YES Problem: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2)) U12(tt(),V2) -> U13(isNat(activate(V2))) U13(tt()) -> tt() U21(tt(),V1) -> U22(isNat(activate(V1))) U22(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> s(plus(activate(N),activate(M))) and(tt(),X) -> activate(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind(n__s(V1)) -> isNatKind(activate(V1)) plus(N,0()) -> U31(and(isNat(N),n__isNatKind(N)),N) plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) isNatKind(X) -> n__isNatKind(X) s(X) -> n__s(X) and(X1,X2) -> n__and(X1,X2) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(X) activate(n__and(X1,X2)) -> and(X1,X2) activate(X) -> X Proof: DP Processor: DPs: U11#(tt(),V1,V2) -> activate#(V2) U11#(tt(),V1,V2) -> activate#(V1) U11#(tt(),V1,V2) -> isNat#(activate(V1)) U11#(tt(),V1,V2) -> U12#(isNat(activate(V1)),activate(V2)) U12#(tt(),V2) -> activate#(V2) U12#(tt(),V2) -> isNat#(activate(V2)) U12#(tt(),V2) -> U13#(isNat(activate(V2))) U21#(tt(),V1) -> activate#(V1) U21#(tt(),V1) -> isNat#(activate(V1)) U21#(tt(),V1) -> U22#(isNat(activate(V1))) U31#(tt(),N) -> activate#(N) U41#(tt(),M,N) -> activate#(M) U41#(tt(),M,N) -> activate#(N) U41#(tt(),M,N) -> plus#(activate(N),activate(M)) U41#(tt(),M,N) -> s#(plus(activate(N),activate(M))) and#(tt(),X) -> activate#(X) 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)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNat#(n__plus(V1,V2)) -> U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) 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)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind#(n__s(V1)) -> activate#(V1) isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) plus#(N,0()) -> isNat#(N) plus#(N,0()) -> and#(isNat(N),n__isNatKind(N)) plus#(N,0()) -> U31#(and(isNat(N),n__isNatKind(N)),N) plus#(N,s(M)) -> isNat#(N) plus#(N,s(M)) -> isNat#(M) plus#(N,s(M)) -> and#(isNat(M),n__isNatKind(M)) plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))) plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) activate#(n__0()) -> 0#() activate#(n__plus(X1,X2)) -> plus#(X1,X2) activate#(n__isNatKind(X)) -> isNatKind#(X) activate#(n__s(X)) -> s#(X) activate#(n__and(X1,X2)) -> and#(X1,X2) TRS: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2)) U12(tt(),V2) -> U13(isNat(activate(V2))) U13(tt()) -> tt() U21(tt(),V1) -> U22(isNat(activate(V1))) U22(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> s(plus(activate(N),activate(M))) and(tt(),X) -> activate(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind(n__s(V1)) -> isNatKind(activate(V1)) plus(N,0()) -> U31(and(isNat(N),n__isNatKind(N)),N) plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) isNatKind(X) -> n__isNatKind(X) s(X) -> n__s(X) and(X1,X2) -> n__and(X1,X2) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(X) activate(n__and(X1,X2)) -> and(X1,X2) activate(X) -> X TDG Processor: DPs: U11#(tt(),V1,V2) -> activate#(V2) U11#(tt(),V1,V2) -> activate#(V1) U11#(tt(),V1,V2) -> isNat#(activate(V1)) U11#(tt(),V1,V2) -> U12#(isNat(activate(V1)),activate(V2)) U12#(tt(),V2) -> activate#(V2) U12#(tt(),V2) -> isNat#(activate(V2)) U12#(tt(),V2) -> U13#(isNat(activate(V2))) U21#(tt(),V1) -> activate#(V1) U21#(tt(),V1) -> isNat#(activate(V1)) U21#(tt(),V1) -> U22#(isNat(activate(V1))) U31#(tt(),N) -> activate#(N) U41#(tt(),M,N) -> activate#(M) U41#(tt(),M,N) -> activate#(N) U41#(tt(),M,N) -> plus#(activate(N),activate(M)) U41#(tt(),M,N) -> s#(plus(activate(N),activate(M))) and#(tt(),X) -> activate#(X) 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)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNat#(n__plus(V1,V2)) -> U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) 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)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind#(n__s(V1)) -> activate#(V1) isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) plus#(N,0()) -> isNat#(N) plus#(N,0()) -> and#(isNat(N),n__isNatKind(N)) plus#(N,0()) -> U31#(and(isNat(N),n__isNatKind(N)),N) plus#(N,s(M)) -> isNat#(N) plus#(N,s(M)) -> isNat#(M) plus#(N,s(M)) -> and#(isNat(M),n__isNatKind(M)) plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))) plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) activate#(n__0()) -> 0#() activate#(n__plus(X1,X2)) -> plus#(X1,X2) activate#(n__isNatKind(X)) -> isNatKind#(X) activate#(n__s(X)) -> s#(X) activate#(n__and(X1,X2)) -> and#(X1,X2) TRS: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2)) U12(tt(),V2) -> U13(isNat(activate(V2))) U13(tt()) -> tt() U21(tt(),V1) -> U22(isNat(activate(V1))) U22(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> s(plus(activate(N),activate(M))) and(tt(),X) -> activate(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind(n__s(V1)) -> isNatKind(activate(V1)) plus(N,0()) -> U31(and(isNat(N),n__isNatKind(N)),N) plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) isNatKind(X) -> n__isNatKind(X) s(X) -> n__s(X) and(X1,X2) -> n__and(X1,X2) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(X) activate(n__and(X1,X2)) -> and(X1,X2) activate(X) -> X graph: 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)) -> and#(isNatKind(activate(V1)),n__isNatKind(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__and(X1,X2)) -> and#(X1,X2) isNatKind#(n__s(V1)) -> activate#(V1) -> activate#(n__s(X)) -> s#(X) isNatKind#(n__s(V1)) -> activate#(V1) -> activate#(n__isNatKind(X)) -> isNatKind#(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)) -> 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)) -> and#(isNatKind(activate(V1)),n__isNatKind(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)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) -> and#(tt(),X) -> activate#(X) isNatKind#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__and(X1,X2)) -> and#(X1,X2) isNatKind#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__s(X)) -> s#(X) isNatKind#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__isNatKind(X)) -> isNatKind#(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__and(X1,X2)) -> and#(X1,X2) isNatKind#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__s(X)) -> s#(X) isNatKind#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__isNatKind(X)) -> isNatKind#(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#() and#(tt(),X) -> activate#(X) -> activate#(n__and(X1,X2)) -> and#(X1,X2) and#(tt(),X) -> activate#(X) -> activate#(n__s(X)) -> s#(X) and#(tt(),X) -> activate#(X) -> activate#(n__isNatKind(X)) -> isNatKind#(X) and#(tt(),X) -> activate#(X) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) and#(tt(),X) -> activate#(X) -> activate#(n__0()) -> 0#() plus#(N,0()) -> and#(isNat(N),n__isNatKind(N)) -> and#(tt(),X) -> activate#(X) plus#(N,0()) -> U31#(and(isNat(N),n__isNatKind(N)),N) -> U31#(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#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) plus#(N,0()) -> isNat#(N) -> isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(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)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))) -> and#(tt(),X) -> activate#(X) plus#(N,s(M)) -> and#(isNat(M),n__isNatKind(M)) -> and#(tt(),X) -> activate#(X) plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) -> U41#(tt(),M,N) -> s#(plus(activate(N),activate(M))) plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) -> U41#(tt(),M,N) -> plus#(activate(N),activate(M)) plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) -> U41#(tt(),M,N) -> activate#(N) plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) -> U41#(tt(),M,N) -> activate#(M) plus#(N,s(M)) -> isNat#(N) -> isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) plus#(N,s(M)) -> isNat#(N) -> isNat#(n__s(V1)) -> isNatKind#(activate(V1)) plus#(N,s(M)) -> isNat#(N) -> isNat#(n__s(V1)) -> activate#(V1) plus#(N,s(M)) -> isNat#(N) -> isNat#(n__plus(V1,V2)) -> U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) plus#(N,s(M)) -> isNat#(N) -> isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) plus#(N,s(M)) -> isNat#(N) -> isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) plus#(N,s(M)) -> isNat#(N) -> isNat#(n__plus(V1,V2)) -> activate#(V1) plus#(N,s(M)) -> isNat#(N) -> isNat#(n__plus(V1,V2)) -> activate#(V2) 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#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) plus#(N,s(M)) -> isNat#(M) -> isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(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) U41#(tt(),M,N) -> plus#(activate(N),activate(M)) -> plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) U41#(tt(),M,N) -> plus#(activate(N),activate(M)) -> plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))) U41#(tt(),M,N) -> plus#(activate(N),activate(M)) -> plus#(N,s(M)) -> and#(isNat(M),n__isNatKind(M)) U41#(tt(),M,N) -> plus#(activate(N),activate(M)) -> plus#(N,s(M)) -> isNat#(M) U41#(tt(),M,N) -> plus#(activate(N),activate(M)) -> plus#(N,s(M)) -> isNat#(N) U41#(tt(),M,N) -> plus#(activate(N),activate(M)) -> plus#(N,0()) -> U31#(and(isNat(N),n__isNatKind(N)),N) U41#(tt(),M,N) -> plus#(activate(N),activate(M)) -> plus#(N,0()) -> and#(isNat(N),n__isNatKind(N)) U41#(tt(),M,N) -> plus#(activate(N),activate(M)) -> plus#(N,0()) -> isNat#(N) U41#(tt(),M,N) -> activate#(N) -> activate#(n__and(X1,X2)) -> and#(X1,X2) U41#(tt(),M,N) -> activate#(N) -> activate#(n__s(X)) -> s#(X) U41#(tt(),M,N) -> activate#(N) -> activate#(n__isNatKind(X)) -> isNatKind#(X) U41#(tt(),M,N) -> activate#(N) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) U41#(tt(),M,N) -> activate#(N) -> activate#(n__0()) -> 0#() U41#(tt(),M,N) -> activate#(M) -> activate#(n__and(X1,X2)) -> and#(X1,X2) U41#(tt(),M,N) -> activate#(M) -> activate#(n__s(X)) -> s#(X) U41#(tt(),M,N) -> activate#(M) -> activate#(n__isNatKind(X)) -> isNatKind#(X) U41#(tt(),M,N) -> activate#(M) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) U41#(tt(),M,N) -> activate#(M) -> activate#(n__0()) -> 0#() U31#(tt(),N) -> activate#(N) -> activate#(n__and(X1,X2)) -> and#(X1,X2) U31#(tt(),N) -> activate#(N) -> activate#(n__s(X)) -> s#(X) U31#(tt(),N) -> activate#(N) -> activate#(n__isNatKind(X)) -> isNatKind#(X) U31#(tt(),N) -> activate#(N) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) U31#(tt(),N) -> activate#(N) -> activate#(n__0()) -> 0#() U21#(tt(),V1) -> isNat#(activate(V1)) -> isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) -> isNat#(activate(V1)) -> isNat#(n__s(V1)) -> isNatKind#(activate(V1)) U21#(tt(),V1) -> isNat#(activate(V1)) -> isNat#(n__s(V1)) -> activate#(V1) U21#(tt(),V1) -> isNat#(activate(V1)) -> isNat#(n__plus(V1,V2)) -> U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) U21#(tt(),V1) -> isNat#(activate(V1)) -> isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) U21#(tt(),V1) -> isNat#(activate(V1)) -> isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) U21#(tt(),V1) -> isNat#(activate(V1)) -> isNat#(n__plus(V1,V2)) -> activate#(V1) U21#(tt(),V1) -> isNat#(activate(V1)) -> isNat#(n__plus(V1,V2)) -> activate#(V2) U21#(tt(),V1) -> activate#(V1) -> activate#(n__and(X1,X2)) -> and#(X1,X2) U21#(tt(),V1) -> activate#(V1) -> activate#(n__s(X)) -> s#(X) U21#(tt(),V1) -> activate#(V1) -> activate#(n__isNatKind(X)) -> isNatKind#(X) U21#(tt(),V1) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) U21#(tt(),V1) -> activate#(V1) -> activate#(n__0()) -> 0#() U12#(tt(),V2) -> isNat#(activate(V2)) -> isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) U12#(tt(),V2) -> isNat#(activate(V2)) -> isNat#(n__s(V1)) -> isNatKind#(activate(V1)) U12#(tt(),V2) -> isNat#(activate(V2)) -> isNat#(n__s(V1)) -> activate#(V1) U12#(tt(),V2) -> isNat#(activate(V2)) -> isNat#(n__plus(V1,V2)) -> U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) U12#(tt(),V2) -> isNat#(activate(V2)) -> isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) U12#(tt(),V2) -> isNat#(activate(V2)) -> isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) U12#(tt(),V2) -> isNat#(activate(V2)) -> isNat#(n__plus(V1,V2)) -> activate#(V1) U12#(tt(),V2) -> isNat#(activate(V2)) -> isNat#(n__plus(V1,V2)) -> activate#(V2) U12#(tt(),V2) -> activate#(V2) -> activate#(n__and(X1,X2)) -> and#(X1,X2) U12#(tt(),V2) -> activate#(V2) -> activate#(n__s(X)) -> s#(X) U12#(tt(),V2) -> activate#(V2) -> activate#(n__isNatKind(X)) -> isNatKind#(X) U12#(tt(),V2) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) U12#(tt(),V2) -> activate#(V2) -> activate#(n__0()) -> 0#() 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)) -> and#(isNatKind(activate(V1)),n__isNatKind(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)) -> U21#(isNatKind(activate(V1)),activate(V1)) -> U21#(tt(),V1) -> U22#(isNat(activate(V1))) isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) -> U21#(tt(),V1) -> isNat#(activate(V1)) isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) -> U21#(tt(),V1) -> activate#(V1) isNat#(n__s(V1)) -> activate#(V1) -> activate#(n__and(X1,X2)) -> and#(X1,X2) isNat#(n__s(V1)) -> activate#(V1) -> activate#(n__s(X)) -> s#(X) isNat#(n__s(V1)) -> activate#(V1) -> activate#(n__isNatKind(X)) -> isNatKind#(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)) -> 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)) -> and#(isNatKind(activate(V1)),n__isNatKind(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)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) -> and#(tt(),X) -> activate#(X) isNat#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__and(X1,X2)) -> and#(X1,X2) isNat#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__s(X)) -> s#(X) isNat#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__isNatKind(X)) -> isNatKind#(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__and(X1,X2)) -> and#(X1,X2) isNat#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__s(X)) -> s#(X) isNat#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__isNatKind(X)) -> isNatKind#(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#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) -> U11#(tt(),V1,V2) -> U12#(isNat(activate(V1)),activate(V2)) isNat#(n__plus(V1,V2)) -> U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) -> U11#(tt(),V1,V2) -> isNat#(activate(V1)) isNat#(n__plus(V1,V2)) -> U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) -> U11#(tt(),V1,V2) -> activate#(V1) isNat#(n__plus(V1,V2)) -> U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) -> U11#(tt(),V1,V2) -> activate#(V2) activate#(n__and(X1,X2)) -> and#(X1,X2) -> and#(tt(),X) -> activate#(X) activate#(n__isNatKind(X)) -> isNatKind#(X) -> isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) activate#(n__isNatKind(X)) -> isNatKind#(X) -> isNatKind#(n__s(V1)) -> activate#(V1) activate#(n__isNatKind(X)) -> isNatKind#(X) -> isNatKind#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) activate#(n__isNatKind(X)) -> isNatKind#(X) -> isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) activate#(n__isNatKind(X)) -> isNatKind#(X) -> isNatKind#(n__plus(V1,V2)) -> activate#(V1) activate#(n__isNatKind(X)) -> isNatKind#(X) -> isNatKind#(n__plus(V1,V2)) -> activate#(V2) activate#(n__plus(X1,X2)) -> plus#(X1,X2) -> plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) activate#(n__plus(X1,X2)) -> plus#(X1,X2) -> plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))) activate#(n__plus(X1,X2)) -> plus#(X1,X2) -> plus#(N,s(M)) -> and#(isNat(M),n__isNatKind(M)) activate#(n__plus(X1,X2)) -> plus#(X1,X2) -> plus#(N,s(M)) -> isNat#(M) activate#(n__plus(X1,X2)) -> plus#(X1,X2) -> plus#(N,s(M)) -> isNat#(N) activate#(n__plus(X1,X2)) -> plus#(X1,X2) -> plus#(N,0()) -> U31#(and(isNat(N),n__isNatKind(N)),N) activate#(n__plus(X1,X2)) -> plus#(X1,X2) -> plus#(N,0()) -> and#(isNat(N),n__isNatKind(N)) activate#(n__plus(X1,X2)) -> plus#(X1,X2) -> plus#(N,0()) -> isNat#(N) U11#(tt(),V1,V2) -> U12#(isNat(activate(V1)),activate(V2)) -> U12#(tt(),V2) -> U13#(isNat(activate(V2))) U11#(tt(),V1,V2) -> U12#(isNat(activate(V1)),activate(V2)) -> U12#(tt(),V2) -> isNat#(activate(V2)) U11#(tt(),V1,V2) -> U12#(isNat(activate(V1)),activate(V2)) -> U12#(tt(),V2) -> activate#(V2) U11#(tt(),V1,V2) -> isNat#(activate(V1)) -> isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) U11#(tt(),V1,V2) -> isNat#(activate(V1)) -> isNat#(n__s(V1)) -> isNatKind#(activate(V1)) U11#(tt(),V1,V2) -> isNat#(activate(V1)) -> isNat#(n__s(V1)) -> activate#(V1) U11#(tt(),V1,V2) -> isNat#(activate(V1)) -> isNat#(n__plus(V1,V2)) -> U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) U11#(tt(),V1,V2) -> isNat#(activate(V1)) -> isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) U11#(tt(),V1,V2) -> isNat#(activate(V1)) -> isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) U11#(tt(),V1,V2) -> isNat#(activate(V1)) -> isNat#(n__plus(V1,V2)) -> activate#(V1) U11#(tt(),V1,V2) -> isNat#(activate(V1)) -> isNat#(n__plus(V1,V2)) -> activate#(V2) U11#(tt(),V1,V2) -> activate#(V2) -> activate#(n__and(X1,X2)) -> and#(X1,X2) U11#(tt(),V1,V2) -> activate#(V2) -> activate#(n__s(X)) -> s#(X) U11#(tt(),V1,V2) -> activate#(V2) -> activate#(n__isNatKind(X)) -> isNatKind#(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__and(X1,X2)) -> and#(X1,X2) U11#(tt(),V1,V2) -> activate#(V1) -> activate#(n__s(X)) -> s#(X) U11#(tt(),V1,V2) -> activate#(V1) -> activate#(n__isNatKind(X)) -> isNatKind#(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: 38 #arcs: 185/1849 DPs: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) -> activate#(V2) activate#(n__plus(X1,X2)) -> plus#(X1,X2) plus#(N,0()) -> isNat#(N) isNat#(n__plus(V1,V2)) -> activate#(V2) activate#(n__isNatKind(X)) -> isNatKind#(X) isNatKind#(n__plus(V1,V2)) -> activate#(V1) activate#(n__and(X1,X2)) -> and#(X1,X2) and#(tt(),X) -> activate#(X) isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind#(n__s(V1)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNat#(n__plus(V1,V2)) -> U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) U11#(tt(),V1,V2) -> activate#(V2) U11#(tt(),V1,V2) -> activate#(V1) U11#(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) -> isNat#(activate(V1)) U11#(tt(),V1,V2) -> U12#(isNat(activate(V1)),activate(V2)) U12#(tt(),V2) -> activate#(V2) U12#(tt(),V2) -> isNat#(activate(V2)) plus#(N,0()) -> and#(isNat(N),n__isNatKind(N)) plus#(N,0()) -> U31#(and(isNat(N),n__isNatKind(N)),N) U31#(tt(),N) -> activate#(N) plus#(N,s(M)) -> isNat#(N) plus#(N,s(M)) -> isNat#(M) plus#(N,s(M)) -> and#(isNat(M),n__isNatKind(M)) plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))) plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) U41#(tt(),M,N) -> activate#(M) U41#(tt(),M,N) -> activate#(N) U41#(tt(),M,N) -> plus#(activate(N),activate(M)) TRS: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2)) U12(tt(),V2) -> U13(isNat(activate(V2))) U13(tt()) -> tt() U21(tt(),V1) -> U22(isNat(activate(V1))) U22(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> s(plus(activate(N),activate(M))) and(tt(),X) -> activate(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind(n__s(V1)) -> isNatKind(activate(V1)) plus(N,0()) -> U31(and(isNat(N),n__isNatKind(N)),N) plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) isNatKind(X) -> n__isNatKind(X) s(X) -> n__s(X) and(X1,X2) -> n__and(X1,X2) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(X) activate(n__and(X1,X2)) -> and(X1,X2) activate(X) -> X KBO Processor: argument filtering: pi(tt) = [] pi(U11) = 2 pi(activate) = [0] pi(isNat) = 0 pi(U12) = 0 pi(U13) = [] pi(U21) = 1 pi(U22) = 0 pi(U31) = [1] pi(U41) = [1,2] pi(plus) = [0,1] pi(s) = [0] pi(and) = [1] pi(n__0) = [] pi(n__plus) = [0,1] pi(isNatKind) = [0] pi(n__isNatKind) = [0] pi(n__s) = [0] pi(0) = [] pi(n__and) = [1] pi(U11#) = [1,2] pi(activate#) = 0 pi(isNat#) = 0 pi(U12#) = [1] pi(U21#) = [1] pi(U31#) = [1] pi(U41#) = [1,2] pi(plus#) = [0,1] pi(and#) = 1 pi(isNatKind#) = [0] usable rules: U31(tt(),N) -> activate(N) U41(tt(),M,N) -> s(plus(activate(N),activate(M))) and(tt(),X) -> activate(X) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind(n__s(V1)) -> isNatKind(activate(V1)) plus(N,0()) -> U31(and(isNat(N),n__isNatKind(N)),N) plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) isNatKind(X) -> n__isNatKind(X) s(X) -> n__s(X) and(X1,X2) -> n__and(X1,X2) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(X) activate(n__and(X1,X2)) -> and(X1,X2) activate(X) -> X weight function: w0 = 1 w(isNatKind#) = w(and#) = w(U31#) = w(U21#) = w(U12#) = w(activate#) = w( n__and) = w(0) = w(n__s) = w(n__isNatKind) = w(isNatKind) = w(n__0) = w( and) = w(s) = w(U41) = w(U31) = w(U21) = w(U13) = w(tt) = 1 w(plus#) = w(U41#) = w(isNat#) = w(U11#) = w(n__plus) = w(plus) = w( U22) = w(U12) = w(isNat) = w(activate) = w(U11) = 0 precedence: U41# ~ activate > plus > isNatKind ~ n__plus ~ U41 > plus# ~ s > n__s > U11# ~ 0 ~ n__isNatKind ~ and > isNatKind# ~ and# ~ U31# ~ U21# ~ U12# ~ isNat# ~ activate# ~ n__and ~ n__0 ~ U31 ~ U22 ~ U21 ~ U13 ~ U12 ~ isNat ~ U11 ~ tt problem: DPs: and#(tt(),X) -> activate#(X) TRS: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2)) U12(tt(),V2) -> U13(isNat(activate(V2))) U13(tt()) -> tt() U21(tt(),V1) -> U22(isNat(activate(V1))) U22(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> s(plus(activate(N),activate(M))) and(tt(),X) -> activate(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind(n__s(V1)) -> isNatKind(activate(V1)) plus(N,0()) -> U31(and(isNat(N),n__isNatKind(N)),N) plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) isNatKind(X) -> n__isNatKind(X) s(X) -> n__s(X) and(X1,X2) -> n__and(X1,X2) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(X) activate(n__and(X1,X2)) -> and(X1,X2) activate(X) -> X Restore Modifier: DPs: and#(tt(),X) -> activate#(X) TRS: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2)) U12(tt(),V2) -> U13(isNat(activate(V2))) U13(tt()) -> tt() U21(tt(),V1) -> U22(isNat(activate(V1))) U22(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> s(plus(activate(N),activate(M))) and(tt(),X) -> activate(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind(n__s(V1)) -> isNatKind(activate(V1)) plus(N,0()) -> U31(and(isNat(N),n__isNatKind(N)),N) plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) isNatKind(X) -> n__isNatKind(X) s(X) -> n__s(X) and(X1,X2) -> n__and(X1,X2) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(X) activate(n__and(X1,X2)) -> and(X1,X2) activate(X) -> X SCC Processor: #sccs: 0 #rules: 0 #arcs: 154/1