/export/starexec/sandbox/solver/bin/starexec_run_ttt2-1.17+nonreach /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/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(X1,X2) activate(n__s(X)) -> s(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)) -> plus#(X1,X2) activate#(n__s(X)) -> s#(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(X1,X2) activate(n__s(X)) -> s(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)) -> plus#(X1,X2) activate#(n__s(X)) -> s#(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(X1,X2) activate(n__s(X)) -> s(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#(X) U64#(tt(),M,N) -> activate#(M) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) U64#(tt(),M,N) -> activate#(M) -> activate#(n__0()) -> 0#() U64#(tt(),M,N) -> activate#(N) -> activate#(n__s(X)) -> s#(X) U64#(tt(),M,N) -> activate#(N) -> activate#(n__plus(X1,X2)) -> plus#(X1,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#(X) U63#(tt(),M,N) -> activate#(M) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) U63#(tt(),M,N) -> activate#(M) -> activate#(n__0()) -> 0#() U63#(tt(),M,N) -> activate#(N) -> activate#(n__s(X)) -> s#(X) U63#(tt(),M,N) -> activate#(N) -> activate#(n__plus(X1,X2)) -> plus#(X1,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#(X) U62#(tt(),M,N) -> activate#(M) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) U62#(tt(),M,N) -> activate#(M) -> activate#(n__0()) -> 0#() U62#(tt(),M,N) -> activate#(N) -> activate#(n__s(X)) -> s#(X) U62#(tt(),M,N) -> activate#(N) -> activate#(n__plus(X1,X2)) -> plus#(X1,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#(X) U61#(tt(),M,N) -> activate#(M) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) U61#(tt(),M,N) -> activate#(M) -> activate#(n__0()) -> 0#() U61#(tt(),M,N) -> activate#(N) -> activate#(n__s(X)) -> s#(X) U61#(tt(),M,N) -> activate#(N) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) U61#(tt(),M,N) -> activate#(N) -> activate#(n__0()) -> 0#() U52#(tt(),N) -> activate#(N) -> activate#(n__s(X)) -> s#(X) U52#(tt(),N) -> activate#(N) -> activate#(n__plus(X1,X2)) -> plus#(X1,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#(X) U51#(tt(),N) -> activate#(N) -> activate#(n__plus(X1,X2)) -> plus#(X1,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#(X) U31#(tt(),V2) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> plus#(X1,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#(X) U22#(tt(),V1) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> plus#(X1,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#(X) U21#(tt(),V1) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> plus#(X1,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#(X) U15#(tt(),V2) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> plus#(X1,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#(X) isNat#(n__s(V1)) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> plus#(X1,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#(X) isNat#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) isNat#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__0()) -> 0#() isNat#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__s(X)) -> s#(X) isNat#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> plus#(X1,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#(X) U14#(tt(),V1,V2) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) U14#(tt(),V1,V2) -> activate#(V2) -> activate#(n__0()) -> 0#() U14#(tt(),V1,V2) -> activate#(V1) -> activate#(n__s(X)) -> s#(X) U14#(tt(),V1,V2) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> plus#(X1,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#(X) U13#(tt(),V1,V2) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) U13#(tt(),V1,V2) -> activate#(V2) -> activate#(n__0()) -> 0#() U13#(tt(),V1,V2) -> activate#(V1) -> activate#(n__s(X)) -> s#(X) U13#(tt(),V1,V2) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> plus#(X1,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#(X) U12#(tt(),V1,V2) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) U12#(tt(),V1,V2) -> activate#(V2) -> activate#(n__0()) -> 0#() U12#(tt(),V1,V2) -> activate#(V1) -> activate#(n__s(X)) -> s#(X) U12#(tt(),V1,V2) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> plus#(X1,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#(X) isNatKind#(n__s(V1)) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> plus#(X1,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#(X) isNatKind#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) isNatKind#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__0()) -> 0#() isNatKind#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__s(X)) -> s#(X) isNatKind#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) isNatKind#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__0()) -> 0#() activate#(n__plus(X1,X2)) -> plus#(X1,X2) -> plus#(N,s(M)) -> U61#(isNat(M),M,N) activate#(n__plus(X1,X2)) -> plus#(X1,X2) -> plus#(N,s(M)) -> isNat#(M) activate#(n__plus(X1,X2)) -> plus#(X1,X2) -> plus#(N,0()) -> U51#(isNat(N),N) activate#(n__plus(X1,X2)) -> plus#(X1,X2) -> plus#(N,0()) -> isNat#(N) 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#(X) U11#(tt(),V1,V2) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) U11#(tt(),V1,V2) -> activate#(V2) -> activate#(n__0()) -> 0#() U11#(tt(),V1,V2) -> activate#(V1) -> activate#(n__s(X)) -> s#(X) U11#(tt(),V1,V2) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) U11#(tt(),V1,V2) -> activate#(V1) -> activate#(n__0()) -> 0#() SCC Processor: #sccs: 1 #rules: 62 #arcs: 266/4761 DPs: plus#(N,0()) -> U51#(isNat(N),N) U51#(tt(),N) -> activate#(N) activate#(n__plus(X1,X2)) -> plus#(X1,X2) plus#(N,0()) -> isNat#(N) isNat#(n__plus(V1,V2)) -> activate#(V2) 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(X1,X2) activate(n__s(X)) -> s(X) activate(X) -> X Matrix Interpretation Processor: dim=3 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(X1,X2) activate(n__s(X)) -> s(X) activate(X) -> X interpretation: [plus#](x0, x1) = [1 1 0]x0 + [1 1 0]x1, [U64#](x0, x1, x2) = [1 1 0]x1 + [1 1 0]x2, [U63#](x0, x1, x2) = [1 1 0]x1 + [1 1 0]x2, [U62#](x0, x1, x2) = [1 1 0]x1 + [1 1 0]x2, [U61#](x0, x1, x2) = [1 1 0]x1 + [1 1 0]x2, [U52#](x0, x1) = [0 1 0]x1 + [1], [U51#](x0, x1) = [1 1 0]x1 + [1], [U31#](x0, x1) = [0 1 0]x1, [U22#](x0, x1) = [1 1 0]x1, [U21#](x0, x1) = [1 1 0]x1, [U15#](x0, x1) = [1 1 0]x1, [isNat#](x0) = [1 1 0]x0, [U14#](x0, x1, x2) = [1 1 0]x1 + [1 1 0]x2, [U13#](x0, x1, x2) = [1 1 0]x1 + [1 1 0]x2 + [1], [U12#](x0, x1, x2) = [1 1 0]x1 + [1 1 0]x2 + [1], [isNatKind#](x0) = [0 1 0]x0, [activate#](x0) = [0 1 0]x0, [U11#](x0, x1, x2) = [1 1 0]x1 + [1 1 0]x2 + [1], [1] [0] = [0] [0], [1 0 0] [n__s](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [0 0 0] [1] [n__plus](x0, x1) = [1 1 0]x0 + [1 1 0]x1 + [0] [1 0 1] [0 0 0] [1], [1] [n__0] = [0] [0], [1 0 0] [s](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [0 0 0] [1] [plus](x0, x1) = [1 1 0]x0 + [1 1 0]x1 + [0] [1 0 1] [0 0 0] [1], [0 0 0] [1 0 0] [1] [U64](x0, x1, x2) = [1 1 0]x1 + [1 1 0]x2 + [0] [0 0 0] [1 0 0] [1], [0 0 0] [1 0 0] [1] [U63](x0, x1, x2) = [1 1 0]x1 + [1 1 0]x2 + [0] [0 0 0] [1 0 0] [1], [0 0 0] [1 0 0] [1] [U62](x0, x1, x2) = [1 1 0]x1 + [1 1 0]x2 + [0] [0 0 0] [1 0 0] [1], [0 0 0] [1 0 0] [1] [U61](x0, x1, x2) = [1 1 0]x1 + [1 1 0]x2 + [0] [0 0 0] [1 0 0] [1], [1 0 0] [0] [U52](x0, x1) = [0 1 0]x1 + [1] [1 0 1] [1], [1 0 0] [0] [U51](x0, x1) = [0 1 0]x1 + [1] [1 0 1] [1], [0] [U41](x0) = [0] [0], [0] [U32](x0) = [0] [0], [0] [U31](x0, x1) = [1] [0], [0] [U23](x0) = [0] [0], [0 0 0] [U22](x0, x1) = [0 1 0]x1 [0 0 0] , [0] [U21](x0, x1) = [0] [0], [1] [U16](x0) = [0] [0], [0] [U15](x0, x1) = [0] [0], [0] [isNat](x0) = [0] [0], [0 0 0] [0] [U14](x0, x1, x2) = [0 0 0]x2 + [0] [1 0 0] [1], [0 0 0] [U13](x0, x1, x2) = [0 0 1]x2 [0 0 0] , [0] [U12](x0, x1, x2) = [0] [0], [0] [isNatKind](x0) = [0] [0], [activate](x0) = x0 , [0 0 0] [U11](x0, x1, x2) = [0 0 0]x2 [0 0 1] , [0] [tt] = [0] [0] orientation: plus#(N,0()) = [1 1 0]N + [1] >= [1 1 0]N + [1] = U51#(isNat(N),N) U51#(tt(),N) = [1 1 0]N + [1] >= [0 1 0]N = activate#(N) activate#(n__plus(X1,X2)) = [1 1 0]X1 + [1 1 0]X2 >= [1 1 0]X1 + [1 1 0]X2 = plus#(X1,X2) plus#(N,0()) = [1 1 0]N + [1] >= [1 1 0]N = isNat#(N) isNat#(n__plus(V1,V2)) = [2 1 0]V1 + [1 1 0]V2 + [1] >= [0 1 0]V2 = activate#(V2) isNat#(n__plus(V1,V2)) = [2 1 0]V1 + [1 1 0]V2 + [1] >= [0 1 0]V1 = activate#(V1) isNat#(n__plus(V1,V2)) = [2 1 0]V1 + [1 1 0]V2 + [1] >= [0 1 0]V1 = isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) = [1 1 0]V1 + [1 1 0]V2 >= [0 1 0]V2 = activate#(V2) isNatKind#(n__plus(V1,V2)) = [1 1 0]V1 + [1 1 0]V2 >= [0 1 0]V1 = activate#(V1) isNatKind#(n__plus(V1,V2)) = [1 1 0]V1 + [1 1 0]V2 >= [0 1 0]V1 = isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) = [1 1 0]V1 + [1 1 0]V2 >= [0 1 0]V2 = U31#(isNatKind(activate(V1)),activate(V2)) U31#(tt(),V2) = [0 1 0]V2 >= [0 1 0]V2 = activate#(V2) U31#(tt(),V2) = [0 1 0]V2 >= [0 1 0]V2 = isNatKind#(activate(V2)) isNatKind#(n__s(V1)) = [0 1 0]V1 >= [0 1 0]V1 = activate#(V1) isNatKind#(n__s(V1)) = [0 1 0]V1 >= [0 1 0]V1 = isNatKind#(activate(V1)) isNat#(n__plus(V1,V2)) = [2 1 0]V1 + [1 1 0]V2 + [1] >= [1 1 0]V1 + [1 1 0]V2 + [1] = U11#(isNatKind(activate(V1)),activate(V1),activate(V2)) U11#(tt(),V1,V2) = [1 1 0]V1 + [1 1 0]V2 + [1] >= [0 1 0]V2 = activate#(V2) U11#(tt(),V1,V2) = [1 1 0]V1 + [1 1 0]V2 + [1] >= [0 1 0]V1 = activate#(V1) U11#(tt(),V1,V2) = [1 1 0]V1 + [1 1 0]V2 + [1] >= [0 1 0]V1 = isNatKind#(activate(V1)) U11#(tt(),V1,V2) = [1 1 0]V1 + [1 1 0]V2 + [1] >= [1 1 0]V1 + [1 1 0]V2 + [1] = U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) U12#(tt(),V1,V2) = [1 1 0]V1 + [1 1 0]V2 + [1] >= [0 1 0]V1 = activate#(V1) U12#(tt(),V1,V2) = [1 1 0]V1 + [1 1 0]V2 + [1] >= [0 1 0]V2 = activate#(V2) U12#(tt(),V1,V2) = [1 1 0]V1 + [1 1 0]V2 + [1] >= [0 1 0]V2 = isNatKind#(activate(V2)) U12#(tt(),V1,V2) = [1 1 0]V1 + [1 1 0]V2 + [1] >= [1 1 0]V1 + [1 1 0]V2 + [1] = U13#(isNatKind(activate(V2)),activate(V1),activate(V2)) U13#(tt(),V1,V2) = [1 1 0]V1 + [1 1 0]V2 + [1] >= [0 1 0]V1 = activate#(V1) U13#(tt(),V1,V2) = [1 1 0]V1 + [1 1 0]V2 + [1] >= [0 1 0]V2 = activate#(V2) U13#(tt(),V1,V2) = [1 1 0]V1 + [1 1 0]V2 + [1] >= [0 1 0]V2 = isNatKind#(activate(V2)) U13#(tt(),V1,V2) = [1 1 0]V1 + [1 1 0]V2 + [1] >= [1 1 0]V1 + [1 1 0]V2 = U14#(isNatKind(activate(V2)),activate(V1),activate(V2)) U14#(tt(),V1,V2) = [1 1 0]V1 + [1 1 0]V2 >= [0 1 0]V2 = activate#(V2) U14#(tt(),V1,V2) = [1 1 0]V1 + [1 1 0]V2 >= [0 1 0]V1 = activate#(V1) U14#(tt(),V1,V2) = [1 1 0]V1 + [1 1 0]V2 >= [1 1 0]V1 = isNat#(activate(V1)) isNat#(n__s(V1)) = [1 1 0]V1 >= [0 1 0]V1 = activate#(V1) isNat#(n__s(V1)) = [1 1 0]V1 >= [0 1 0]V1 = isNatKind#(activate(V1)) isNat#(n__s(V1)) = [1 1 0]V1 >= [1 1 0]V1 = U21#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) = [1 1 0]V1 >= [0 1 0]V1 = activate#(V1) U21#(tt(),V1) = [1 1 0]V1 >= [0 1 0]V1 = isNatKind#(activate(V1)) U21#(tt(),V1) = [1 1 0]V1 >= [1 1 0]V1 = U22#(isNatKind(activate(V1)),activate(V1)) U22#(tt(),V1) = [1 1 0]V1 >= [0 1 0]V1 = activate#(V1) U22#(tt(),V1) = [1 1 0]V1 >= [1 1 0]V1 = isNat#(activate(V1)) U14#(tt(),V1,V2) = [1 1 0]V1 + [1 1 0]V2 >= [1 1 0]V2 = U15#(isNat(activate(V1)),activate(V2)) U15#(tt(),V2) = [1 1 0]V2 >= [0 1 0]V2 = activate#(V2) U15#(tt(),V2) = [1 1 0]V2 >= [1 1 0]V2 = isNat#(activate(V2)) plus#(N,s(M)) = [1 1 0]M + [1 1 0]N >= [1 1 0]M = isNat#(M) plus#(N,s(M)) = [1 1 0]M + [1 1 0]N >= [1 1 0]M + [1 1 0]N = U61#(isNat(M),M,N) U61#(tt(),M,N) = [1 1 0]M + [1 1 0]N >= [0 1 0]N = activate#(N) U61#(tt(),M,N) = [1 1 0]M + [1 1 0]N >= [0 1 0]M = activate#(M) U61#(tt(),M,N) = [1 1 0]M + [1 1 0]N >= [0 1 0]M = isNatKind#(activate(M)) U61#(tt(),M,N) = [1 1 0]M + [1 1 0]N >= [1 1 0]M + [1 1 0]N = U62#(isNatKind(activate(M)),activate(M),activate(N)) U62#(tt(),M,N) = [1 1 0]M + [1 1 0]N >= [0 1 0]M = activate#(M) U62#(tt(),M,N) = [1 1 0]M + [1 1 0]N >= [0 1 0]N = activate#(N) U62#(tt(),M,N) = [1 1 0]M + [1 1 0]N >= [1 1 0]N = isNat#(activate(N)) U62#(tt(),M,N) = [1 1 0]M + [1 1 0]N >= [1 1 0]M + [1 1 0]N = U63#(isNat(activate(N)),activate(M),activate(N)) U63#(tt(),M,N) = [1 1 0]M + [1 1 0]N >= [0 1 0]M = activate#(M) U63#(tt(),M,N) = [1 1 0]M + [1 1 0]N >= [0 1 0]N = activate#(N) U63#(tt(),M,N) = [1 1 0]M + [1 1 0]N >= [0 1 0]N = isNatKind#(activate(N)) U63#(tt(),M,N) = [1 1 0]M + [1 1 0]N >= [1 1 0]M + [1 1 0]N = U64#(isNatKind(activate(N)),activate(M),activate(N)) U64#(tt(),M,N) = [1 1 0]M + [1 1 0]N >= [0 1 0]M = activate#(M) U64#(tt(),M,N) = [1 1 0]M + [1 1 0]N >= [0 1 0]N = activate#(N) U64#(tt(),M,N) = [1 1 0]M + [1 1 0]N >= [1 1 0]M + [1 1 0]N = plus#(activate(N),activate(M)) U51#(tt(),N) = [1 1 0]N + [1] >= [0 1 0]N = isNatKind#(activate(N)) U51#(tt(),N) = [1 1 0]N + [1] >= [0 1 0]N + [1] = U52#(isNatKind(activate(N)),activate(N)) U52#(tt(),N) = [0 1 0]N + [1] >= [0 1 0]N = activate#(N) [0 0 0] [0] U11(tt(),V1,V2) = [0 0 0]V2 >= [0] = U12(isNatKind(activate(V1)),activate(V1),activate(V2)) [0 0 1] [0] [0] [0 0 0] U12(tt(),V1,V2) = [0] >= [0 0 1]V2 = U13(isNatKind(activate(V2)),activate(V1),activate(V2)) [0] [0 0 0] [0 0 0] [0 0 0] [0] U13(tt(),V1,V2) = [0 0 1]V2 >= [0 0 0]V2 + [0] = U14(isNatKind(activate(V2)),activate(V1),activate(V2)) [0 0 0] [1 0 0] [1] [0 0 0] [0] [0] U14(tt(),V1,V2) = [0 0 0]V2 + [0] >= [0] = U15(isNat(activate(V1)),activate(V2)) [1 0 0] [1] [0] [0] [1] U15(tt(),V2) = [0] >= [0] = U16(isNat(activate(V2))) [0] [0] [1] [0] U16(tt()) = [0] >= [0] = tt() [0] [0] [0] [0 0 0] U21(tt(),V1) = [0] >= [0 1 0]V1 = U22(isNatKind(activate(V1)),activate(V1)) [0] [0 0 0] [0 0 0] [0] U22(tt(),V1) = [0 1 0]V1 >= [0] = U23(isNat(activate(V1))) [0 0 0] [0] [0] [0] U23(tt()) = [0] >= [0] = tt() [0] [0] [0] [0] U31(tt(),V2) = [1] >= [0] = U32(isNatKind(activate(V2))) [0] [0] [0] [0] U32(tt()) = [0] >= [0] = tt() [0] [0] [0] [0] U41(tt()) = [0] >= [0] = tt() [0] [0] [1 0 0] [0] [1 0 0] [0] U51(tt(),N) = [0 1 0]N + [1] >= [0 1 0]N + [1] = U52(isNatKind(activate(N)),activate(N)) [1 0 1] [1] [1 0 1] [1] [1 0 0] [0] U52(tt(),N) = [0 1 0]N + [1] >= N = activate(N) [1 0 1] [1] [0 0 0] [1 0 0] [1] [0 0 0] [1 0 0] [1] U61(tt(),M,N) = [1 1 0]M + [1 1 0]N + [0] >= [1 1 0]M + [1 1 0]N + [0] = U62(isNatKind(activate(M)),activate(M),activate(N)) [0 0 0] [1 0 0] [1] [0 0 0] [1 0 0] [1] [0 0 0] [1 0 0] [1] [0 0 0] [1 0 0] [1] U62(tt(),M,N) = [1 1 0]M + [1 1 0]N + [0] >= [1 1 0]M + [1 1 0]N + [0] = U63(isNat(activate(N)),activate(M),activate(N)) [0 0 0] [1 0 0] [1] [0 0 0] [1 0 0] [1] [0 0 0] [1 0 0] [1] [0 0 0] [1 0 0] [1] U63(tt(),M,N) = [1 1 0]M + [1 1 0]N + [0] >= [1 1 0]M + [1 1 0]N + [0] = U64(isNatKind(activate(N)),activate(M),activate(N)) [0 0 0] [1 0 0] [1] [0 0 0] [1 0 0] [1] [0 0 0] [1 0 0] [1] [0 0 0] [1 0 0] [1] U64(tt(),M,N) = [1 1 0]M + [1 1 0]N + [0] >= [1 1 0]M + [1 1 0]N + [0] = s(plus(activate(N),activate(M))) [0 0 0] [1 0 0] [1] [0 0 0] [0 0 0] [0] [0] [0] isNat(n__0()) = [0] >= [0] = tt() [0] [0] [0] [0 0 0] isNat(n__plus(V1,V2)) = [0] >= [0 0 0]V2 = U11(isNatKind(activate(V1)),activate(V1),activate(V2)) [0] [0 0 1] [0] [0] isNat(n__s(V1)) = [0] >= [0] = U21(isNatKind(activate(V1)),activate(V1)) [0] [0] [0] [0] isNatKind(n__0()) = [0] >= [0] = tt() [0] [0] [0] [0] isNatKind(n__plus(V1,V2)) = [0] >= [1] = U31(isNatKind(activate(V1)),activate(V2)) [0] [0] [0] [0] isNatKind(n__s(V1)) = [0] >= [0] = U41(isNatKind(activate(V1))) [0] [0] [1 0 0] [1] [1 0 0] [0] plus(N,0()) = [1 1 0]N + [1] >= [0 1 0]N + [1] = U51(isNat(N),N) [1 0 1] [1] [1 0 1] [1] [0 0 0] [1 0 0] [1] [0 0 0] [1 0 0] [1] plus(N,s(M)) = [1 1 0]M + [1 1 0]N + [0] >= [1 1 0]M + [1 1 0]N + [0] = U61(isNat(M),M,N) [0 0 0] [1 0 1] [1] [0 0 0] [1 0 0] [1] [1] [1] 0() = [0] >= [0] = n__0() [0] [0] [1 0 0] [0 0 0] [1] [1 0 0] [0 0 0] [1] plus(X1,X2) = [1 1 0]X1 + [1 1 0]X2 + [0] >= [1 1 0]X1 + [1 1 0]X2 + [0] = n__plus(X1,X2) [1 0 1] [0 0 0] [1] [1 0 1] [0 0 0] [1] [1 0 0] [1 0 0] s(X) = [0 1 0]X >= [0 1 0]X = n__s(X) [0 0 0] [0 0 0] [1] [1] activate(n__0()) = [0] >= [0] = 0() [0] [0] [1 0 0] [0 0 0] [1] [1 0 0] [0 0 0] [1] activate(n__plus(X1,X2)) = [1 1 0]X1 + [1 1 0]X2 + [0] >= [1 1 0]X1 + [1 1 0]X2 + [0] = plus(X1,X2) [1 0 1] [0 0 0] [1] [1 0 1] [0 0 0] [1] [1 0 0] [1 0 0] activate(n__s(X)) = [0 1 0]X >= [0 1 0]X = s(X) [0 0 0] [0 0 0] activate(X) = X >= X = X problem: DPs: plus#(N,0()) -> U51#(isNat(N),N) activate#(n__plus(X1,X2)) -> plus#(X1,X2) 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) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) U12#(tt(),V1,V2) -> U13#(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) -> U52#(isNatKind(activate(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(X1,X2) activate(n__s(X)) -> s(X) activate(X) -> X Restore Modifier: DPs: plus#(N,0()) -> U51#(isNat(N),N) activate#(n__plus(X1,X2)) -> plus#(X1,X2) 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) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)) U12#(tt(),V1,V2) -> U13#(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) -> U52#(isNatKind(activate(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(X1,X2) activate(n__s(X)) -> s(X) activate(X) -> X SCC Processor: #sccs: 1 #rules: 34 #arcs: 194/2025 DPs: plus#(N,s(M)) -> U61#(isNat(M),M,N) U61#(tt(),M,N) -> activate#(N) activate#(n__plus(X1,X2)) -> plus#(X1,X2) plus#(N,s(M)) -> isNat#(M) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> 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__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)) 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)) 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(X1,X2) activate(n__s(X)) -> s(X) activate(X) -> X Matrix Interpretation Processor: dim=3 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(X1,X2) activate(n__s(X)) -> s(X) activate(X) -> X interpretation: [plus#](x0, x1) = [1 0 0]x0 + [1 0 0]x1, [U64#](x0, x1, x2) = [1 0 0]x1 + [1 0 0]x2, [U63#](x0, x1, x2) = [1 0 0]x1 + [1 0 0]x2, [U62#](x0, x1, x2) = [1 0 0]x1 + [1 0 0]x2, [U61#](x0, x1, x2) = [1 0 0]x1 + [1 0 0]x2, [U31#](x0, x1) = [1 0 0]x1, [U22#](x0, x1) = [1 0 0]x1, [U21#](x0, x1) = [1 0 0]x1, [isNat#](x0) = [1 0 0]x0, [isNatKind#](x0) = [1 0 0]x0, [activate#](x0) = [1 0 0]x0, [0] [0] = [0] [0], [1 0 0] [1] [n__s](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [1 0 0] [1 0 0] [n__plus](x0, x1) = [0 1 0]x0 + [0 0 1]x1 [1 1 1] [1 0 1] , [0] [n__0] = [0] [0], [1 0 0] [1] [s](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [1 0 0] [1 0 0] [plus](x0, x1) = [0 1 0]x0 + [0 0 1]x1 [1 1 1] [1 0 1] , [1 0 0] [1 0 0] [1] [U64](x0, x1, x2) = [0 0 0]x1 + [0 0 0]x2 + [0] [0 0 0] [0 1 1] [1], [1 0 0] [1 0 0] [1] [U63](x0, x1, x2) = [0 0 0]x1 + [0 0 0]x2 + [0] [0 0 0] [0 1 1] [1], [1 0 0] [1 0 0] [1] [U62](x0, x1, x2) = [0 0 0]x1 + [0 0 0]x2 + [0] [0 0 0] [1 1 1] [1], [1 0 0] [1 0 0] [1] [U61](x0, x1, x2) = [0 0 0]x1 + [0 1 0]x2 + [0] [1 0 0] [1 1 1] [1], [U52](x0, x1) = x1 , [1 0 0] [U51](x0, x1) = [0 1 0]x1 [0 1 1] , [0] [U41](x0) = [0] [0], [0] [U32](x0) = [0] [1], [1] [U31](x0, x1) = [0] [0], [0] [U23](x0) = [0] [0], [0 1 0] [U22](x0, x1) = [0 0 0]x0 [0 0 0] , [1] [U21](x0, x1) = [0] [0], [0] [U16](x0) = [0] [0], [0] [U15](x0, x1) = [0] [0], [0] [isNat](x0) = [1] [0], [0] [U14](x0, x1, x2) = [0] [0], [0 0 0] [U13](x0, x1, x2) = [0 1 0]x2 [0 0 0] , [0 0 0] [U12](x0, x1, x2) = [0 1 0]x0 [0 0 0] , [0 0 0] [isNatKind](x0) = [0 1 0]x0 [1 0 0] , [activate](x0) = x0 , [0 1 0] [0 0 0] [U11](x0, x1, x2) = [0 0 1]x1 + [0 1 0]x2 [0 0 0] [0 0 0] , [0] [tt] = [0] [0] orientation: plus#(N,s(M)) = [1 0 0]M + [1 0 0]N + [1] >= [1 0 0]M + [1 0 0]N = U61#(isNat(M),M,N) U61#(tt(),M,N) = [1 0 0]M + [1 0 0]N >= [1 0 0]N = activate#(N) activate#(n__plus(X1,X2)) = [1 0 0]X1 + [1 0 0]X2 >= [1 0 0]X1 + [1 0 0]X2 = plus#(X1,X2) plus#(N,s(M)) = [1 0 0]M + [1 0 0]N + [1] >= [1 0 0]M = isNat#(M) isNat#(n__s(V1)) = [1 0 0]V1 + [1] >= [1 0 0]V1 = activate#(V1) isNat#(n__s(V1)) = [1 0 0]V1 + [1] >= [1 0 0]V1 = isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) = [1 0 0]V1 + [1 0 0]V2 >= [1 0 0]V2 = activate#(V2) isNatKind#(n__plus(V1,V2)) = [1 0 0]V1 + [1 0 0]V2 >= [1 0 0]V1 = activate#(V1) isNatKind#(n__plus(V1,V2)) = [1 0 0]V1 + [1 0 0]V2 >= [1 0 0]V1 = isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) = [1 0 0]V1 + [1 0 0]V2 >= [1 0 0]V2 = U31#(isNatKind(activate(V1)),activate(V2)) U31#(tt(),V2) = [1 0 0]V2 >= [1 0 0]V2 = activate#(V2) U31#(tt(),V2) = [1 0 0]V2 >= [1 0 0]V2 = isNatKind#(activate(V2)) isNatKind#(n__s(V1)) = [1 0 0]V1 + [1] >= [1 0 0]V1 = activate#(V1) isNatKind#(n__s(V1)) = [1 0 0]V1 + [1] >= [1 0 0]V1 = isNatKind#(activate(V1)) isNat#(n__s(V1)) = [1 0 0]V1 + [1] >= [1 0 0]V1 = U21#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) = [1 0 0]V1 >= [1 0 0]V1 = activate#(V1) U21#(tt(),V1) = [1 0 0]V1 >= [1 0 0]V1 = isNatKind#(activate(V1)) U21#(tt(),V1) = [1 0 0]V1 >= [1 0 0]V1 = U22#(isNatKind(activate(V1)),activate(V1)) U22#(tt(),V1) = [1 0 0]V1 >= [1 0 0]V1 = activate#(V1) U22#(tt(),V1) = [1 0 0]V1 >= [1 0 0]V1 = isNat#(activate(V1)) U61#(tt(),M,N) = [1 0 0]M + [1 0 0]N >= [1 0 0]M = activate#(M) U61#(tt(),M,N) = [1 0 0]M + [1 0 0]N >= [1 0 0]M = isNatKind#(activate(M)) U61#(tt(),M,N) = [1 0 0]M + [1 0 0]N >= [1 0 0]M + [1 0 0]N = U62#(isNatKind(activate(M)),activate(M),activate(N)) U62#(tt(),M,N) = [1 0 0]M + [1 0 0]N >= [1 0 0]M = activate#(M) U62#(tt(),M,N) = [1 0 0]M + [1 0 0]N >= [1 0 0]N = activate#(N) U62#(tt(),M,N) = [1 0 0]M + [1 0 0]N >= [1 0 0]N = isNat#(activate(N)) U62#(tt(),M,N) = [1 0 0]M + [1 0 0]N >= [1 0 0]M + [1 0 0]N = U63#(isNat(activate(N)),activate(M),activate(N)) U63#(tt(),M,N) = [1 0 0]M + [1 0 0]N >= [1 0 0]M = activate#(M) U63#(tt(),M,N) = [1 0 0]M + [1 0 0]N >= [1 0 0]N = activate#(N) U63#(tt(),M,N) = [1 0 0]M + [1 0 0]N >= [1 0 0]N = isNatKind#(activate(N)) U63#(tt(),M,N) = [1 0 0]M + [1 0 0]N >= [1 0 0]M + [1 0 0]N = U64#(isNatKind(activate(N)),activate(M),activate(N)) U64#(tt(),M,N) = [1 0 0]M + [1 0 0]N >= [1 0 0]M = activate#(M) U64#(tt(),M,N) = [1 0 0]M + [1 0 0]N >= [1 0 0]N = activate#(N) U64#(tt(),M,N) = [1 0 0]M + [1 0 0]N >= [1 0 0]M + [1 0 0]N = plus#(activate(N),activate(M)) [0 1 0] [0 0 0] [0 0 0] U11(tt(),V1,V2) = [0 0 1]V1 + [0 1 0]V2 >= [0 1 0]V1 = U12(isNatKind(activate(V1)),activate(V1),activate(V2)) [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] U12(tt(),V1,V2) = [0] >= [0 1 0]V2 = U13(isNatKind(activate(V2)),activate(V1),activate(V2)) [0] [0 0 0] [0 0 0] [0] U13(tt(),V1,V2) = [0 1 0]V2 >= [0] = U14(isNatKind(activate(V2)),activate(V1),activate(V2)) [0 0 0] [0] [0] [0] U14(tt(),V1,V2) = [0] >= [0] = U15(isNat(activate(V1)),activate(V2)) [0] [0] [0] [0] U15(tt(),V2) = [0] >= [0] = U16(isNat(activate(V2))) [0] [0] [0] [0] U16(tt()) = [0] >= [0] = tt() [0] [0] [1] [0 1 0] U21(tt(),V1) = [0] >= [0 0 0]V1 = U22(isNatKind(activate(V1)),activate(V1)) [0] [0 0 0] [0] [0] U22(tt(),V1) = [0] >= [0] = U23(isNat(activate(V1))) [0] [0] [0] [0] U23(tt()) = [0] >= [0] = tt() [0] [0] [1] [0] U31(tt(),V2) = [0] >= [0] = U32(isNatKind(activate(V2))) [0] [1] [0] [0] U32(tt()) = [0] >= [0] = tt() [1] [0] [0] [0] U41(tt()) = [0] >= [0] = tt() [0] [0] [1 0 0] U51(tt(),N) = [0 1 0]N >= N = U52(isNatKind(activate(N)),activate(N)) [0 1 1] U52(tt(),N) = N >= N = activate(N) [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] U61(tt(),M,N) = [0 0 0]M + [0 1 0]N + [0] >= [0 0 0]M + [0 0 0]N + [0] = U62(isNatKind(activate(M)),activate(M),activate(N)) [1 0 0] [1 1 1] [1] [0 0 0] [1 1 1] [1] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] U62(tt(),M,N) = [0 0 0]M + [0 0 0]N + [0] >= [0 0 0]M + [0 0 0]N + [0] = U63(isNat(activate(N)),activate(M),activate(N)) [0 0 0] [1 1 1] [1] [0 0 0] [0 1 1] [1] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] U63(tt(),M,N) = [0 0 0]M + [0 0 0]N + [0] >= [0 0 0]M + [0 0 0]N + [0] = U64(isNatKind(activate(N)),activate(M),activate(N)) [0 0 0] [0 1 1] [1] [0 0 0] [0 1 1] [1] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] U64(tt(),M,N) = [0 0 0]M + [0 0 0]N + [0] >= [0 0 0]M + [0 0 0]N + [0] = s(plus(activate(N),activate(M))) [0 0 0] [0 1 1] [1] [0 0 0] [0 0 0] [0] [0] [0] isNat(n__0()) = [1] >= [0] = tt() [0] [0] [0] [0 1 0] [0 0 0] isNat(n__plus(V1,V2)) = [1] >= [0 0 1]V1 + [0 1 0]V2 = U11(isNatKind(activate(V1)),activate(V1),activate(V2)) [0] [0 0 0] [0 0 0] [0] [1] isNat(n__s(V1)) = [1] >= [0] = U21(isNatKind(activate(V1)),activate(V1)) [0] [0] [0] [0] isNatKind(n__0()) = [0] >= [0] = tt() [0] [0] [0 0 0] [0 0 0] [1] isNatKind(n__plus(V1,V2)) = [0 1 0]V1 + [0 0 1]V2 >= [0] = U31(isNatKind(activate(V1)),activate(V2)) [1 0 0] [1 0 0] [0] [0 0 0] [0] [0] isNatKind(n__s(V1)) = [0 0 0]V1 + [0] >= [0] = U41(isNatKind(activate(V1))) [1 0 0] [1] [0] [1 0 0] [1 0 0] plus(N,0()) = [0 1 0]N >= [0 1 0]N = U51(isNat(N),N) [1 1 1] [0 1 1] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] plus(N,s(M)) = [0 0 0]M + [0 1 0]N + [0] >= [0 0 0]M + [0 1 0]N + [0] = U61(isNat(M),M,N) [1 0 0] [1 1 1] [1] [1 0 0] [1 1 1] [1] [0] [0] 0() = [0] >= [0] = n__0() [0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] plus(X1,X2) = [0 1 0]X1 + [0 0 1]X2 >= [0 1 0]X1 + [0 0 1]X2 = n__plus(X1,X2) [1 1 1] [1 0 1] [1 1 1] [1 0 1] [1 0 0] [1] [1 0 0] [1] s(X) = [0 0 0]X + [0] >= [0 0 0]X + [0] = n__s(X) [0 0 0] [0] [0 0 0] [0] [0] [0] activate(n__0()) = [0] >= [0] = 0() [0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] activate(n__plus(X1,X2)) = [0 1 0]X1 + [0 0 1]X2 >= [0 1 0]X1 + [0 0 1]X2 = plus(X1,X2) [1 1 1] [1 0 1] [1 1 1] [1 0 1] [1 0 0] [1] [1 0 0] [1] activate(n__s(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = s(X) [0 0 0] [0] [0 0 0] [0] activate(X) = X >= X = X problem: DPs: U61#(tt(),M,N) -> activate#(N) activate#(n__plus(X1,X2)) -> plus#(X1,X2) 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)) 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)) 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)) 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(X1,X2) activate(n__s(X)) -> s(X) activate(X) -> X Restore Modifier: DPs: U61#(tt(),M,N) -> activate#(N) activate#(n__plus(X1,X2)) -> plus#(X1,X2) 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)) 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)) 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)) 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(X1,X2) activate(n__s(X)) -> s(X) activate(X) -> X SCC Processor: #sccs: 1 #rules: 3 #arcs: 92/729 DPs: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)) U31#(tt(),V2) -> isNatKind#(activate(V2)) 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(X1,X2) activate(n__s(X)) -> s(X) activate(X) -> X KBO Processor: argument filtering: pi(tt) = [] pi(U11) = [] pi(activate) = [0] pi(isNatKind) = 0 pi(U12) = [] pi(U13) = [] pi(U14) = [] pi(isNat) = [] pi(U15) = [] pi(U16) = [] pi(U21) = [] pi(U22) = [] pi(U23) = [] pi(U31) = [] pi(U32) = [] pi(U41) = [] pi(U51) = [0,1] pi(U52) = [1] pi(U61) = [0,2] pi(U62) = [] pi(U63) = [] pi(U64) = [] pi(plus) = [0,1] pi(s) = [] pi(n__0) = [] pi(n__plus) = [0,1] pi(n__s) = [] pi(0) = [] pi(isNatKind#) = 0 pi(U31#) = [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(X1,X2) activate(n__s(X)) -> s(X) activate(X) -> X weight function: w0 = 1 w(U31#) = w(0) = w(n__s) = w(n__plus) = w(n__0) = w(s) = w( plus) = w(U64) = w(U63) = w(U62) = w(U61) = w(U52) = w(U51) = w( U41) = w(U32) = w(U31) = w(U23) = w(U22) = w(U21) = w(U16) = w( U15) = w(isNat) = w(U14) = w(U13) = w(U12) = w(activate) = w( U11) = w(tt) = 1 w(isNatKind#) = w(isNatKind) = 0 precedence: 0 ~ plus > isNat > U62 ~ U11 > U63 ~ U21 ~ U12 > n__plus ~ U13 > U14 > U64 ~ U51 ~ U22 ~ U15 > U31# ~ s ~ U52 ~ U23 ~ U16 > isNatKind# ~ n__s ~ n__0 ~ U61 ~ U41 ~ U32 ~ U31 ~ isNatKind ~ activate ~ tt 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(X1,X2) activate(n__s(X)) -> s(X) activate(X) -> X Qed