13#(tt(),V1,V2) = 4V1 + 2V2 + 1 >= 4V1 + 2V2 + 1 = U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) U14#(tt(),V1,V2) = 4V1 + 2V2 + 1 >= 2V2 = activate#(V2) U14#(tt(),V1,V2) = 4V1 + 2V2 + 1 >= 2V1 = activate#(V1) U14#(tt(),V1,V2) = 4V1 + 2V2 + 1 >= 2V1 = isNat#(activate(V1)) isNat#(n__s(V1)) = 2V1 >= 2V1 = activate#(V1) isNat#(n__s(V1)) = 2V1 >= 2V1 = isNatKind#(activate(V1)) isNat#(n__s(V1)) = 2V1 >= 2V1 = U21#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) = 2V1 >= 2V1 = activate#(V1) U21#(tt(),V1) = 2V1 >= 2V1 = isNatKind#(activate(V1)) U21#(tt(),V1) = 2V1 >= 2V1 = U22#(isNatKind(activate(V1)),activate(V1)) U22#(tt(),V1) = 2V1 >= 2V1 = activate#(V1) U22#(tt(),V1) = 2V1 >= 2V1 = isNat#(activate(V1)) U14#(tt(),V1,V2) = 4V1 + 2V2 + 1 >= 2V2 + 1 = U15#(isNat(activate(V1)),activate(V2)) U15#(tt(),V2) = 2V2 + 1 >= 2V2 = activate#(V2) U15#(tt(),V2) = 2V2 + 1 >= 2V2 = isNat#(activate(V2)) plus#(N,s(M)) = 2M + 4N >= 2M = isNat#(M) plus#(N,s(M)) = 2M + 4N >= 2M + 4N = U61#(isNat(M),M,N) U61#(tt(),M,N) = 2M + 4N >= 2N = activate#(N) U61#(tt(),M,N) = 2M + 4N >= 2M = activate#(M) U61#(tt(),M,N) = 2M + 4N >= 2M = isNatKind#(activate(M)) U61#(tt(),M,N) = 2M + 4N >= 2M + 4N = U62#(isNatKind(activate(M)),activate(M),activate(N)) U62#(tt(),M,N) = 2M + 4N >= 2M = activate#(M) U62#(tt(),M,N) = 2M + 4N >= 2N = activate#(N) U62#(tt(),M,N) = 2M + 4N >= 2N = isNat#(activate(N)) U62#(tt(),M,N) = 2M + 4N >= 2M + 4N = U63#(isNat(activate(N)),activate(M),activate(N)) U63#(tt(),M,N) = 2M + 4N >= 2M = activate#(M) U63#(tt(),M,N) = 2M + 4N >= 2N = activate#(N) U63#(tt(),M,N) = 2M + 4N >= 2N = isNatKind#(activate(N)) U63#(tt(),M,N) = 2M + 4N >= 2M + 4N = U64#(isNatKind(activate(N)),activate(M),activate(N)) U64#(tt(),M,N) = 2M + 4N >= 2M = activate#(M) U64#(tt(),M,N) = 2M + 4N >= 2N = activate#(N) U64#(tt(),M,N) = 2M + 4N >= 2M + 4N = plus#(activate(N),activate(M)) U51#(tt(),N) = 2N >= 2N = isNatKind#(activate(N)) U51#(tt(),N) = 2N >= 2N = U52#(isNatKind(activate(N)),activate(N)) U52#(tt(),N) = 2N >= 2N = activate#(N) U11(tt(),V1,V2) = 6V1 + V2 + 4 >= 4V1 + 4 = U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) = 4V1 + 4 >= 4 = U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) = 4 >= 4 = U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) = 4 >= 4 = U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) = 4 >= 4 = U16(isNat(activate(V2))) U16(tt()) = 4 >= 0 = tt() U21(tt(),V1) = 2V1 >= 2V1 = U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) = 2V1 >= 0 = U23(isNat(activate(V1))) U23(tt()) = 0 >= 0 = tt() U31(tt(),V2) = 0 >= 0 = U32(isNatKind(activate(V2))) U32(tt()) = 0 >= 0 = tt() U41(tt()) = 5 >= 0 = tt() U51(tt(),N) = 3N >= 2N = U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) = 2N >= N = activate(N) U61(tt(),M,N) = M + 3N + 2 >= M + 3N + 2 = U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) = M + 3N + 2 >= M + 3N + 2 = U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) = M + 3N + 2 >= M + 3N + 2 = U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) = M + 3N + 2 >= M + 3N + 2 = s(plus(activate(N),activate(M))) isNat(n__0()) = 0 >= 0 = tt() isNat(n__plus(V1,V2)) = 6V1 + 2V2 + 4 >= 6V1 + V2 + 4 = U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) = 2V1 >= 2V1 = U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) = 0 >= 0 = tt() isNatKind(n__plus(V1,V2)) = 0 >= 0 = U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) = 0 >= 5 = U41(isNatKind(activate(V1))) plus(N,0()) = 3N + 2 >= 3N = U51(isNat(N),N) plus(N,s(M)) = M + 3N + 2 >= M + 3N + 2 = U61(isNat(M),M,N) 0() = 0 >= 0 = n__0() plus(X1,X2) = 3X1 + X2 + 2 >= 3X1 + X2 + 2 = n__plus(X1,X2) s(X) = X >= X = n__s(X) activate(n__0()) = 0 >= 0 = 0() activate(n__plus(X1,X2)) = 3X1 + X2 + 2 >= 3X1 + X2 + 2 = plus(activate(X1),activate(X2)) activate(n__s(X)) = X >= X = s(activate(X)) activate(X) = X >= X = X problem: DPs: plus#(N,0()) -> U51#(isNat(N),N) U51#(tt(),N) -> activate#(N) plus#(N,0()) -> isNat#(N) activate#(n__s(X)) -> activate#(X) isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)) isNatKind#(n__s(V1)) -> activate#(V1) isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) -> activate#(V1) U21#(tt(),V1) -> isNatKind#(activate(V1)) U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)) U22#(tt(),V1) -> activate#(V1) U22#(tt(),V1) -> isNat#(activate(V1)) U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)) plus#(N,s(M)) -> isNat#(M) plus#(N,s(M)) -> U61#(isNat(M),M,N) U61#(tt(),M,N) -> activate#(N) U61#(tt(),M,N) -> activate#(M) U61#(tt(),M,N) -> isNatKind#(activate(M)) U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N)) U62#(tt(),M,N) -> activate#(M) U62#(tt(),M,N) -> activate#(N) U62#(tt(),M,N) -> isNat#(activate(N)) U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)) U63#(tt(),M,N) -> activate#(M) U63#(tt(),M,N) -> activate#(N) U63#(tt(),M,N) -> isNatKind#(activate(N)) U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) U64#(tt(),M,N) -> activate#(M) U64#(tt(),M,N) -> activate#(N) U64#(tt(),M,N) -> plus#(activate(N),activate(M)) U51#(tt(),N) -> isNatKind#(activate(N)) U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N)) U52#(tt(),N) -> activate#(N) TRS: U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X Restore Modifier: DPs: plus#(N,0()) -> U51#(isNat(N),N) U51#(tt(),N) -> activate#(N) plus#(N,0()) -> isNat#(N) activate#(n__s(X)) -> activate#(X) isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)) isNatKind#(n__s(V1)) -> activate#(V1) isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) -> activate#(V1) U21#(tt(),V1) -> isNatKind#(activate(V1)) U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)) U22#(tt(),V1) -> activate#(V1) U22#(tt(),V1) -> isNat#(activate(V1)) U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)) plus#(N,s(M)) -> isNat#(M) plus#(N,s(M)) -> U61#(isNat(M),M,N) U61#(tt(),M,N) -> activate#(N) U61#(tt(),M,N) -> activate#(M) U61#(tt(),M,N) -> isNatKind#(activate(M)) U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N)) U62#(tt(),M,N) -> activate#(M) U62#(tt(),M,N) -> activate#(N) U62#(tt(),M,N) -> isNat#(activate(N)) U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)) U63#(tt(),M,N) -> activate#(M) U63#(tt(),M,N) -> activate#(N) U63#(tt(),M,N) -> isNatKind#(activate(N)) U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) U64#(tt(),M,N) -> activate#(M) U64#(tt(),M,N) -> activate#(N) U64#(tt(),M,N) -> plus#(activate(N),activate(M)) U51#(tt(),N) -> isNatKind#(activate(N)) U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N)) U52#(tt(),N) -> activate#(N) TRS: U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X SCC Processor: #sccs: 4 #rules: 10 #arcs: 290/1521 DPs: plus#(N,s(M)) -> U61#(isNat(M),M,N) U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N)) U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)) U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) U64#(tt(),M,N) -> plus#(activate(N),activate(M)) TRS: U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X Matrix Interpretation Processor: dim=1 usable rules: U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X interpretation: [plus#](x0, x1) = 2x1, [U64#](x0, x1, x2) = 2x1, [U63#](x0, x1, x2) = 2x1, [U62#](x0, x1, x2) = 2x1 + 1/2, [U61#](x0, x1, x2) = 2x1 + 1, [0] = 0, [n__s](x0) = x0 + 1/2, [n__plus](x0, x1) = x0 + x1, [n__0] = 0, [s](x0) = x0 + 1/2, [plus](x0, x1) = x0 + x1, [U64](x0, x1, x2) = x1 + x2 + 1/2, [U63](x0, x1, x2) = x1 + x2 + 1/2, [U62](x0, x1, x2) = x1 + x2 + 1/2, [U61](x0, x1, x2) = x1 + x2 + 1/2, [U52](x0, x1) = x1, [U51](x0, x1) = x1, [U41](x0) = 0, [U32](x0) = 0, [U31](x0, x1) = 0, [U23](x0) = 2, [U22](x0, x1) = 3/2x1, [U21](x0, x1) = 2, [U16](x0) = 0, [U15](x0, x1) = 0, [isNat](x0) = 0, [U14](x0, x1, x2) = x2, [U13](x0, x1, x2) = 0, [U12](x0, x1, x2) = 0, [isNatKind](x0) = 0, [activate](x0) = x0, [U11](x0, x1, x2) = 1/2x2 + 3, [tt] = 0 orientation: plus#(N,s(M)) = 2M + 1 >= 2M + 1 = U61#(isNat(M),M,N) U61#(tt(),M,N) = 2M + 1 >= 2M + 1/2 = U62#(isNatKind(activate(M)),activate(M),activate(N)) U62#(tt(),M,N) = 2M + 1/2 >= 2M = U63#(isNat(activate(N)),activate(M),activate(N)) U63#(tt(),M,N) = 2M >= 2M = U64#(isNatKind(activate(N)),activate(M),activate(N)) U64#(tt(),M,N) = 2M >= 2M = plus#(activate(N),activate(M)) U11(tt(),V1,V2) = 1/2V2 + 3 >= 0 = U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) = 0 >= 0 = U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) = 0 >= V2 = U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) = V2 >= 0 = U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) = 0 >= 0 = U16(isNat(activate(V2))) U16(tt()) = 0 >= 0 = tt() U21(tt(),V1) = 2 >= 3/2V1 = U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) = 3/2V1 >= 2 = U23(isNat(activate(V1))) U23(tt()) = 2 >= 0 = tt() U31(tt(),V2) = 0 >= 0 = U32(isNatKind(activate(V2))) U32(tt()) = 0 >= 0 = tt() U41(tt()) = 0 >= 0 = tt() U51(tt(),N) = N >= N = U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) = N >= N = activate(N) U61(tt(),M,N) = M + N + 1/2 >= M + N + 1/2 = U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) = M + N + 1/2 >= M + N + 1/2 = U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) = M + N + 1/2 >= M + N + 1/2 = U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) = M + N + 1/2 >= M + N + 1/2 = s(plus(activate(N),activate(M))) isNat(n__0()) = 0 >= 0 = tt() isNat(n__plus(V1,V2)) = 0 >= 1/2V2 + 3 = U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) = 0 >= 2 = U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) = 0 >= 0 = tt() isNatKind(n__plus(V1,V2)) = 0 >= 0 = U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) = 0 >= 0 = U41(isNatKind(activate(V1))) plus(N,0()) = N >= N = U51(isNat(N),N) plus(N,s(M)) = M + N + 1/2 >= M + N + 1/2 = U61(isNat(M),M,N) 0() = 0 >= 0 = n__0() plus(X1,X2) = X1 + X2 >= X1 + X2 = n__plus(X1,X2) s(X) = X + 1/2 >= X + 1/2 = n__s(X) activate(n__0()) = 0 >= 0 = 0() activate(n__plus(X1,X2)) = X1 + X2 >= X1 + X2 = plus(activate(X1),activate(X2)) activate(n__s(X)) = X + 1/2 >= X + 1/2 = s(activate(X)) activate(X) = X >= X = X problem: DPs: plus#(N,s(M)) -> U61#(isNat(M),M,N) U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) U64#(tt(),M,N) -> plus#(activate(N),activate(M)) TRS: U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X Restore Modifier: DPs: plus#(N,s(M)) -> U61#(isNat(M),M,N) U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)) U64#(tt(),M,N) -> plus#(activate(N),activate(M)) TRS: U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X SCC Processor: #sccs: 0 #rules: 0 #arcs: 5/9 DPs: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)) U22#(tt(),V1) -> isNat#(activate(V1)) TRS: U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X Matrix Interpretation Processor: dim=1 usable rules: U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X interpretation: [U22#](x0, x1) = 2x1 + 4, [U21#](x0, x1) = 2x1 + 4, [isNat#](x0) = 2x0 + 4, [0] = 0, [n__s](x0) = x0 + 1, [n__plus](x0, x1) = 4x0 + 2x1, [n__0] = 0, [s](x0) = x0 + 1, [plus](x0, x1) = 4x0 + 2x1, [U64](x0, x1, x2) = 2x1 + 4x2 + 1, [U63](x0, x1, x2) = 2x1 + 4x2 + 1, [U62](x0, x1, x2) = 2x1 + 4x2 + 1, [U61](x0, x1, x2) = 2x1 + 4x2 + 1, [U52](x0, x1) = 4x1, [U51](x0, x1) = 4x1, [U41](x0) = 6, [U32](x0) = 0, [U31](x0, x1) = x0 + 1, [U23](x0) = 0, [U22](x0, x1) = 4x0 + x1, [U21](x0, x1) = 2x1, [U16](x0) = 4, [U15](x0, x1) = 2, [isNat](x0) = 0, [U14](x0, x1, x2) = 4, [U13](x0, x1, x2) = 0, [U12](x0, x1, x2) = 6x1 + 4x2, [isNatKind](x0) = 4x0, [activate](x0) = x0, [U11](x0, x1, x2) = 0, [tt] = 0 orientation: isNat#(n__s(V1)) = 2V1 + 6 >= 2V1 + 4 = U21#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) = 2V1 + 4 >= 2V1 + 4 = U22#(isNatKind(activate(V1)),activate(V1)) U22#(tt(),V1) = 2V1 + 4 >= 2V1 + 4 = isNat#(activate(V1)) U11(tt(),V1,V2) = 0 >= 6V1 + 4V2 = U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) = 6V1 + 4V2 >= 0 = U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) = 0 >= 4 = U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) = 4 >= 2 = U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) = 2 >= 4 = U16(isNat(activate(V2))) U16(tt()) = 4 >= 0 = tt() U21(tt(),V1) = 2V1 >= 17V1 = U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) = V1 >= 0 = U23(isNat(activate(V1))) U23(tt()) = 0 >= 0 = tt() U31(tt(),V2) = 1 >= 0 = U32(isNatKind(activate(V2))) U32(tt()) = 0 >= 0 = tt() U41(tt()) = 6 >= 0 = tt() U51(tt(),N) = 4N >= 4N = U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) = 4N >= N = activate(N) U61(tt(),M,N) = 2M + 4N + 1 >= 2M + 4N + 1 = U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) = 2M + 4N + 1 >= 2M + 4N + 1 = U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) = 2M + 4N + 1 >= 2M + 4N + 1 = U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) = 2M + 4N + 1 >= 2M + 4N + 1 = s(plus(activate(N),activate(M))) isNat(n__0()) = 0 >= 0 = tt() isNat(n__plus(V1,V2)) = 0 >= 0 = U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) = 0 >= 2V1 = U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) = 0 >= 0 = tt() isNatKind(n__plus(V1,V2)) = 16V1 + 8V2 >= 4V1 + 1 = U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) = 4V1 + 4 >= 6 = U41(isNatKind(activate(V1))) plus(N,0()) = 4N >= 4N = U51(isNat(N),N) plus(N,s(M)) = 2M + 4N + 2 >= 2M + 4N + 1 = U61(isNat(M),M,N) 0() = 0 >= 0 = n__0() plus(X1,X2) = 4X1 + 2X2 >= 4X1 + 2X2 = n__plus(X1,X2) s(X) = X + 1 >= X + 1 = n__s(X) activate(n__0()) = 0 >= 0 = 0() activate(n__plus(X1,X2)) = 4X1 + 2X2 >= 4X1 + 2X2 = plus(activate(X1),activate(X2)) activate(n__s(X)) = X + 1 >= X + 1 = s(activate(X)) activate(X) = X >= X = X problem: DPs: U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)) U22#(tt(),V1) -> isNat#(activate(V1)) TRS: U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X Restore Modifier: DPs: U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)) U22#(tt(),V1) -> isNat#(activate(V1)) TRS: U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X SCC Processor: #sccs: 0 #rules: 0 #arcs: 3/4 DPs: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) TRS: U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X Matrix Interpretation Processor: dim=1 usable rules: U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X interpretation: [isNatKind#](x0) = 2x0 + 7, [0] = 0, [n__s](x0) = x0 + 4, [n__plus](x0, x1) = 4x0 + 2x1, [n__0] = 0, [s](x0) = x0 + 4, [plus](x0, x1) = 4x0 + 2x1, [U64](x0, x1, x2) = 2x1 + 4x2 + 4, [U63](x0, x1, x2) = 2x1 + 4x2 + 6, [U62](x0, x1, x2) = 2x1 + 4x2 + 6, [U61](x0, x1, x2) = 2x1 + 4x2 + 6, [U52](x0, x1) = x1, [U51](x0, x1) = 2x1, [U41](x0) = 0, [U32](x0) = 2x0, [U31](x0, x1) = 6x1, [U23](x0) = 1, [U22](x0, x1) = 4x0 + x1 + 2, [U21](x0, x1) = 4x0 + x1 + 4, [U16](x0) = x0, [U15](x0, x1) = x0 + 3, [isNat](x0) = 1, [U14](x0, x1, x2) = 2x0 + 2x2, [U13](x0, x1, x2) = 2x0 + 2, [U12](x0, x1, x2) = 0, [isNatKind](x0) = 1, [activate](x0) = x0, [U11](x0, x1, x2) = 6x1, [tt] = 0 orientation: isNatKind#(n__s(V1)) = 2V1 + 15 >= 2V1 + 7 = isNatKind#(activate(V1)) U11(tt(),V1,V2) = 6V1 >= 0 = U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) = 0 >= 4 = U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) = 2 >= 2V2 + 2 = U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) = 2V2 >= 4 = U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) = 3 >= 1 = U16(isNat(activate(V2))) U16(tt()) = 0 >= 0 = tt() U21(tt(),V1) = V1 + 4 >= V1 + 6 = U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) = V1 + 2 >= 1 = U23(isNat(activate(V1))) U23(tt()) = 1 >= 0 = tt() U31(tt(),V2) = 6V2 >= 2 = U32(isNatKind(activate(V2))) U32(tt()) = 0 >= 0 = tt() U41(tt()) = 0 >= 0 = tt() U51(tt(),N) = 2N >= N = U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) = N >= N = activate(N) U61(tt(),M,N) = 2M + 4N + 6 >= 2M + 4N + 6 = U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) = 2M + 4N + 6 >= 2M + 4N + 6 = U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) = 2M + 4N + 6 >= 2M + 4N + 4 = U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) = 2M + 4N + 4 >= 2M + 4N + 4 = s(plus(activate(N),activate(M))) isNat(n__0()) = 1 >= 0 = tt() isNat(n__plus(V1,V2)) = 1 >= 6V1 = U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) = 1 >= V1 + 8 = U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) = 1 >= 0 = tt() isNatKind(n__plus(V1,V2)) = 1 >= 6V2 = U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) = 1 >= 0 = U41(isNatKind(activate(V1))) plus(N,0()) = 4N >= 2N = U51(isNat(N),N) plus(N,s(M)) = 2M + 4N + 8 >= 2M + 4N + 6 = U61(isNat(M),M,N) 0() = 0 >= 0 = n__0() plus(X1,X2) = 4X1 + 2X2 >= 4X1 + 2X2 = n__plus(X1,X2) s(X) = X + 4 >= X + 4 = n__s(X) activate(n__0()) = 0 >= 0 = 0() activate(n__plus(X1,X2)) = 4X1 + 2X2 >= 4X1 + 2X2 = plus(activate(X1),activate(X2)) activate(n__s(X)) = X + 4 >= X + 4 = s(activate(X)) activate(X) = X >= X = X problem: DPs: TRS: U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X Qed DPs: activate#(n__s(X)) -> activate#(X) TRS: U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X Size-Change Termination Processor: DPs: TRS: U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2)) U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2)) U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2)) U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2)) U15(tt(),V2) -> U16(isNat(activate(V2))) U16(tt()) -> tt() U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1)) U22(tt(),V1) -> U23(isNat(activate(V1))) U23(tt()) -> tt() U31(tt(),V2) -> U32(isNatKind(activate(V2))) U32(tt()) -> tt() U41(tt()) -> tt() U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N)) U52(tt(),N) -> activate(N) U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N)) U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N)) U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N)) U64(tt(),M,N) -> s(plus(activate(N),activate(M))) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2)) isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1))) plus(N,0()) -> U51(isNat(N),N) plus(N,s(M)) -> U61(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X The DP: activate#(n__s(X)) -> activate#(X) has the edges: 0 > 0 Qed