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(n__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) isNat(X) -> n__isNat(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(activate(X)) activate(n__and(X1,X2)) -> and(activate(X1),X2) activate(n__isNat(X)) -> isNat(X) 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#(M) plus#(N,s(M)) -> and#(isNat(M),n__isNatKind(M)) plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))) plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,N) activate#(n__0()) -> 0#() activate#(n__plus(X1,X2)) -> activate#(X2) activate#(n__plus(X1,X2)) -> activate#(X1) activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) activate#(n__isNatKind(X)) -> isNatKind#(X) activate#(n__s(X)) -> activate#(X) activate#(n__s(X)) -> s#(activate(X)) activate#(n__and(X1,X2)) -> activate#(X1) activate#(n__and(X1,X2)) -> and#(activate(X1),X2) activate#(n__isNat(X)) -> isNat#(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(n__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) isNat(X) -> n__isNat(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(activate(X)) activate(n__and(X1,X2)) -> and(activate(X1),X2) activate(n__isNat(X)) -> isNat(X) 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#(M) plus#(N,s(M)) -> and#(isNat(M),n__isNatKind(M)) plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))) plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,N) activate#(n__0()) -> 0#() activate#(n__plus(X1,X2)) -> activate#(X2) activate#(n__plus(X1,X2)) -> activate#(X1) activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) activate#(n__isNatKind(X)) -> isNatKind#(X) activate#(n__s(X)) -> activate#(X) activate#(n__s(X)) -> s#(activate(X)) activate#(n__and(X1,X2)) -> activate#(X1) activate#(n__and(X1,X2)) -> and#(activate(X1),X2) activate#(n__isNat(X)) -> isNat#(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(n__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) isNat(X) -> n__isNat(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(activate(X)) activate(n__and(X1,X2)) -> and(activate(X1),X2) activate(n__isNat(X)) -> isNat(X) 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__isNat(X)) -> isNat#(X) isNatKind#(n__s(V1)) -> activate#(V1) -> activate#(n__and(X1,X2)) -> and#(activate(X1),X2) isNatKind#(n__s(V1)) -> activate#(V1) -> activate#(n__and(X1,X2)) -> activate#(X1) isNatKind#(n__s(V1)) -> activate#(V1) -> activate#(n__s(X)) -> s#(activate(X)) isNatKind#(n__s(V1)) -> activate#(V1) -> activate#(n__s(X)) -> activate#(X) isNatKind#(n__s(V1)) -> activate#(V1) -> activate#(n__isNatKind(X)) -> isNatKind#(X) isNatKind#(n__s(V1)) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) isNatKind#(n__s(V1)) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> activate#(X1) isNatKind#(n__s(V1)) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> activate#(X2) isNatKind#(n__s(V1)) -> activate#(V1) -> activate#(n__0()) -> 0#() isNatKind#(n__plus(V1,V2)) -> 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__isNat(X)) -> isNat#(X) isNatKind#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__and(X1,X2)) -> and#(activate(X1),X2) isNatKind#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__and(X1,X2)) -> activate#(X1) isNatKind#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__s(X)) -> s#(activate(X)) isNatKind#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__s(X)) -> activate#(X) isNatKind#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__isNatKind(X)) -> isNatKind#(X) isNatKind#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) isNatKind#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> activate#(X1) isNatKind#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> activate#(X2) isNatKind#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__0()) -> 0#() isNatKind#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__isNat(X)) -> isNat#(X) isNatKind#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__and(X1,X2)) -> and#(activate(X1),X2) isNatKind#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__and(X1,X2)) -> activate#(X1) isNatKind#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__s(X)) -> s#(activate(X)) isNatKind#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__s(X)) -> activate#(X) isNatKind#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__isNatKind(X)) -> isNatKind#(X) isNatKind#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) isNatKind#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> activate#(X1) isNatKind#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> activate#(X2) isNatKind#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__0()) -> 0#() and#(tt(),X) -> activate#(X) -> activate#(n__isNat(X)) -> isNat#(X) and#(tt(),X) -> activate#(X) -> activate#(n__and(X1,X2)) -> and#(activate(X1),X2) and#(tt(),X) -> activate#(X) -> activate#(n__and(X1,X2)) -> activate#(X1) and#(tt(),X) -> activate#(X) -> activate#(n__s(X)) -> s#(activate(X)) and#(tt(),X) -> activate#(X) -> activate#(n__s(X)) -> activate#(X) and#(tt(),X) -> activate#(X) -> activate#(n__isNatKind(X)) -> isNatKind#(X) and#(tt(),X) -> activate#(X) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) and#(tt(),X) -> activate#(X) -> activate#(n__plus(X1,X2)) -> activate#(X1) and#(tt(),X) -> activate#(X) -> activate#(n__plus(X1,X2)) -> activate#(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(n__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(n__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(n__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(n__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(n__isNat(N),n__isNatKind(N))),M,N) -> U41#(tt(),M,N) -> activate#(M) 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(n__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(n__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,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#(M) -> activate#(n__isNat(X)) -> isNat#(X) U41#(tt(),M,N) -> activate#(M) -> activate#(n__and(X1,X2)) -> and#(activate(X1),X2) U41#(tt(),M,N) -> activate#(M) -> activate#(n__and(X1,X2)) -> activate#(X1) U41#(tt(),M,N) -> activate#(M) -> activate#(n__s(X)) -> s#(activate(X)) U41#(tt(),M,N) -> activate#(M) -> activate#(n__s(X)) -> activate#(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#(activate(X1),activate(X2)) U41#(tt(),M,N) -> activate#(M) -> activate#(n__plus(X1,X2)) -> activate#(X1) U41#(tt(),M,N) -> activate#(M) -> activate#(n__plus(X1,X2)) -> activate#(X2) U41#(tt(),M,N) -> activate#(M) -> activate#(n__0()) -> 0#() U41#(tt(),M,N) -> activate#(N) -> activate#(n__isNat(X)) -> isNat#(X) U41#(tt(),M,N) -> activate#(N) -> activate#(n__and(X1,X2)) -> and#(activate(X1),X2) U41#(tt(),M,N) -> activate#(N) -> activate#(n__and(X1,X2)) -> activate#(X1) U41#(tt(),M,N) -> activate#(N) -> activate#(n__s(X)) -> s#(activate(X)) U41#(tt(),M,N) -> activate#(N) -> activate#(n__s(X)) -> activate#(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#(activate(X1),activate(X2)) U41#(tt(),M,N) -> activate#(N) -> activate#(n__plus(X1,X2)) -> activate#(X1) U41#(tt(),M,N) -> activate#(N) -> activate#(n__plus(X1,X2)) -> activate#(X2) U41#(tt(),M,N) -> activate#(N) -> activate#(n__0()) -> 0#() U31#(tt(),N) -> activate#(N) -> activate#(n__isNat(X)) -> isNat#(X) U31#(tt(),N) -> activate#(N) -> activate#(n__and(X1,X2)) -> and#(activate(X1),X2) U31#(tt(),N) -> activate#(N) -> activate#(n__and(X1,X2)) -> activate#(X1) U31#(tt(),N) -> activate#(N) -> activate#(n__s(X)) -> s#(activate(X)) U31#(tt(),N) -> activate#(N) -> activate#(n__s(X)) -> activate#(X) U31#(tt(),N) -> activate#(N) -> activate#(n__isNatKind(X)) -> isNatKind#(X) U31#(tt(),N) -> activate#(N) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) U31#(tt(),N) -> activate#(N) -> activate#(n__plus(X1,X2)) -> activate#(X1) U31#(tt(),N) -> activate#(N) -> activate#(n__plus(X1,X2)) -> activate#(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__isNat(X)) -> isNat#(X) U21#(tt(),V1) -> activate#(V1) -> activate#(n__and(X1,X2)) -> and#(activate(X1),X2) U21#(tt(),V1) -> activate#(V1) -> activate#(n__and(X1,X2)) -> activate#(X1) U21#(tt(),V1) -> activate#(V1) -> activate#(n__s(X)) -> s#(activate(X)) U21#(tt(),V1) -> activate#(V1) -> activate#(n__s(X)) -> activate#(X) U21#(tt(),V1) -> activate#(V1) -> activate#(n__isNatKind(X)) -> isNatKind#(X) U21#(tt(),V1) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) U21#(tt(),V1) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> activate#(X1) U21#(tt(),V1) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> activate#(X2) U21#(tt(),V1) -> activate#(V1) -> activate#(n__0()) -> 0#() 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__isNat(X)) -> isNat#(X) U12#(tt(),V2) -> activate#(V2) -> activate#(n__and(X1,X2)) -> and#(activate(X1),X2) U12#(tt(),V2) -> activate#(V2) -> activate#(n__and(X1,X2)) -> activate#(X1) U12#(tt(),V2) -> activate#(V2) -> activate#(n__s(X)) -> s#(activate(X)) U12#(tt(),V2) -> activate#(V2) -> activate#(n__s(X)) -> activate#(X) U12#(tt(),V2) -> activate#(V2) -> activate#(n__isNatKind(X)) -> isNatKind#(X) U12#(tt(),V2) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) U12#(tt(),V2) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> activate#(X1) U12#(tt(),V2) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> activate#(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__isNat(X)) -> isNat#(X) isNat#(n__s(V1)) -> activate#(V1) -> activate#(n__and(X1,X2)) -> and#(activate(X1),X2) isNat#(n__s(V1)) -> activate#(V1) -> activate#(n__and(X1,X2)) -> activate#(X1) isNat#(n__s(V1)) -> activate#(V1) -> activate#(n__s(X)) -> s#(activate(X)) isNat#(n__s(V1)) -> activate#(V1) -> activate#(n__s(X)) -> activate#(X) isNat#(n__s(V1)) -> activate#(V1) -> activate#(n__isNatKind(X)) -> isNatKind#(X) isNat#(n__s(V1)) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) isNat#(n__s(V1)) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> activate#(X1) isNat#(n__s(V1)) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> activate#(X2) isNat#(n__s(V1)) -> activate#(V1) -> activate#(n__0()) -> 0#() isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) -> isNatKind#(n__s(V1)) -> 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__isNat(X)) -> isNat#(X) isNat#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__and(X1,X2)) -> and#(activate(X1),X2) isNat#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__and(X1,X2)) -> activate#(X1) isNat#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__s(X)) -> s#(activate(X)) isNat#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__s(X)) -> activate#(X) isNat#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__isNatKind(X)) -> isNatKind#(X) isNat#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) isNat#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> activate#(X1) isNat#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> activate#(X2) isNat#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__0()) -> 0#() isNat#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__isNat(X)) -> isNat#(X) isNat#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__and(X1,X2)) -> and#(activate(X1),X2) isNat#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__and(X1,X2)) -> activate#(X1) isNat#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__s(X)) -> s#(activate(X)) isNat#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__s(X)) -> activate#(X) isNat#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__isNatKind(X)) -> isNatKind#(X) isNat#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) isNat#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> activate#(X1) isNat#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> activate#(X2) isNat#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__0()) -> 0#() isNat#(n__plus(V1,V2)) -> U11#(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#(activate(X1),X2) -> and#(tt(),X) -> activate#(X) activate#(n__and(X1,X2)) -> activate#(X1) -> activate#(n__isNat(X)) -> isNat#(X) activate#(n__and(X1,X2)) -> activate#(X1) -> activate#(n__and(X1,X2)) -> and#(activate(X1),X2) activate#(n__and(X1,X2)) -> activate#(X1) -> activate#(n__and(X1,X2)) -> activate#(X1) activate#(n__and(X1,X2)) -> activate#(X1) -> activate#(n__s(X)) -> s#(activate(X)) activate#(n__and(X1,X2)) -> activate#(X1) -> activate#(n__s(X)) -> activate#(X) activate#(n__and(X1,X2)) -> activate#(X1) -> activate#(n__isNatKind(X)) -> isNatKind#(X) activate#(n__and(X1,X2)) -> activate#(X1) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) activate#(n__and(X1,X2)) -> activate#(X1) -> activate#(n__plus(X1,X2)) -> activate#(X1) activate#(n__and(X1,X2)) -> activate#(X1) -> activate#(n__plus(X1,X2)) -> activate#(X2) activate#(n__and(X1,X2)) -> activate#(X1) -> activate#(n__0()) -> 0#() activate#(n__isNat(X)) -> isNat#(X) -> isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) activate#(n__isNat(X)) -> isNat#(X) -> isNat#(n__s(V1)) -> isNatKind#(activate(V1)) activate#(n__isNat(X)) -> isNat#(X) -> isNat#(n__s(V1)) -> activate#(V1) activate#(n__isNat(X)) -> isNat#(X) -> isNat#(n__plus(V1,V2)) -> U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) activate#(n__isNat(X)) -> isNat#(X) -> isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) activate#(n__isNat(X)) -> isNat#(X) -> isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) activate#(n__isNat(X)) -> isNat#(X) -> isNat#(n__plus(V1,V2)) -> activate#(V1) activate#(n__isNat(X)) -> isNat#(X) -> isNat#(n__plus(V1,V2)) -> activate#(V2) activate#(n__s(X)) -> activate#(X) -> activate#(n__isNat(X)) -> isNat#(X) activate#(n__s(X)) -> activate#(X) -> activate#(n__and(X1,X2)) -> and#(activate(X1),X2) activate#(n__s(X)) -> activate#(X) -> activate#(n__and(X1,X2)) -> activate#(X1) activate#(n__s(X)) -> activate#(X) -> activate#(n__s(X)) -> s#(activate(X)) activate#(n__s(X)) -> activate#(X) -> activate#(n__s(X)) -> activate#(X) activate#(n__s(X)) -> activate#(X) -> activate#(n__isNatKind(X)) -> isNatKind#(X) activate#(n__s(X)) -> activate#(X) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) activate#(n__s(X)) -> activate#(X) -> activate#(n__plus(X1,X2)) -> activate#(X1) activate#(n__s(X)) -> activate#(X) -> activate#(n__plus(X1,X2)) -> activate#(X2) activate#(n__s(X)) -> activate#(X) -> activate#(n__0()) -> 0#() activate#(n__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#(activate(X1),activate(X2)) -> plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,N) activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) -> plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))) activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) -> plus#(N,s(M)) -> and#(isNat(M),n__isNatKind(M)) activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) -> plus#(N,s(M)) -> isNat#(M) activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) -> plus#(N,0()) -> U31#(and(isNat(N),n__isNatKind(N)),N) activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) -> plus#(N,0()) -> and#(isNat(N),n__isNatKind(N)) activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) -> plus#(N,0()) -> isNat#(N) activate#(n__plus(X1,X2)) -> activate#(X2) -> activate#(n__isNat(X)) -> isNat#(X) activate#(n__plus(X1,X2)) -> activate#(X2) -> activate#(n__and(X1,X2)) -> and#(activate(X1),X2) activate#(n__plus(X1,X2)) -> activate#(X2) -> activate#(n__and(X1,X2)) -> activate#(X1) activate#(n__plus(X1,X2)) -> activate#(X2) -> activate#(n__s(X)) -> s#(activate(X)) activate#(n__plus(X1,X2)) -> activate#(X2) -> activate#(n__s(X)) -> activate#(X) activate#(n__plus(X1,X2)) -> activate#(X2) -> activate#(n__isNatKind(X)) -> isNatKind#(X) activate#(n__plus(X1,X2)) -> activate#(X2) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) activate#(n__plus(X1,X2)) -> activate#(X2) -> activate#(n__plus(X1,X2)) -> activate#(X1) activate#(n__plus(X1,X2)) -> activate#(X2) -> activate#(n__plus(X1,X2)) -> activate#(X2) activate#(n__plus(X1,X2)) -> activate#(X2) -> activate#(n__0()) -> 0#() activate#(n__plus(X1,X2)) -> activate#(X1) -> activate#(n__isNat(X)) -> isNat#(X) activate#(n__plus(X1,X2)) -> activate#(X1) -> activate#(n__and(X1,X2)) -> and#(activate(X1),X2) activate#(n__plus(X1,X2)) -> activate#(X1) -> activate#(n__and(X1,X2)) -> activate#(X1) activate#(n__plus(X1,X2)) -> activate#(X1) -> activate#(n__s(X)) -> s#(activate(X)) activate#(n__plus(X1,X2)) -> activate#(X1) -> activate#(n__s(X)) -> activate#(X) activate#(n__plus(X1,X2)) -> activate#(X1) -> activate#(n__isNatKind(X)) -> isNatKind#(X) activate#(n__plus(X1,X2)) -> activate#(X1) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) activate#(n__plus(X1,X2)) -> activate#(X1) -> activate#(n__plus(X1,X2)) -> activate#(X1) activate#(n__plus(X1,X2)) -> activate#(X1) -> activate#(n__plus(X1,X2)) -> activate#(X2) activate#(n__plus(X1,X2)) -> activate#(X1) -> activate#(n__0()) -> 0#() U11#(tt(),V1,V2) -> U12#(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__isNat(X)) -> isNat#(X) U11#(tt(),V1,V2) -> activate#(V2) -> activate#(n__and(X1,X2)) -> and#(activate(X1),X2) U11#(tt(),V1,V2) -> activate#(V2) -> activate#(n__and(X1,X2)) -> activate#(X1) U11#(tt(),V1,V2) -> activate#(V2) -> activate#(n__s(X)) -> s#(activate(X)) U11#(tt(),V1,V2) -> activate#(V2) -> activate#(n__s(X)) -> activate#(X) U11#(tt(),V1,V2) -> activate#(V2) -> activate#(n__isNatKind(X)) -> isNatKind#(X) U11#(tt(),V1,V2) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) U11#(tt(),V1,V2) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> activate#(X1) U11#(tt(),V1,V2) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> activate#(X2) U11#(tt(),V1,V2) -> activate#(V2) -> activate#(n__0()) -> 0#() U11#(tt(),V1,V2) -> activate#(V1) -> activate#(n__isNat(X)) -> isNat#(X) U11#(tt(),V1,V2) -> activate#(V1) -> activate#(n__and(X1,X2)) -> and#(activate(X1),X2) U11#(tt(),V1,V2) -> activate#(V1) -> activate#(n__and(X1,X2)) -> activate#(X1) U11#(tt(),V1,V2) -> activate#(V1) -> activate#(n__s(X)) -> s#(activate(X)) U11#(tt(),V1,V2) -> activate#(V1) -> activate#(n__s(X)) -> activate#(X) U11#(tt(),V1,V2) -> activate#(V1) -> activate#(n__isNatKind(X)) -> isNatKind#(X) U11#(tt(),V1,V2) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) U11#(tt(),V1,V2) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> activate#(X1) U11#(tt(),V1,V2) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> activate#(X2) U11#(tt(),V1,V2) -> activate#(V1) -> activate#(n__0()) -> 0#() SCC Processor: #sccs: 1 #rules: 42 #arcs: 293/2209 DPs: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) -> activate#(V2) activate#(n__plus(X1,X2)) -> activate#(X2) activate#(n__plus(X1,X2)) -> activate#(X1) activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) plus#(N,0()) -> isNat#(N) isNat#(n__plus(V1,V2)) -> activate#(V2) activate#(n__isNatKind(X)) -> isNatKind#(X) isNatKind#(n__plus(V1,V2)) -> activate#(V1) activate#(n__s(X)) -> activate#(X) activate#(n__and(X1,X2)) -> activate#(X1) activate#(n__and(X1,X2)) -> and#(activate(X1),X2) and#(tt(),X) -> activate#(X) activate#(n__isNat(X)) -> isNat#(X) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> isNatKind#(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) 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#(M) plus#(N,s(M)) -> and#(isNat(M),n__isNatKind(M)) plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))) plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(n__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(n__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) isNat(X) -> n__isNat(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(activate(X)) activate(n__and(X1,X2)) -> and(activate(X1),X2) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Arctic Interpretation Processor: dimension: 1 usable rules: 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(n__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) isNat(X) -> n__isNat(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(activate(X)) activate(n__and(X1,X2)) -> and(activate(X1),X2) activate(n__isNat(X)) -> isNat(X) activate(X) -> X interpretation: [0] = 0, [U12](x0, x1) = x0 + x1 + 0, [isNatKind](x0) = x0, [U31](x0, x1) = x0 + 7x1 + 3, [U11](x0, x1, x2) = x0 + 4x1 + x2 + 0, [isNat](x0) = x0 + 0, [n__isNatKind](x0) = x0, [U11#](x0, x1, x2) = 4x0 + 4x1 + 6x2, [U21#](x0, x1) = x1 + 0, [n__isNat](x0) = x0 + 0, [and#](x0, x1) = x0 + 2x1 + 0, [n__s](x0) = x0 + 0, [activate#](x0) = x0 + 0, [U31#](x0, x1) = 2x0 + 1x1 + 0, [U13](x0) = x0 + 0, [isNatKind#](x0) = x0 + 0, [plus](x0, x1) = 7x0 + 6x1, [tt] = 0, [U22](x0) = 0, [n__and](x0, x1) = 2x0 + 2x1, [plus#](x0, x1) = 4x0 + 4x1 + 0, [n__plus](x0, x1) = 7x0 + 6x1, [and](x0, x1) = 2x0 + 2x1, [isNat#](x0) = x0 + 0, [U12#](x0, x1) = 4x0 + 6x1 + 0, [s](x0) = x0 + 0, [U41](x0, x1, x2) = 6x1 + 7x2 + 4, [n__0] = 0, [U21](x0, x1) = x0 + x1, [U41#](x0, x1, x2) = x0 + 4x1 + 4x2 + 0, [activate](x0) = x0 orientation: isNatKind#(n__s(V1)) = V1 + 0 >= V1 + 0 = isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) = 7V1 + 6V2 + 0 >= V2 + 0 = activate#(V2) activate#(n__plus(X1,X2)) = 7X1 + 6X2 + 0 >= X2 + 0 = activate#(X2) activate#(n__plus(X1,X2)) = 7X1 + 6X2 + 0 >= X1 + 0 = activate#(X1) activate#(n__plus(X1,X2)) = 7X1 + 6X2 + 0 >= 4X1 + 4X2 + 0 = plus#(activate(X1),activate(X2)) plus#(N,0()) = 4N + 4 >= N + 0 = isNat#(N) isNat#(n__plus(V1,V2)) = 7V1 + 6V2 + 0 >= V2 + 0 = activate#(V2) activate#(n__isNatKind(X)) = X + 0 >= X + 0 = isNatKind#(X) isNatKind#(n__plus(V1,V2)) = 7V1 + 6V2 + 0 >= V1 + 0 = activate#(V1) activate#(n__s(X)) = X + 0 >= X + 0 = activate#(X) activate#(n__and(X1,X2)) = 2X1 + 2X2 + 0 >= X1 + 0 = activate#(X1) activate#(n__and(X1,X2)) = 2X1 + 2X2 + 0 >= X1 + 2X2 + 0 = and#(activate(X1),X2) and#(tt(),X) = 2X + 0 >= X + 0 = activate#(X) activate#(n__isNat(X)) = X + 0 >= X + 0 = isNat#(X) isNat#(n__plus(V1,V2)) = 7V1 + 6V2 + 0 >= V1 + 0 = activate#(V1) isNat#(n__plus(V1,V2)) = 7V1 + 6V2 + 0 >= V1 + 0 = isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) = 7V1 + 6V2 + 0 >= V1 + 0 = isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) = 7V1 + 6V2 + 0 >= V1 + 2V2 + 0 = and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind#(n__s(V1)) = V1 + 0 >= V1 + 0 = activate#(V1) isNat#(n__plus(V1,V2)) = 7V1 + 6V2 + 0 >= V1 + 2V2 + 0 = and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNat#(n__plus(V1,V2)) = 7V1 + 6V2 + 0 >= 6V1 + 6V2 = U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1), activate(V2)) U11#(tt(),V1,V2) = 4V1 + 6V2 + 4 >= V2 + 0 = activate#(V2) U11#(tt(),V1,V2) = 4V1 + 6V2 + 4 >= V1 + 0 = activate#(V1) U11#(tt(),V1,V2) = 4V1 + 6V2 + 4 >= V1 + 0 = isNat#(activate(V1)) isNat#(n__s(V1)) = V1 + 0 >= V1 + 0 = activate#(V1) isNat#(n__s(V1)) = V1 + 0 >= V1 + 0 = isNatKind#(activate(V1)) isNat#(n__s(V1)) = V1 + 0 >= V1 + 0 = U21#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) = V1 + 0 >= V1 + 0 = activate#(V1) U21#(tt(),V1) = V1 + 0 >= V1 + 0 = isNat#(activate(V1)) U11#(tt(),V1,V2) = 4V1 + 6V2 + 4 >= 4V1 + 6V2 + 4 = U12#(isNat(activate(V1)),activate(V2)) U12#(tt(),V2) = 6V2 + 4 >= V2 + 0 = activate#(V2) U12#(tt(),V2) = 6V2 + 4 >= V2 + 0 = isNat#(activate(V2)) plus#(N,0()) = 4N + 4 >= 2N + 0 = and#(isNat(N),n__isNatKind(N)) plus#(N,0()) = 4N + 4 >= 4N + 4 = U31#(and(isNat(N),n__isNatKind(N)),N) U31#(tt(),N) = 1N + 2 >= N + 0 = activate#(N) plus#(N,s(M)) = 4M + 4N + 4 >= M + 0 = isNat#(M) plus#(N,s(M)) = 4M + 4N + 4 >= 2M + 0 = and#(isNat(M),n__isNatKind(M)) plus#(N,s(M)) = 4M + 4N + 4 >= 2M + 4N + 4 = and#(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))) plus#(N,s(M)) = 4M + 4N + 4 >= 4M + 4N + 4 = U41#(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))), M,N) U41#(tt(),M,N) = 4M + 4N + 0 >= M + 0 = activate#(M) U41#(tt(),M,N) = 4M + 4N + 0 >= N + 0 = activate#(N) U41#(tt(),M,N) = 4M + 4N + 0 >= 4M + 4N + 0 = plus#(activate(N),activate(M)) U11(tt(),V1,V2) = 4V1 + V2 + 0 >= V1 + V2 + 0 = U12(isNat(activate(V1)),activate(V2)) U12(tt(),V2) = V2 + 0 >= V2 + 0 = U13(isNat(activate(V2))) U13(tt()) = 0 >= 0 = tt() U21(tt(),V1) = V1 + 0 >= 0 = U22(isNat(activate(V1))) U22(tt()) = 0 >= 0 = tt() U31(tt(),N) = 7N + 3 >= N = activate(N) U41(tt(),M,N) = 6M + 7N + 4 >= 6M + 7N + 0 = s(plus(activate(N),activate(M))) and(tt(),X) = 2X + 2 >= X = activate(X) isNat(n__0()) = 0 >= 0 = tt() isNat(n__plus(V1,V2)) = 7V1 + 6V2 + 0 >= 4V1 + 2V2 + 0 = U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1), activate(V2)) isNat(n__s(V1)) = V1 + 0 >= V1 = U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) = 0 >= 0 = tt() isNatKind(n__plus(V1,V2)) = 7V1 + 6V2 >= 2V1 + 2V2 = and(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind(n__s(V1)) = V1 + 0 >= V1 = isNatKind(activate(V1)) plus(N,0()) = 7N + 6 >= 7N + 3 = U31(and(isNat(N),n__isNatKind(N)),N) plus(N,s(M)) = 6M + 7N + 6 >= 6M + 7N + 4 = U41(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))), M,N) 0() = 0 >= 0 = n__0() plus(X1,X2) = 7X1 + 6X2 >= 7X1 + 6X2 = n__plus(X1,X2) isNatKind(X) = X >= X = n__isNatKind(X) s(X) = X + 0 >= X + 0 = n__s(X) and(X1,X2) = 2X1 + 2X2 >= 2X1 + 2X2 = n__and(X1,X2) isNat(X) = X + 0 >= X + 0 = n__isNat(X) activate(n__0()) = 0 >= 0 = 0() activate(n__plus(X1,X2)) = 7X1 + 6X2 >= 7X1 + 6X2 = plus(activate(X1),activate(X2)) activate(n__isNatKind(X)) = X >= X = isNatKind(X) activate(n__s(X)) = X + 0 >= X + 0 = s(activate(X)) activate(n__and(X1,X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = and(activate(X1),X2) activate(n__isNat(X)) = X + 0 >= X + 0 = isNat(X) activate(X) = X >= X = X problem: DPs: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) -> activate#(V2) activate#(n__plus(X1,X2)) -> activate#(X2) activate#(n__plus(X1,X2)) -> activate#(X1) activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) isNat#(n__plus(V1,V2)) -> activate#(V2) activate#(n__isNatKind(X)) -> isNatKind#(X) isNatKind#(n__plus(V1,V2)) -> activate#(V1) activate#(n__s(X)) -> activate#(X) activate#(n__and(X1,X2)) -> activate#(X1) activate#(n__and(X1,X2)) -> and#(activate(X1),X2) and#(tt(),X) -> activate#(X) activate#(n__isNat(X)) -> isNat#(X) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> isNatKind#(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) 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)) U21#(tt(),V1) -> activate#(V1) U21#(tt(),V1) -> isNat#(activate(V1)) U11#(tt(),V1,V2) -> U12#(isNat(activate(V1)),activate(V2)) plus#(N,0()) -> U31#(and(isNat(N),n__isNatKind(N)),N) plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))) plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(n__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(n__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) isNat(X) -> n__isNat(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(activate(X)) activate(n__and(X1,X2)) -> and(activate(X1),X2) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Restore Modifier: DPs: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) -> activate#(V2) activate#(n__plus(X1,X2)) -> activate#(X2) activate#(n__plus(X1,X2)) -> activate#(X1) activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) isNat#(n__plus(V1,V2)) -> activate#(V2) activate#(n__isNatKind(X)) -> isNatKind#(X) isNatKind#(n__plus(V1,V2)) -> activate#(V1) activate#(n__s(X)) -> activate#(X) activate#(n__and(X1,X2)) -> activate#(X1) activate#(n__and(X1,X2)) -> and#(activate(X1),X2) and#(tt(),X) -> activate#(X) activate#(n__isNat(X)) -> isNat#(X) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> isNatKind#(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) 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)) U21#(tt(),V1) -> activate#(V1) U21#(tt(),V1) -> isNat#(activate(V1)) U11#(tt(),V1,V2) -> U12#(isNat(activate(V1)),activate(V2)) plus#(N,0()) -> U31#(and(isNat(N),n__isNatKind(N)),N) plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))) plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(n__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(n__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) isNat(X) -> n__isNat(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(activate(X)) activate(n__and(X1,X2)) -> and(activate(X1),X2) activate(n__isNat(X)) -> isNat(X) activate(X) -> X SCC Processor: #sccs: 1 #rules: 29 #arcs: 254/1024 DPs: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) -> activate#(V2) activate#(n__plus(X1,X2)) -> activate#(X2) activate#(n__plus(X1,X2)) -> activate#(X1) activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))) and#(tt(),X) -> activate#(X) activate#(n__isNatKind(X)) -> isNatKind#(X) isNatKind#(n__plus(V1,V2)) -> activate#(V1) activate#(n__s(X)) -> activate#(X) activate#(n__and(X1,X2)) -> activate#(X1) activate#(n__and(X1,X2)) -> and#(activate(X1),X2) activate#(n__isNat(X)) -> isNat#(X) 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)) -> 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)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) -> activate#(V1) U21#(tt(),V1) -> isNat#(activate(V1)) plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(n__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(n__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) isNat(X) -> n__isNat(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(activate(X)) activate(n__and(X1,X2)) -> and(activate(X1),X2) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Arctic Interpretation Processor: dimension: 1 usable rules: 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(n__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) isNat(X) -> n__isNat(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(activate(X)) activate(n__and(X1,X2)) -> and(activate(X1),X2) activate(n__isNat(X)) -> isNat(X) activate(X) -> X interpretation: [0] = 0, [U12](x0, x1) = x1 + 0, [isNatKind](x0) = x0 + 0, [U31](x0, x1) = x1 + 0, [U11](x0, x1, x2) = x0 + x1 + x2 + 0, [isNat](x0) = x0 + 0, [n__isNatKind](x0) = x0 + 0, [U21#](x0, x1) = x0 + x1 + 0, [n__isNat](x0) = x0 + 0, [and#](x0, x1) = x0 + x1 + 0, [n__s](x0) = x0 + 0, [activate#](x0) = x0 + 0, [U13](x0) = 0, [isNatKind#](x0) = x0 + 0, [plus](x0, x1) = 4x0 + x1 + 4, [tt] = 0, [U22](x0) = 0, [n__and](x0, x1) = x0 + x1 + 0, [plus#](x0, x1) = x0 + x1 + 0, [n__plus](x0, x1) = 4x0 + x1 + 4, [and](x0, x1) = x0 + x1 + 0, [isNat#](x0) = x0, [s](x0) = x0 + 0, [U41](x0, x1, x2) = x0 + x1 + 4x2 + 4, [n__0] = 0, [U21](x0, x1) = x1 + 0, [U41#](x0, x1, x2) = x0 + x1 + x2 + 0, [activate](x0) = x0 + 0 orientation: isNatKind#(n__s(V1)) = V1 + 0 >= V1 + 0 = isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) = 4V1 + V2 + 4 >= V2 + 0 = activate#(V2) activate#(n__plus(X1,X2)) = 4X1 + X2 + 4 >= X2 + 0 = activate#(X2) activate#(n__plus(X1,X2)) = 4X1 + X2 + 4 >= X1 + 0 = activate#(X1) activate#(n__plus(X1,X2)) = 4X1 + X2 + 4 >= X1 + X2 + 0 = plus#(activate(X1),activate(X2)) plus#(N,s(M)) = M + N + 0 >= M + N + 0 = and#(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))) and#(tt(),X) = X + 0 >= X + 0 = activate#(X) activate#(n__isNatKind(X)) = X + 0 >= X + 0 = isNatKind#(X) isNatKind#(n__plus(V1,V2)) = 4V1 + V2 + 4 >= V1 + 0 = activate#(V1) activate#(n__s(X)) = X + 0 >= X + 0 = activate#(X) activate#(n__and(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = activate#(X1) activate#(n__and(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = and#(activate(X1),X2) activate#(n__isNat(X)) = X + 0 >= X = isNat#(X) isNat#(n__plus(V1,V2)) = 4V1 + V2 + 4 >= V2 + 0 = activate#(V2) isNat#(n__plus(V1,V2)) = 4V1 + V2 + 4 >= V1 + 0 = activate#(V1) isNat#(n__plus(V1,V2)) = 4V1 + V2 + 4 >= V1 + 0 = isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) = 4V1 + V2 + 4 >= V1 + 0 = isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) = 4V1 + V2 + 4 >= V1 + V2 + 0 = and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind#(n__s(V1)) = V1 + 0 >= V1 + 0 = activate#(V1) isNat#(n__plus(V1,V2)) = 4V1 + V2 + 4 >= V1 + V2 + 0 = and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNat#(n__s(V1)) = V1 + 0 >= V1 + 0 = activate#(V1) isNat#(n__s(V1)) = V1 + 0 >= V1 + 0 = isNatKind#(activate(V1)) isNat#(n__s(V1)) = V1 + 0 >= V1 + 0 = U21#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) = V1 + 0 >= V1 + 0 = activate#(V1) U21#(tt(),V1) = V1 + 0 >= V1 + 0 = isNat#(activate(V1)) plus#(N,s(M)) = M + N + 0 >= M + N + 0 = U41#(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))), M,N) U41#(tt(),M,N) = M + N + 0 >= M + 0 = activate#(M) U41#(tt(),M,N) = M + N + 0 >= N + 0 = activate#(N) U41#(tt(),M,N) = M + N + 0 >= M + N + 0 = plus#(activate(N),activate(M)) U11(tt(),V1,V2) = V1 + V2 + 0 >= V2 + 0 = U12(isNat(activate(V1)),activate(V2)) U12(tt(),V2) = V2 + 0 >= 0 = U13(isNat(activate(V2))) U13(tt()) = 0 >= 0 = tt() U21(tt(),V1) = V1 + 0 >= 0 = U22(isNat(activate(V1))) U22(tt()) = 0 >= 0 = tt() U31(tt(),N) = N + 0 >= N + 0 = activate(N) U41(tt(),M,N) = M + 4N + 4 >= M + 4N + 4 = s(plus(activate(N),activate(M))) and(tt(),X) = X + 0 >= X + 0 = activate(X) isNat(n__0()) = 0 >= 0 = tt() isNat(n__plus(V1,V2)) = 4V1 + V2 + 4 >= V1 + V2 + 0 = U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1), activate(V2)) isNat(n__s(V1)) = V1 + 0 >= V1 + 0 = U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) = 0 >= 0 = tt() isNatKind(n__plus(V1,V2)) = 4V1 + V2 + 4 >= V1 + V2 + 0 = and(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind(n__s(V1)) = V1 + 0 >= V1 + 0 = isNatKind(activate(V1)) plus(N,0()) = 4N + 4 >= N + 0 = U31(and(isNat(N),n__isNatKind(N)),N) plus(N,s(M)) = M + 4N + 4 >= M + 4N + 4 = U41(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))), M,N) 0() = 0 >= 0 = n__0() plus(X1,X2) = 4X1 + X2 + 4 >= 4X1 + X2 + 4 = n__plus(X1,X2) isNatKind(X) = X + 0 >= X + 0 = n__isNatKind(X) s(X) = X + 0 >= X + 0 = n__s(X) and(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = n__and(X1,X2) isNat(X) = X + 0 >= X + 0 = n__isNat(X) activate(n__0()) = 0 >= 0 = 0() activate(n__plus(X1,X2)) = 4X1 + X2 + 4 >= 4X1 + X2 + 4 = plus(activate(X1),activate(X2)) activate(n__isNatKind(X)) = X + 0 >= X + 0 = isNatKind(X) activate(n__s(X)) = X + 0 >= X + 0 = s(activate(X)) activate(n__and(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = and(activate(X1),X2) activate(n__isNat(X)) = X + 0 >= X + 0 = isNat(X) activate(X) = X + 0 >= X = X problem: DPs: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) -> activate#(V2) activate#(n__plus(X1,X2)) -> activate#(X2) activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))) and#(tt(),X) -> activate#(X) activate#(n__isNatKind(X)) -> isNatKind#(X) activate#(n__s(X)) -> activate#(X) activate#(n__and(X1,X2)) -> activate#(X1) activate#(n__and(X1,X2)) -> and#(activate(X1),X2) activate#(n__isNat(X)) -> isNat#(X) isNat#(n__plus(V1,V2)) -> activate#(V2) isNatKind#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind#(n__s(V1)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) -> activate#(V1) U21#(tt(),V1) -> isNat#(activate(V1)) plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(n__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(n__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) isNat(X) -> n__isNat(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(activate(X)) activate(n__and(X1,X2)) -> and(activate(X1),X2) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Restore Modifier: DPs: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) -> activate#(V2) activate#(n__plus(X1,X2)) -> activate#(X2) activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))) and#(tt(),X) -> activate#(X) activate#(n__isNatKind(X)) -> isNatKind#(X) activate#(n__s(X)) -> activate#(X) activate#(n__and(X1,X2)) -> activate#(X1) activate#(n__and(X1,X2)) -> and#(activate(X1),X2) activate#(n__isNat(X)) -> isNat#(X) isNat#(n__plus(V1,V2)) -> activate#(V2) isNatKind#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind#(n__s(V1)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) -> activate#(V1) U21#(tt(),V1) -> isNat#(activate(V1)) plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(n__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(n__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) isNat(X) -> n__isNat(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(activate(X)) activate(n__and(X1,X2)) -> and(activate(X1),X2) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Arctic Interpretation Processor: dimension: 1 usable rules: 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(n__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) isNat(X) -> n__isNat(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(activate(X)) activate(n__and(X1,X2)) -> and(activate(X1),X2) activate(n__isNat(X)) -> isNat(X) activate(X) -> X interpretation: [0] = 0, [U12](x0, x1) = x0 + 4x1 + 0, [isNatKind](x0) = 2x0 + 0, [U31](x0, x1) = x0 + 1x1 + 0, [U11](x0, x1, x2) = 6x1 + 4x2 + 3, [isNat](x0) = 3x0 + 0, [n__isNatKind](x0) = 2x0 + 0, [U21#](x0, x1) = x0 + 3x1 + 0, [n__isNat](x0) = 3x0 + 0, [and#](x0, x1) = x0 + x1 + 0, [n__s](x0) = x0, [activate#](x0) = x0, [U13](x0) = x0 + 0, [isNatKind#](x0) = x0 + 0, [plus](x0, x1) = 3x0 + 4x1 + 0, [tt] = 0, [U22](x0) = x0 + 0, [n__and](x0, x1) = x0 + x1 + 0, [plus#](x0, x1) = 3x0 + 3x1 + 0, [n__plus](x0, x1) = 3x0 + 4x1 + 0, [and](x0, x1) = x0 + x1 + 0, [isNat#](x0) = 3x0 + 0, [s](x0) = x0, [U41](x0, x1, x2) = 4x1 + 3x2 + 0, [n__0] = 0, [U21](x0, x1) = x0 + 3x1 + 0, [U41#](x0, x1, x2) = x0 + 3x1 + 3x2 + 0, [activate](x0) = x0 orientation: isNatKind#(n__s(V1)) = V1 + 0 >= V1 + 0 = isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) = 3V1 + 4V2 + 0 >= V2 = activate#(V2) activate#(n__plus(X1,X2)) = 3X1 + 4X2 + 0 >= X2 = activate#(X2) activate#(n__plus(X1,X2)) = 3X1 + 4X2 + 0 >= 3X1 + 3X2 + 0 = plus#(activate(X1),activate(X2)) plus#(N,s(M)) = 3M + 3N + 0 >= 3M + 3N + 0 = and#(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))) and#(tt(),X) = X + 0 >= X = activate#(X) activate#(n__isNatKind(X)) = 2X + 0 >= X + 0 = isNatKind#(X) activate#(n__s(X)) = X >= X = activate#(X) activate#(n__and(X1,X2)) = X1 + X2 + 0 >= X1 = activate#(X1) activate#(n__and(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = and#(activate(X1),X2) activate#(n__isNat(X)) = 3X + 0 >= 3X + 0 = isNat#(X) isNat#(n__plus(V1,V2)) = 6V1 + 7V2 + 3 >= V2 = activate#(V2) isNatKind#(n__plus(V1,V2)) = 3V1 + 4V2 + 0 >= 2V1 + 2V2 + 0 = and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind#(n__s(V1)) = V1 + 0 >= V1 = activate#(V1) isNat#(n__plus(V1,V2)) = 6V1 + 7V2 + 3 >= 2V1 + 2V2 + 0 = and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNat#(n__s(V1)) = 3V1 + 0 >= V1 = activate#(V1) isNat#(n__s(V1)) = 3V1 + 0 >= V1 + 0 = isNatKind#(activate(V1)) isNat#(n__s(V1)) = 3V1 + 0 >= 3V1 + 0 = U21#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) = 3V1 + 0 >= V1 = activate#(V1) U21#(tt(),V1) = 3V1 + 0 >= 3V1 + 0 = isNat#(activate(V1)) plus#(N,s(M)) = 3M + 3N + 0 >= 3M + 3N + 0 = U41#(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))), M,N) U41#(tt(),M,N) = 3M + 3N + 0 >= M = activate#(M) U41#(tt(),M,N) = 3M + 3N + 0 >= N = activate#(N) U41#(tt(),M,N) = 3M + 3N + 0 >= 3M + 3N + 0 = plus#(activate(N),activate(M)) U11(tt(),V1,V2) = 6V1 + 4V2 + 3 >= 3V1 + 4V2 + 0 = U12(isNat(activate(V1)),activate(V2)) U12(tt(),V2) = 4V2 + 0 >= 3V2 + 0 = U13(isNat(activate(V2))) U13(tt()) = 0 >= 0 = tt() U21(tt(),V1) = 3V1 + 0 >= 3V1 + 0 = U22(isNat(activate(V1))) U22(tt()) = 0 >= 0 = tt() U31(tt(),N) = 1N + 0 >= N = activate(N) U41(tt(),M,N) = 4M + 3N + 0 >= 4M + 3N + 0 = s(plus(activate(N),activate(M))) and(tt(),X) = X + 0 >= X = activate(X) isNat(n__0()) = 3 >= 0 = tt() isNat(n__plus(V1,V2)) = 6V1 + 7V2 + 3 >= 6V1 + 4V2 + 3 = U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1), activate(V2)) isNat(n__s(V1)) = 3V1 + 0 >= 3V1 + 0 = U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) = 2 >= 0 = tt() isNatKind(n__plus(V1,V2)) = 5V1 + 6V2 + 2 >= 2V1 + 2V2 + 0 = and(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind(n__s(V1)) = 2V1 + 0 >= 2V1 + 0 = isNatKind(activate(V1)) plus(N,0()) = 3N + 4 >= 3N + 0 = U31(and(isNat(N),n__isNatKind(N)),N) plus(N,s(M)) = 4M + 3N + 0 >= 4M + 3N + 0 = U41(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))), M,N) 0() = 0 >= 0 = n__0() plus(X1,X2) = 3X1 + 4X2 + 0 >= 3X1 + 4X2 + 0 = n__plus(X1,X2) isNatKind(X) = 2X + 0 >= 2X + 0 = n__isNatKind(X) s(X) = X >= X = n__s(X) and(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = n__and(X1,X2) isNat(X) = 3X + 0 >= 3X + 0 = n__isNat(X) activate(n__0()) = 0 >= 0 = 0() activate(n__plus(X1,X2)) = 3X1 + 4X2 + 0 >= 3X1 + 4X2 + 0 = plus(activate(X1),activate(X2)) activate(n__isNatKind(X)) = 2X + 0 >= 2X + 0 = isNatKind(X) activate(n__s(X)) = X >= X = s(activate(X)) activate(n__and(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = and(activate(X1),X2) activate(n__isNat(X)) = 3X + 0 >= 3X + 0 = isNat(X) activate(X) = X >= X = X problem: DPs: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))) and#(tt(),X) -> activate#(X) activate#(n__isNatKind(X)) -> isNatKind#(X) activate#(n__s(X)) -> activate#(X) activate#(n__and(X1,X2)) -> activate#(X1) activate#(n__and(X1,X2)) -> and#(activate(X1),X2) activate#(n__isNat(X)) -> isNat#(X) isNatKind#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind#(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) -> isNat#(activate(V1)) plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,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(n__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) isNat(X) -> n__isNat(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(activate(X)) activate(n__and(X1,X2)) -> and(activate(X1),X2) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Restore Modifier: DPs: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))) and#(tt(),X) -> activate#(X) activate#(n__isNatKind(X)) -> isNatKind#(X) activate#(n__s(X)) -> activate#(X) activate#(n__and(X1,X2)) -> activate#(X1) activate#(n__and(X1,X2)) -> and#(activate(X1),X2) activate#(n__isNat(X)) -> isNat#(X) isNatKind#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind#(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) -> isNat#(activate(V1)) plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,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(n__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) isNat(X) -> n__isNat(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(activate(X)) activate(n__and(X1,X2)) -> and(activate(X1),X2) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Arctic Interpretation Processor: dimension: 1 usable rules: 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(n__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) isNat(X) -> n__isNat(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(activate(X)) activate(n__and(X1,X2)) -> and(activate(X1),X2) activate(n__isNat(X)) -> isNat(X) activate(X) -> X interpretation: [0] = 1, [U12](x0, x1) = 3x1, [isNatKind](x0) = 3x0, [U31](x0, x1) = 4x1, [U11](x0, x1, x2) = x1 + 4x2, [isNat](x0) = 3x0, [n__isNatKind](x0) = 3x0, [U21#](x0, x1) = x1 + 0, [n__isNat](x0) = 3x0, [and#](x0, x1) = x0 + x1, [n__s](x0) = x0 + 0, [activate#](x0) = x0, [U13](x0) = x0, [isNatKind#](x0) = x0, [plus](x0, x1) = 4x0 + 4x1 + 0, [tt] = 0, [U22](x0) = 0, [n__and](x0, x1) = 1x0 + 1x1, [plus#](x0, x1) = 4x0 + 4x1 + 0, [n__plus](x0, x1) = 4x0 + 4x1 + 0, [and](x0, x1) = 1x0 + 1x1, [isNat#](x0) = x0, [s](x0) = x0 + 0, [U41](x0, x1, x2) = 4x1 + 4x2 + 0, [n__0] = 1, [U21](x0, x1) = x0 + 2x1, [U41#](x0, x1, x2) = 4x1 + 4x2 + 4, [activate](x0) = x0 orientation: isNatKind#(n__s(V1)) = V1 + 0 >= V1 = isNatKind#(activate(V1)) activate#(n__plus(X1,X2)) = 4X1 + 4X2 + 0 >= 4X1 + 4X2 + 0 = plus#(activate(X1),activate(X2)) plus#(N,s(M)) = 4M + 4N + 4 >= 4M + 4N = and#(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))) and#(tt(),X) = X + 0 >= X = activate#(X) activate#(n__isNatKind(X)) = 3X >= X = isNatKind#(X) activate#(n__s(X)) = X + 0 >= X = activate#(X) activate#(n__and(X1,X2)) = 1X1 + 1X2 >= X1 = activate#(X1) activate#(n__and(X1,X2)) = 1X1 + 1X2 >= X1 + X2 = and#(activate(X1),X2) activate#(n__isNat(X)) = 3X >= X = isNat#(X) isNatKind#(n__plus(V1,V2)) = 4V1 + 4V2 + 0 >= 3V1 + 3V2 = and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind#(n__s(V1)) = V1 + 0 >= V1 = activate#(V1) isNat#(n__s(V1)) = V1 + 0 >= V1 = isNatKind#(activate(V1)) isNat#(n__s(V1)) = V1 + 0 >= V1 + 0 = U21#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) = V1 + 0 >= V1 = isNat#(activate(V1)) plus#(N,s(M)) = 4M + 4N + 4 >= 4M + 4N + 4 = U41#(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))), M,N) U41#(tt(),M,N) = 4M + 4N + 4 >= 4M + 4N + 0 = plus#(activate(N),activate(M)) U11(tt(),V1,V2) = V1 + 4V2 >= 3V2 = U12(isNat(activate(V1)),activate(V2)) U12(tt(),V2) = 3V2 >= 3V2 = U13(isNat(activate(V2))) U13(tt()) = 0 >= 0 = tt() U21(tt(),V1) = 2V1 + 0 >= 0 = U22(isNat(activate(V1))) U22(tt()) = 0 >= 0 = tt() U31(tt(),N) = 4N >= N = activate(N) U41(tt(),M,N) = 4M + 4N + 0 >= 4M + 4N + 0 = s(plus(activate(N),activate(M))) and(tt(),X) = 1X + 1 >= X = activate(X) isNat(n__0()) = 4 >= 0 = tt() isNat(n__plus(V1,V2)) = 7V1 + 7V2 + 3 >= V1 + 4V2 = U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1), activate(V2)) isNat(n__s(V1)) = 3V1 + 3 >= 3V1 = U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) = 4 >= 0 = tt() isNatKind(n__plus(V1,V2)) = 7V1 + 7V2 + 3 >= 4V1 + 4V2 = and(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind(n__s(V1)) = 3V1 + 3 >= 3V1 = isNatKind(activate(V1)) plus(N,0()) = 4N + 5 >= 4N = U31(and(isNat(N),n__isNatKind(N)),N) plus(N,s(M)) = 4M + 4N + 4 >= 4M + 4N + 0 = U41(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))), M,N) 0() = 1 >= 1 = n__0() plus(X1,X2) = 4X1 + 4X2 + 0 >= 4X1 + 4X2 + 0 = n__plus(X1,X2) isNatKind(X) = 3X >= 3X = n__isNatKind(X) s(X) = X + 0 >= X + 0 = n__s(X) and(X1,X2) = 1X1 + 1X2 >= 1X1 + 1X2 = n__and(X1,X2) isNat(X) = 3X >= 3X = n__isNat(X) activate(n__0()) = 1 >= 1 = 0() activate(n__plus(X1,X2)) = 4X1 + 4X2 + 0 >= 4X1 + 4X2 + 0 = plus(activate(X1),activate(X2)) activate(n__isNatKind(X)) = 3X >= 3X = isNatKind(X) activate(n__s(X)) = X + 0 >= X + 0 = s(activate(X)) activate(n__and(X1,X2)) = 1X1 + 1X2 >= 1X1 + 1X2 = and(activate(X1),X2) activate(n__isNat(X)) = 3X >= 3X = isNat(X) activate(X) = X >= X = X problem: DPs: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))) and#(tt(),X) -> activate#(X) activate#(n__s(X)) -> activate#(X) isNatKind#(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) -> isNat#(activate(V1)) plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,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(n__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) isNat(X) -> n__isNat(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(activate(X)) activate(n__and(X1,X2)) -> and(activate(X1),X2) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Restore Modifier: DPs: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))) and#(tt(),X) -> activate#(X) activate#(n__s(X)) -> activate#(X) isNatKind#(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) -> isNat#(activate(V1)) plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,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(n__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) isNat(X) -> n__isNat(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(activate(X)) activate(n__and(X1,X2)) -> and(activate(X1),X2) activate(n__isNat(X)) -> isNat(X) activate(X) -> X SCC Processor: #sccs: 3 #rules: 9 #arcs: 169/121 DPs: U21#(tt(),V1) -> isNat#(activate(V1)) isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) 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(n__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) isNat(X) -> n__isNat(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(activate(X)) activate(n__and(X1,X2)) -> and(activate(X1),X2) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Matrix Interpretation Processor: dim=1 usable rules: 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(n__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) isNat(X) -> n__isNat(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(activate(X)) activate(n__and(X1,X2)) -> and(activate(X1),X2) activate(n__isNat(X)) -> isNat(X) activate(X) -> X interpretation: [0] = 5, [U12](x0, x1) = x0, [isNatKind](x0) = 6, [U31](x0, x1) = 3x0 + x1 + 7, [U11](x0, x1, x2) = 4, [isNat](x0) = 4, [n__isNatKind](x0) = 6, [U21#](x0, x1) = 2x0 + 2x1, [n__isNat](x0) = 4, [n__s](x0) = x0 + 4, [U13](x0) = 3, [plus](x0, x1) = x0 + 5x1, [tt] = 3, [U22](x0) = 4, [n__and](x0, x1) = x1, [n__plus](x0, x1) = x0 + 5x1, [and](x0, x1) = x1, [isNat#](x0) = 2x0 + 6, [s](x0) = x0 + 4, [U41](x0, x1, x2) = 5x1 + x2 + 4, [n__0] = 5, [U21](x0, x1) = 4, [activate](x0) = x0 orientation: U21#(tt(),V1) = 2V1 + 6 >= 2V1 + 6 = isNat#(activate(V1)) isNat#(n__s(V1)) = 2V1 + 14 >= 2V1 + 12 = U21#(isNatKind(activate(V1)),activate(V1)) U11(tt(),V1,V2) = 4 >= 4 = U12(isNat(activate(V1)),activate(V2)) U12(tt(),V2) = 3 >= 3 = U13(isNat(activate(V2))) U13(tt()) = 3 >= 3 = tt() U21(tt(),V1) = 4 >= 4 = U22(isNat(activate(V1))) U22(tt()) = 4 >= 3 = tt() U31(tt(),N) = N + 16 >= N = activate(N) U41(tt(),M,N) = 5M + N + 4 >= 5M + N + 4 = s(plus(activate(N),activate(M))) and(tt(),X) = X >= X = activate(X) isNat(n__0()) = 4 >= 3 = tt() isNat(n__plus(V1,V2)) = 4 >= 4 = U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1), activate(V2)) isNat(n__s(V1)) = 4 >= 4 = U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) = 6 >= 3 = tt() isNatKind(n__plus(V1,V2)) = 6 >= 6 = and(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind(n__s(V1)) = 6 >= 6 = isNatKind(activate(V1)) plus(N,0()) = N + 25 >= N + 25 = U31(and(isNat(N),n__isNatKind(N)),N) plus(N,s(M)) = 5M + N + 20 >= 5M + N + 4 = U41(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))), M,N) 0() = 5 >= 5 = n__0() plus(X1,X2) = X1 + 5X2 >= X1 + 5X2 = n__plus(X1,X2) isNatKind(X) = 6 >= 6 = n__isNatKind(X) s(X) = X + 4 >= X + 4 = n__s(X) and(X1,X2) = X2 >= X2 = n__and(X1,X2) isNat(X) = 4 >= 4 = n__isNat(X) activate(n__0()) = 5 >= 5 = 0() activate(n__plus(X1,X2)) = X1 + 5X2 >= X1 + 5X2 = plus(activate(X1),activate(X2)) activate(n__isNatKind(X)) = 6 >= 6 = isNatKind(X) activate(n__s(X)) = X + 4 >= X + 4 = s(activate(X)) activate(n__and(X1,X2)) = X2 >= X2 = and(activate(X1),X2) activate(n__isNat(X)) = 4 >= 4 = isNat(X) activate(X) = X >= X = X problem: DPs: U21#(tt(),V1) -> isNat#(activate(V1)) 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(n__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) isNat(X) -> n__isNat(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(activate(X)) activate(n__and(X1,X2)) -> and(activate(X1),X2) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Restore Modifier: DPs: U21#(tt(),V1) -> isNat#(activate(V1)) 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(n__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) isNat(X) -> n__isNat(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(activate(X)) activate(n__and(X1,X2)) -> and(activate(X1),X2) activate(n__isNat(X)) -> isNat(X) activate(X) -> X SCC Processor: #sccs: 0 #rules: 0 #arcs: 2/1 DPs: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) 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(n__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) isNat(X) -> n__isNat(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(activate(X)) activate(n__and(X1,X2)) -> and(activate(X1),X2) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Matrix Interpretation Processor: dim=1 usable rules: 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(n__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) isNat(X) -> n__isNat(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(activate(X)) activate(n__and(X1,X2)) -> and(activate(X1),X2) activate(n__isNat(X)) -> isNat(X) activate(X) -> X interpretation: [0] = 0, [U12](x0, x1) = 0, [isNatKind](x0) = 2x0, [U31](x0, x1) = x1 + 1, [U11](x0, x1, x2) = 0, [isNat](x0) = 0, [n__isNatKind](x0) = 2x0, [n__isNat](x0) = 0, [n__s](x0) = x0 + 2, [U13](x0) = 0, [isNatKind#](x0) = 4x0 + 4, [plus](x0, x1) = x0 + x1 + 1, [tt] = 0, [U22](x0) = 0, [n__and](x0, x1) = x1 + 2, [n__plus](x0, x1) = x0 + x1 + 1, [and](x0, x1) = x1 + 2, [s](x0) = x0 + 2, [U41](x0, x1, x2) = x1 + x2 + 3, [n__0] = 0, [U21](x0, x1) = 0, [activate](x0) = x0 orientation: isNatKind#(n__s(V1)) = 4V1 + 12 >= 4V1 + 4 = isNatKind#(activate(V1)) U11(tt(),V1,V2) = 0 >= 0 = U12(isNat(activate(V1)),activate(V2)) U12(tt(),V2) = 0 >= 0 = U13(isNat(activate(V2))) U13(tt()) = 0 >= 0 = tt() U21(tt(),V1) = 0 >= 0 = U22(isNat(activate(V1))) U22(tt()) = 0 >= 0 = tt() U31(tt(),N) = N + 1 >= N = activate(N) U41(tt(),M,N) = M + N + 3 >= M + N + 3 = s(plus(activate(N),activate(M))) and(tt(),X) = X + 2 >= X = activate(X) isNat(n__0()) = 0 >= 0 = tt() isNat(n__plus(V1,V2)) = 0 >= 0 = U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1), activate(V2)) isNat(n__s(V1)) = 0 >= 0 = U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) = 0 >= 0 = tt() isNatKind(n__plus(V1,V2)) = 2V1 + 2V2 + 2 >= 2V2 + 2 = and(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind(n__s(V1)) = 2V1 + 4 >= 2V1 = isNatKind(activate(V1)) plus(N,0()) = N + 1 >= N + 1 = U31(and(isNat(N),n__isNatKind(N)),N) plus(N,s(M)) = M + N + 3 >= M + N + 3 = U41(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))), M,N) 0() = 0 >= 0 = n__0() plus(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = n__plus(X1,X2) isNatKind(X) = 2X >= 2X = n__isNatKind(X) s(X) = X + 2 >= X + 2 = n__s(X) and(X1,X2) = X2 + 2 >= X2 + 2 = n__and(X1,X2) isNat(X) = 0 >= 0 = n__isNat(X) activate(n__0()) = 0 >= 0 = 0() activate(n__plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(activate(X1),activate(X2)) activate(n__isNatKind(X)) = 2X >= 2X = isNatKind(X) activate(n__s(X)) = X + 2 >= X + 2 = s(activate(X)) activate(n__and(X1,X2)) = X2 + 2 >= X2 + 2 = and(activate(X1),X2) activate(n__isNat(X)) = 0 >= 0 = isNat(X) activate(X) = X >= X = X problem: DPs: 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(n__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) isNat(X) -> n__isNat(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(activate(X)) activate(n__and(X1,X2)) -> and(activate(X1),X2) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Qed DPs: activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))) and#(tt(),X) -> activate#(X) activate#(n__s(X)) -> activate#(X) plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,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(n__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) isNat(X) -> n__isNat(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(activate(X)) activate(n__and(X1,X2)) -> and(activate(X1),X2) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Arctic Interpretation Processor: dimension: 1 usable rules: interpretation: [0] = 0, [U12](x0, x1) = x0 + 0, [isNatKind](x0) = 6x0 + 0, [U31](x0, x1) = 2x0 + 12x1 + 3, [U11](x0, x1, x2) = 2x0 + 4x1 + 6x2 + 0, [isNat](x0) = 0, [n__isNatKind](x0) = 0, [n__isNat](x0) = 0, [and#](x0, x1) = 1x1 + 0, [n__s](x0) = x0, [activate#](x0) = 1x0 + 0, [U13](x0) = 2x0 + 4, [plus](x0, x1) = 1x0 + 1x1 + 0, [tt] = 2, [U22](x0) = 2x0 + 1, [n__and](x0, x1) = x0 + x1 + 0, [plus#](x0, x1) = 2, [n__plus](x0, x1) = x0 + 1x1 + 1, [and](x0, x1) = x1, [s](x0) = 2x0 + 0, [U41](x0, x1, x2) = x0 + x1 + 4x2 + 0, [n__0] = 1, [U21](x0, x1) = x0 + 6x1 + 0, [U41#](x0, x1, x2) = 2, [activate](x0) = x0 + 0 orientation: activate#(n__plus(X1,X2)) = 1X1 + 2X2 + 2 >= 2 = plus#(activate(X1),activate(X2)) plus#(N,s(M)) = 2 >= 1 = and#(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))) and#(tt(),X) = 1X + 0 >= 1X + 0 = activate#(X) activate#(n__s(X)) = 1X + 0 >= 1X + 0 = activate#(X) plus#(N,s(M)) = 2 >= 2 = U41#(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))), M,N) U41#(tt(),M,N) = 2 >= 2 = plus#(activate(N),activate(M)) U11(tt(),V1,V2) = 4V1 + 6V2 + 4 >= 0 = U12(isNat(activate(V1)),activate(V2)) U12(tt(),V2) = 2 >= 4 = U13(isNat(activate(V2))) U13(tt()) = 4 >= 2 = tt() U21(tt(),V1) = 6V1 + 2 >= 2 = U22(isNat(activate(V1))) U22(tt()) = 4 >= 2 = tt() U31(tt(),N) = 12N + 4 >= N + 0 = activate(N) U41(tt(),M,N) = M + 4N + 2 >= 3M + 3N + 3 = s(plus(activate(N),activate(M))) and(tt(),X) = X >= X + 0 = activate(X) isNat(n__0()) = 0 >= 2 = tt() isNat(n__plus(V1,V2)) = 0 >= 4V1 + 6V2 + 6 = U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1), activate(V2)) isNat(n__s(V1)) = 0 >= 6V1 + 6 = U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) = 7 >= 2 = tt() isNatKind(n__plus(V1,V2)) = 6V1 + 7V2 + 7 >= 0 = and(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind(n__s(V1)) = 6V1 + 0 >= 6V1 + 6 = isNatKind(activate(V1)) plus(N,0()) = 1N + 1 >= 12N + 3 = U31(and(isNat(N),n__isNatKind(N)),N) plus(N,s(M)) = 3M + 1N + 1 >= M + 4N + 0 = U41(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))), M,N) 0() = 0 >= 1 = n__0() plus(X1,X2) = 1X1 + 1X2 + 0 >= X1 + 1X2 + 1 = n__plus(X1,X2) isNatKind(X) = 6X + 0 >= 0 = n__isNatKind(X) s(X) = 2X + 0 >= X = n__s(X) and(X1,X2) = X2 >= X1 + X2 + 0 = n__and(X1,X2) isNat(X) = 0 >= 0 = n__isNat(X) activate(n__0()) = 1 >= 0 = 0() activate(n__plus(X1,X2)) = X1 + 1X2 + 1 >= 1X1 + 1X2 + 1 = plus(activate(X1),activate(X2)) activate(n__isNatKind(X)) = 0 >= 6X + 0 = isNatKind(X) activate(n__s(X)) = X + 0 >= 2X + 2 = s(activate(X)) activate(n__and(X1,X2)) = X1 + X2 + 0 >= X2 = and(activate(X1),X2) activate(n__isNat(X)) = 0 >= 0 = isNat(X) activate(X) = X + 0 >= X = X problem: DPs: activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) and#(tt(),X) -> activate#(X) activate#(n__s(X)) -> activate#(X) plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,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(n__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) isNat(X) -> n__isNat(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(activate(X)) activate(n__and(X1,X2)) -> and(activate(X1),X2) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Restore Modifier: DPs: activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) and#(tt(),X) -> activate#(X) activate#(n__s(X)) -> activate#(X) plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,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(n__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) isNat(X) -> n__isNat(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(activate(X)) activate(n__and(X1,X2)) -> and(activate(X1),X2) activate(n__isNat(X)) -> isNat(X) activate(X) -> X SCC Processor: #sccs: 2 #rules: 3 #arcs: 10/25 DPs: activate#(n__s(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(n__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) isNat(X) -> n__isNat(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(activate(X)) activate(n__and(X1,X2)) -> and(activate(X1),X2) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Size-Change Termination Processor: DPs: 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(n__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) isNat(X) -> n__isNat(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(activate(X)) activate(n__and(X1,X2)) -> and(activate(X1),X2) activate(n__isNat(X)) -> isNat(X) activate(X) -> X The DP: activate#(n__s(X)) -> activate#(X) has the edges: 0 > 0 Qed DPs: plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,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(n__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) isNat(X) -> n__isNat(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(activate(X)) activate(n__and(X1,X2)) -> and(activate(X1),X2) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Matrix Interpretation Processor: dim=1 usable rules: 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(n__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) isNat(X) -> n__isNat(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(activate(X)) activate(n__and(X1,X2)) -> and(activate(X1),X2) activate(n__isNat(X)) -> isNat(X) activate(X) -> X interpretation: [0] = 0, [U12](x0, x1) = x0, [isNatKind](x0) = 0, [U31](x0, x1) = x1, [U11](x0, x1, x2) = x1, [isNat](x0) = x0, [n__isNatKind](x0) = 0, [n__isNat](x0) = x0, [n__s](x0) = x0 + 1, [U13](x0) = 0, [plus](x0, x1) = x0 + x1, [tt] = 0, [U22](x0) = 0, [n__and](x0, x1) = 2x1, [plus#](x0, x1) = 4x0 + x1 + 1, [n__plus](x0, x1) = x0 + x1, [and](x0, x1) = 2x1, [s](x0) = x0 + 1, [U41](x0, x1, x2) = x1 + x2 + 1, [n__0] = 0, [U21](x0, x1) = x1, [U41#](x0, x1, x2) = x1 + 4x2 + 2, [activate](x0) = x0 orientation: plus#(N,s(M)) = M + 4N + 2 >= M + 4N + 2 = U41#(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))), M,N) U41#(tt(),M,N) = M + 4N + 2 >= M + 4N + 1 = plus#(activate(N),activate(M)) U11(tt(),V1,V2) = V1 >= V1 = U12(isNat(activate(V1)),activate(V2)) U12(tt(),V2) = 0 >= 0 = U13(isNat(activate(V2))) U13(tt()) = 0 >= 0 = tt() U21(tt(),V1) = V1 >= 0 = U22(isNat(activate(V1))) U22(tt()) = 0 >= 0 = tt() U31(tt(),N) = N >= N = activate(N) U41(tt(),M,N) = M + N + 1 >= M + N + 1 = s(plus(activate(N),activate(M))) and(tt(),X) = 2X >= X = activate(X) isNat(n__0()) = 0 >= 0 = tt() isNat(n__plus(V1,V2)) = V1 + V2 >= V1 = U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1), activate(V2)) isNat(n__s(V1)) = V1 + 1 >= V1 = U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) = 0 >= 0 = tt() isNatKind(n__plus(V1,V2)) = 0 >= 0 = and(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind(n__s(V1)) = 0 >= 0 = isNatKind(activate(V1)) plus(N,0()) = N >= N = U31(and(isNat(N),n__isNatKind(N)),N) plus(N,s(M)) = M + N + 1 >= M + N + 1 = U41(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))), M,N) 0() = 0 >= 0 = n__0() plus(X1,X2) = X1 + X2 >= X1 + X2 = n__plus(X1,X2) isNatKind(X) = 0 >= 0 = n__isNatKind(X) s(X) = X + 1 >= X + 1 = n__s(X) and(X1,X2) = 2X2 >= 2X2 = n__and(X1,X2) isNat(X) = X >= X = n__isNat(X) activate(n__0()) = 0 >= 0 = 0() activate(n__plus(X1,X2)) = X1 + X2 >= X1 + X2 = plus(activate(X1),activate(X2)) activate(n__isNatKind(X)) = 0 >= 0 = isNatKind(X) activate(n__s(X)) = X + 1 >= X + 1 = s(activate(X)) activate(n__and(X1,X2)) = 2X2 >= 2X2 = and(activate(X1),X2) activate(n__isNat(X)) = X >= X = isNat(X) activate(X) = X >= X = X problem: DPs: plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,N) 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(n__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) isNat(X) -> n__isNat(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(activate(X)) activate(n__and(X1,X2)) -> and(activate(X1),X2) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Restore Modifier: DPs: plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,N) 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(n__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) isNat(X) -> n__isNat(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(activate(X)) activate(n__and(X1,X2)) -> and(activate(X1),X2) activate(n__isNat(X)) -> isNat(X) activate(X) -> X SCC Processor: #sccs: 0 #rules: 0 #arcs: 2/1