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#(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#() 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#() 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#(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#() 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#() 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#(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#() 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#() 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#(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#() 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#() 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 KBO Processor: argument filtering: pi(tt) = [] pi(U11) = 1 pi(activate) = [0] pi(isNatKind) = 0 pi(U12) = 1 pi(U13) = 0 pi(U14) = 0 pi(isNat) = [] pi(U15) = 0 pi(U16) = 0 pi(U21) = 1 pi(U22) = 0 pi(U23) = 0 pi(U31) = 1 pi(U32) = [] pi(U41) = 0 pi(U51) = [1] pi(U52) = [1] pi(U61) = [1,2] pi(U62) = [1,2] pi(U63) = [1,2] pi(U64) = [1,2] pi(plus) = [0,1] pi(s) = [0] pi(n__0) = [] pi(n__plus) = [0,1] pi(n__s) = [0] pi(0) = [] pi(U11#) = [1,2] pi(activate#) = [0] pi(isNatKind#) = 0 pi(U12#) = [1,2] pi(U13#) = [1,2] pi(U14#) = [1,2] pi(isNat#) = [0] pi(U15#) = [1] pi(U21#) = [1] pi(U22#) = [1] pi(U31#) = [1] pi(U51#) = [1] pi(U52#) = [1] pi(U61#) = [1,2] pi(U62#) = [1,2] pi(U63#) = [1,2] pi(U64#) = [1,2] pi(plus#) = [0,1] 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(X1,X2) activate(n__s(X)) -> s(X) activate(X) -> X weight function: w0 = 1 w(U62#) = w(U61#) = w(U52#) = w(U51#) = w(U31#) = w(U22#) = w(U21#) = w( U15#) = w(isNat#) = w(U13#) = w(U12#) = w(isNatKind#) = w(activate#) = w( U11#) = w(0) = w(n__s) = w(n__0) = w(s) = w(U64) = w(U63) = w(U62) = w( U61) = w(U52) = w(U51) = w(U41) = w(U32) = w(U23) = w(isNat) = w( U13) = w(tt) = 1 w(plus#) = w(U64#) = w(U63#) = w(U14#) = w(n__plus) = w(plus) = w( U31) = w(U22) = w(U21) = w(U16) = w(U15) = w(U14) = w(U12) = w( isNatKind) = w(activate) = w(U11) = 0 precedence: U63# ~ U21# ~ activate ~ tt > U64# ~ 0 > plus# > U14# > U22# ~ U15# ~ plus > U61 > U62 > isNat# ~ n__plus ~ U63 > U64 > U61# ~ U51# ~ s > U11# > U52# ~ U12# > U31# ~ n__s ~ U51 > U62# ~ U13# ~ isNatKind# ~ activate# ~ n__0 ~ U52 ~ U41 ~ U32 ~ U31 ~ U23 ~ U22 ~ U21 ~ U16 ~ U15 ~ isNat ~ U14 ~ U13 ~ U12 ~ isNatKind ~ U11 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