YES Problem: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2)) U12(tt(),V2) -> U13(isNat(activate(V2))) U13(tt()) -> tt() U21(tt(),V1) -> U22(isNat(activate(V1))) U22(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> s(plus(activate(N),activate(M))) and(tt(),X) -> activate(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind(n__s(V1)) -> isNatKind(activate(V1)) plus(N,0()) -> U31(and(isNat(N),n__isNatKind(N)),N) plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) isNatKind(X) -> n__isNatKind(X) s(X) -> n__s(X) and(X1,X2) -> n__and(X1,X2) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(X) activate(n__and(X1,X2)) -> and(X1,X2) activate(X) -> X Proof: DP Processor: DPs: U11#(tt(),V1,V2) -> activate#(V2) U11#(tt(),V1,V2) -> activate#(V1) U11#(tt(),V1,V2) -> isNat#(activate(V1)) U11#(tt(),V1,V2) -> U12#(isNat(activate(V1)),activate(V2)) U12#(tt(),V2) -> activate#(V2) U12#(tt(),V2) -> isNat#(activate(V2)) U12#(tt(),V2) -> U13#(isNat(activate(V2))) U21#(tt(),V1) -> activate#(V1) U21#(tt(),V1) -> isNat#(activate(V1)) U21#(tt(),V1) -> U22#(isNat(activate(V1))) U31#(tt(),N) -> activate#(N) U41#(tt(),M,N) -> activate#(M) U41#(tt(),M,N) -> activate#(N) U41#(tt(),M,N) -> plus#(activate(N),activate(M)) U41#(tt(),M,N) -> s#(plus(activate(N),activate(M))) and#(tt(),X) -> activate#(X) isNat#(n__plus(V1,V2)) -> activate#(V2) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNat#(n__plus(V1,V2)) -> U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) isNatKind#(n__plus(V1,V2)) -> activate#(V2) isNatKind#(n__plus(V1,V2)) -> activate#(V1) isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind#(n__s(V1)) -> activate#(V1) isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) plus#(N,0()) -> isNat#(N) plus#(N,0()) -> and#(isNat(N),n__isNatKind(N)) plus#(N,0()) -> U31#(and(isNat(N),n__isNatKind(N)),N) plus#(N,s(M)) -> isNat#(N) plus#(N,s(M)) -> isNat#(M) plus#(N,s(M)) -> and#(isNat(M),n__isNatKind(M)) plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))) plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) activate#(n__0()) -> 0#() activate#(n__plus(X1,X2)) -> plus#(X1,X2) activate#(n__isNatKind(X)) -> isNatKind#(X) activate#(n__s(X)) -> s#(X) activate#(n__and(X1,X2)) -> and#(X1,X2) TRS: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2)) U12(tt(),V2) -> U13(isNat(activate(V2))) U13(tt()) -> tt() U21(tt(),V1) -> U22(isNat(activate(V1))) U22(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> s(plus(activate(N),activate(M))) and(tt(),X) -> activate(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind(n__s(V1)) -> isNatKind(activate(V1)) plus(N,0()) -> U31(and(isNat(N),n__isNatKind(N)),N) plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) isNatKind(X) -> n__isNatKind(X) s(X) -> n__s(X) and(X1,X2) -> n__and(X1,X2) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(X) activate(n__and(X1,X2)) -> and(X1,X2) activate(X) -> X TDG Processor: DPs: U11#(tt(),V1,V2) -> activate#(V2) U11#(tt(),V1,V2) -> activate#(V1) U11#(tt(),V1,V2) -> isNat#(activate(V1)) U11#(tt(),V1,V2) -> U12#(isNat(activate(V1)),activate(V2)) U12#(tt(),V2) -> activate#(V2) U12#(tt(),V2) -> isNat#(activate(V2)) U12#(tt(),V2) -> U13#(isNat(activate(V2))) U21#(tt(),V1) -> activate#(V1) U21#(tt(),V1) -> isNat#(activate(V1)) U21#(tt(),V1) -> U22#(isNat(activate(V1))) U31#(tt(),N) -> activate#(N) U41#(tt(),M,N) -> activate#(M) U41#(tt(),M,N) -> activate#(N) U41#(tt(),M,N) -> plus#(activate(N),activate(M)) U41#(tt(),M,N) -> s#(plus(activate(N),activate(M))) and#(tt(),X) -> activate#(X) isNat#(n__plus(V1,V2)) -> activate#(V2) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNat#(n__plus(V1,V2)) -> U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) isNatKind#(n__plus(V1,V2)) -> activate#(V2) isNatKind#(n__plus(V1,V2)) -> activate#(V1) isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind#(n__s(V1)) -> activate#(V1) isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) plus#(N,0()) -> isNat#(N) plus#(N,0()) -> and#(isNat(N),n__isNatKind(N)) plus#(N,0()) -> U31#(and(isNat(N),n__isNatKind(N)),N) plus#(N,s(M)) -> isNat#(N) plus#(N,s(M)) -> isNat#(M) plus#(N,s(M)) -> and#(isNat(M),n__isNatKind(M)) plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))) plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) activate#(n__0()) -> 0#() activate#(n__plus(X1,X2)) -> plus#(X1,X2) activate#(n__isNatKind(X)) -> isNatKind#(X) activate#(n__s(X)) -> s#(X) activate#(n__and(X1,X2)) -> and#(X1,X2) TRS: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2)) U12(tt(),V2) -> U13(isNat(activate(V2))) U13(tt()) -> tt() U21(tt(),V1) -> U22(isNat(activate(V1))) U22(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> s(plus(activate(N),activate(M))) and(tt(),X) -> activate(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind(n__s(V1)) -> isNatKind(activate(V1)) plus(N,0()) -> U31(and(isNat(N),n__isNatKind(N)),N) plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) isNatKind(X) -> n__isNatKind(X) s(X) -> n__s(X) and(X1,X2) -> n__and(X1,X2) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(X) activate(n__and(X1,X2)) -> and(X1,X2) activate(X) -> X graph: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) -> isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) -> isNatKind#(n__s(V1)) -> activate#(V1) isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V1) isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V2) isNatKind#(n__s(V1)) -> activate#(V1) -> activate#(n__and(X1,X2)) -> and#(X1,X2) isNatKind#(n__s(V1)) -> activate#(V1) -> activate#(n__s(X)) -> s#(X) isNatKind#(n__s(V1)) -> activate#(V1) -> activate#(n__isNatKind(X)) -> isNatKind#(X) isNatKind#(n__s(V1)) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) isNatKind#(n__s(V1)) -> activate#(V1) -> activate#(n__0()) -> 0#() isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) -> isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) -> isNatKind#(n__s(V1)) -> activate#(V1) isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V1) isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V2) isNatKind#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) -> and#(tt(),X) -> activate#(X) isNatKind#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__and(X1,X2)) -> and#(X1,X2) isNatKind#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__s(X)) -> s#(X) isNatKind#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__isNatKind(X)) -> isNatKind#(X) isNatKind#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) isNatKind#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__0()) -> 0#() isNatKind#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__and(X1,X2)) -> and#(X1,X2) isNatKind#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__s(X)) -> s#(X) isNatKind#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__isNatKind(X)) -> isNatKind#(X) isNatKind#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) isNatKind#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__0()) -> 0#() and#(tt(),X) -> activate#(X) -> activate#(n__and(X1,X2)) -> and#(X1,X2) and#(tt(),X) -> activate#(X) -> activate#(n__s(X)) -> s#(X) and#(tt(),X) -> activate#(X) -> activate#(n__isNatKind(X)) -> isNatKind#(X) and#(tt(),X) -> activate#(X) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) and#(tt(),X) -> activate#(X) -> activate#(n__0()) -> 0#() plus#(N,0()) -> and#(isNat(N),n__isNatKind(N)) -> and#(tt(),X) -> activate#(X) plus#(N,0()) -> U31#(and(isNat(N),n__isNatKind(N)),N) -> U31#(tt(),N) -> activate#(N) plus#(N,0()) -> isNat#(N) -> isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) plus#(N,0()) -> isNat#(N) -> isNat#(n__s(V1)) -> isNatKind#(activate(V1)) plus#(N,0()) -> isNat#(N) -> isNat#(n__s(V1)) -> activate#(V1) plus#(N,0()) -> isNat#(N) -> isNat#(n__plus(V1,V2)) -> U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) plus#(N,0()) -> isNat#(N) -> isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) plus#(N,0()) -> isNat#(N) -> isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) plus#(N,0()) -> isNat#(N) -> isNat#(n__plus(V1,V2)) -> activate#(V1) plus#(N,0()) -> isNat#(N) -> isNat#(n__plus(V1,V2)) -> activate#(V2) plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))) -> and#(tt(),X) -> activate#(X) plus#(N,s(M)) -> and#(isNat(M),n__isNatKind(M)) -> and#(tt(),X) -> activate#(X) plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) -> U41#(tt(),M,N) -> s#(plus(activate(N),activate(M))) plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) -> U41#(tt(),M,N) -> plus#(activate(N),activate(M)) plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) -> U41#(tt(),M,N) -> activate#(N) plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) -> U41#(tt(),M,N) -> activate#(M) plus#(N,s(M)) -> isNat#(N) -> isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) plus#(N,s(M)) -> isNat#(N) -> isNat#(n__s(V1)) -> isNatKind#(activate(V1)) plus#(N,s(M)) -> isNat#(N) -> isNat#(n__s(V1)) -> activate#(V1) plus#(N,s(M)) -> isNat#(N) -> isNat#(n__plus(V1,V2)) -> U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) plus#(N,s(M)) -> isNat#(N) -> isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) plus#(N,s(M)) -> isNat#(N) -> isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) plus#(N,s(M)) -> isNat#(N) -> isNat#(n__plus(V1,V2)) -> activate#(V1) plus#(N,s(M)) -> isNat#(N) -> isNat#(n__plus(V1,V2)) -> activate#(V2) plus#(N,s(M)) -> isNat#(M) -> isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) plus#(N,s(M)) -> isNat#(M) -> isNat#(n__s(V1)) -> isNatKind#(activate(V1)) plus#(N,s(M)) -> isNat#(M) -> isNat#(n__s(V1)) -> activate#(V1) plus#(N,s(M)) -> isNat#(M) -> isNat#(n__plus(V1,V2)) -> U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) plus#(N,s(M)) -> isNat#(M) -> isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) plus#(N,s(M)) -> isNat#(M) -> isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) plus#(N,s(M)) -> isNat#(M) -> isNat#(n__plus(V1,V2)) -> activate#(V1) plus#(N,s(M)) -> isNat#(M) -> isNat#(n__plus(V1,V2)) -> activate#(V2) U41#(tt(),M,N) -> plus#(activate(N),activate(M)) -> plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) U41#(tt(),M,N) -> plus#(activate(N),activate(M)) -> plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))) U41#(tt(),M,N) -> plus#(activate(N),activate(M)) -> plus#(N,s(M)) -> and#(isNat(M),n__isNatKind(M)) U41#(tt(),M,N) -> plus#(activate(N),activate(M)) -> plus#(N,s(M)) -> isNat#(M) U41#(tt(),M,N) -> plus#(activate(N),activate(M)) -> plus#(N,s(M)) -> isNat#(N) U41#(tt(),M,N) -> plus#(activate(N),activate(M)) -> plus#(N,0()) -> U31#(and(isNat(N),n__isNatKind(N)),N) U41#(tt(),M,N) -> plus#(activate(N),activate(M)) -> plus#(N,0()) -> and#(isNat(N),n__isNatKind(N)) U41#(tt(),M,N) -> plus#(activate(N),activate(M)) -> plus#(N,0()) -> isNat#(N) U41#(tt(),M,N) -> activate#(N) -> activate#(n__and(X1,X2)) -> and#(X1,X2) U41#(tt(),M,N) -> activate#(N) -> activate#(n__s(X)) -> s#(X) U41#(tt(),M,N) -> activate#(N) -> activate#(n__isNatKind(X)) -> isNatKind#(X) U41#(tt(),M,N) -> activate#(N) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) U41#(tt(),M,N) -> activate#(N) -> activate#(n__0()) -> 0#() U41#(tt(),M,N) -> activate#(M) -> activate#(n__and(X1,X2)) -> and#(X1,X2) U41#(tt(),M,N) -> activate#(M) -> activate#(n__s(X)) -> s#(X) U41#(tt(),M,N) -> activate#(M) -> activate#(n__isNatKind(X)) -> isNatKind#(X) U41#(tt(),M,N) -> activate#(M) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) U41#(tt(),M,N) -> activate#(M) -> activate#(n__0()) -> 0#() U31#(tt(),N) -> activate#(N) -> activate#(n__and(X1,X2)) -> and#(X1,X2) U31#(tt(),N) -> activate#(N) -> activate#(n__s(X)) -> s#(X) U31#(tt(),N) -> activate#(N) -> activate#(n__isNatKind(X)) -> isNatKind#(X) U31#(tt(),N) -> activate#(N) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) U31#(tt(),N) -> activate#(N) -> activate#(n__0()) -> 0#() U21#(tt(),V1) -> isNat#(activate(V1)) -> isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) -> isNat#(activate(V1)) -> isNat#(n__s(V1)) -> isNatKind#(activate(V1)) U21#(tt(),V1) -> isNat#(activate(V1)) -> isNat#(n__s(V1)) -> activate#(V1) U21#(tt(),V1) -> isNat#(activate(V1)) -> isNat#(n__plus(V1,V2)) -> U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) U21#(tt(),V1) -> isNat#(activate(V1)) -> isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) U21#(tt(),V1) -> isNat#(activate(V1)) -> isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) U21#(tt(),V1) -> isNat#(activate(V1)) -> isNat#(n__plus(V1,V2)) -> activate#(V1) U21#(tt(),V1) -> isNat#(activate(V1)) -> isNat#(n__plus(V1,V2)) -> activate#(V2) U21#(tt(),V1) -> activate#(V1) -> activate#(n__and(X1,X2)) -> and#(X1,X2) U21#(tt(),V1) -> activate#(V1) -> activate#(n__s(X)) -> s#(X) U21#(tt(),V1) -> activate#(V1) -> activate#(n__isNatKind(X)) -> isNatKind#(X) U21#(tt(),V1) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) U21#(tt(),V1) -> activate#(V1) -> activate#(n__0()) -> 0#() U12#(tt(),V2) -> isNat#(activate(V2)) -> isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) U12#(tt(),V2) -> isNat#(activate(V2)) -> isNat#(n__s(V1)) -> isNatKind#(activate(V1)) U12#(tt(),V2) -> isNat#(activate(V2)) -> isNat#(n__s(V1)) -> activate#(V1) U12#(tt(),V2) -> isNat#(activate(V2)) -> isNat#(n__plus(V1,V2)) -> U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) U12#(tt(),V2) -> isNat#(activate(V2)) -> isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) U12#(tt(),V2) -> isNat#(activate(V2)) -> isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) U12#(tt(),V2) -> isNat#(activate(V2)) -> isNat#(n__plus(V1,V2)) -> activate#(V1) U12#(tt(),V2) -> isNat#(activate(V2)) -> isNat#(n__plus(V1,V2)) -> activate#(V2) U12#(tt(),V2) -> activate#(V2) -> activate#(n__and(X1,X2)) -> and#(X1,X2) U12#(tt(),V2) -> activate#(V2) -> activate#(n__s(X)) -> s#(X) U12#(tt(),V2) -> activate#(V2) -> activate#(n__isNatKind(X)) -> isNatKind#(X) U12#(tt(),V2) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) U12#(tt(),V2) -> activate#(V2) -> activate#(n__0()) -> 0#() isNat#(n__s(V1)) -> isNatKind#(activate(V1)) -> isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) -> isNatKind#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V2) isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) -> U21#(tt(),V1) -> U22#(isNat(activate(V1))) isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) -> U21#(tt(),V1) -> isNat#(activate(V1)) isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) -> U21#(tt(),V1) -> activate#(V1) isNat#(n__s(V1)) -> activate#(V1) -> activate#(n__and(X1,X2)) -> and#(X1,X2) isNat#(n__s(V1)) -> activate#(V1) -> activate#(n__s(X)) -> s#(X) isNat#(n__s(V1)) -> activate#(V1) -> activate#(n__isNatKind(X)) -> isNatKind#(X) isNat#(n__s(V1)) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) isNat#(n__s(V1)) -> activate#(V1) -> activate#(n__0()) -> 0#() isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) -> isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) -> isNatKind#(n__s(V1)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) -> isNatKind#(n__plus(V1,V2)) -> activate#(V2) isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) -> and#(tt(),X) -> activate#(X) isNat#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__and(X1,X2)) -> and#(X1,X2) isNat#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__s(X)) -> s#(X) isNat#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__isNatKind(X)) -> isNatKind#(X) isNat#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) isNat#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__0()) -> 0#() isNat#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__and(X1,X2)) -> and#(X1,X2) isNat#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__s(X)) -> s#(X) isNat#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__isNatKind(X)) -> isNatKind#(X) isNat#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) isNat#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__0()) -> 0#() isNat#(n__plus(V1,V2)) -> U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) -> U11#(tt(),V1,V2) -> U12#(isNat(activate(V1)),activate(V2)) isNat#(n__plus(V1,V2)) -> U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) -> U11#(tt(),V1,V2) -> isNat#(activate(V1)) isNat#(n__plus(V1,V2)) -> U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) -> U11#(tt(),V1,V2) -> activate#(V1) isNat#(n__plus(V1,V2)) -> U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) -> U11#(tt(),V1,V2) -> activate#(V2) activate#(n__and(X1,X2)) -> and#(X1,X2) -> and#(tt(),X) -> activate#(X) activate#(n__isNatKind(X)) -> isNatKind#(X) -> isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) activate#(n__isNatKind(X)) -> isNatKind#(X) -> isNatKind#(n__s(V1)) -> activate#(V1) activate#(n__isNatKind(X)) -> isNatKind#(X) -> isNatKind#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) activate#(n__isNatKind(X)) -> isNatKind#(X) -> isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) activate#(n__isNatKind(X)) -> isNatKind#(X) -> isNatKind#(n__plus(V1,V2)) -> activate#(V1) activate#(n__isNatKind(X)) -> isNatKind#(X) -> isNatKind#(n__plus(V1,V2)) -> activate#(V2) activate#(n__plus(X1,X2)) -> plus#(X1,X2) -> plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) activate#(n__plus(X1,X2)) -> plus#(X1,X2) -> plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))) activate#(n__plus(X1,X2)) -> plus#(X1,X2) -> plus#(N,s(M)) -> and#(isNat(M),n__isNatKind(M)) activate#(n__plus(X1,X2)) -> plus#(X1,X2) -> plus#(N,s(M)) -> isNat#(M) activate#(n__plus(X1,X2)) -> plus#(X1,X2) -> plus#(N,s(M)) -> isNat#(N) activate#(n__plus(X1,X2)) -> plus#(X1,X2) -> plus#(N,0()) -> U31#(and(isNat(N),n__isNatKind(N)),N) activate#(n__plus(X1,X2)) -> plus#(X1,X2) -> plus#(N,0()) -> and#(isNat(N),n__isNatKind(N)) activate#(n__plus(X1,X2)) -> plus#(X1,X2) -> plus#(N,0()) -> isNat#(N) U11#(tt(),V1,V2) -> U12#(isNat(activate(V1)),activate(V2)) -> U12#(tt(),V2) -> U13#(isNat(activate(V2))) U11#(tt(),V1,V2) -> U12#(isNat(activate(V1)),activate(V2)) -> U12#(tt(),V2) -> isNat#(activate(V2)) U11#(tt(),V1,V2) -> U12#(isNat(activate(V1)),activate(V2)) -> U12#(tt(),V2) -> activate#(V2) U11#(tt(),V1,V2) -> isNat#(activate(V1)) -> isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) U11#(tt(),V1,V2) -> isNat#(activate(V1)) -> isNat#(n__s(V1)) -> isNatKind#(activate(V1)) U11#(tt(),V1,V2) -> isNat#(activate(V1)) -> isNat#(n__s(V1)) -> activate#(V1) U11#(tt(),V1,V2) -> isNat#(activate(V1)) -> isNat#(n__plus(V1,V2)) -> U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) U11#(tt(),V1,V2) -> isNat#(activate(V1)) -> isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) U11#(tt(),V1,V2) -> isNat#(activate(V1)) -> isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) U11#(tt(),V1,V2) -> isNat#(activate(V1)) -> isNat#(n__plus(V1,V2)) -> activate#(V1) U11#(tt(),V1,V2) -> isNat#(activate(V1)) -> isNat#(n__plus(V1,V2)) -> activate#(V2) U11#(tt(),V1,V2) -> activate#(V2) -> activate#(n__and(X1,X2)) -> and#(X1,X2) U11#(tt(),V1,V2) -> activate#(V2) -> activate#(n__s(X)) -> s#(X) U11#(tt(),V1,V2) -> activate#(V2) -> activate#(n__isNatKind(X)) -> isNatKind#(X) U11#(tt(),V1,V2) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) U11#(tt(),V1,V2) -> activate#(V2) -> activate#(n__0()) -> 0#() U11#(tt(),V1,V2) -> activate#(V1) -> activate#(n__and(X1,X2)) -> and#(X1,X2) U11#(tt(),V1,V2) -> activate#(V1) -> activate#(n__s(X)) -> s#(X) U11#(tt(),V1,V2) -> activate#(V1) -> activate#(n__isNatKind(X)) -> isNatKind#(X) U11#(tt(),V1,V2) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) U11#(tt(),V1,V2) -> activate#(V1) -> activate#(n__0()) -> 0#() SCC Processor: #sccs: 1 #rules: 38 #arcs: 185/1849 DPs: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) -> activate#(V2) activate#(n__plus(X1,X2)) -> plus#(X1,X2) plus#(N,0()) -> isNat#(N) isNat#(n__plus(V1,V2)) -> activate#(V2) activate#(n__isNatKind(X)) -> isNatKind#(X) isNatKind#(n__plus(V1,V2)) -> activate#(V1) activate#(n__and(X1,X2)) -> and#(X1,X2) and#(tt(),X) -> activate#(X) isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind#(n__s(V1)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNat#(n__plus(V1,V2)) -> U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) U11#(tt(),V1,V2) -> activate#(V2) U11#(tt(),V1,V2) -> activate#(V1) U11#(tt(),V1,V2) -> isNat#(activate(V1)) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNatKind#(activate(V1)) isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) -> activate#(V1) U21#(tt(),V1) -> isNat#(activate(V1)) U11#(tt(),V1,V2) -> U12#(isNat(activate(V1)),activate(V2)) U12#(tt(),V2) -> activate#(V2) U12#(tt(),V2) -> isNat#(activate(V2)) plus#(N,0()) -> and#(isNat(N),n__isNatKind(N)) plus#(N,0()) -> U31#(and(isNat(N),n__isNatKind(N)),N) U31#(tt(),N) -> activate#(N) plus#(N,s(M)) -> isNat#(N) plus#(N,s(M)) -> isNat#(M) plus#(N,s(M)) -> and#(isNat(M),n__isNatKind(M)) plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))) plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) U41#(tt(),M,N) -> activate#(M) U41#(tt(),M,N) -> activate#(N) U41#(tt(),M,N) -> plus#(activate(N),activate(M)) TRS: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2)) U12(tt(),V2) -> U13(isNat(activate(V2))) U13(tt()) -> tt() U21(tt(),V1) -> U22(isNat(activate(V1))) U22(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> s(plus(activate(N),activate(M))) and(tt(),X) -> activate(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind(n__s(V1)) -> isNatKind(activate(V1)) plus(N,0()) -> U31(and(isNat(N),n__isNatKind(N)),N) plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) isNatKind(X) -> n__isNatKind(X) s(X) -> n__s(X) and(X1,X2) -> n__and(X1,X2) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(X) activate(n__and(X1,X2)) -> and(X1,X2) activate(X) -> X Arctic Interpretation Processor: dimension: 1 usable rules: U31(tt(),N) -> activate(N) U41(tt(),M,N) -> s(plus(activate(N),activate(M))) and(tt(),X) -> activate(X) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind(n__s(V1)) -> isNatKind(activate(V1)) plus(N,0()) -> U31(and(isNat(N),n__isNatKind(N)),N) plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) isNatKind(X) -> n__isNatKind(X) s(X) -> n__s(X) and(X1,X2) -> n__and(X1,X2) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(X) activate(n__and(X1,X2)) -> and(X1,X2) activate(X) -> X interpretation: [isNatKind#](x0) = x0, [and#](x0, x1) = x1, [plus#](x0, x1) = x0 + x1, [U41#](x0, x1, x2) = x0 + x1 + x2, [U31#](x0, x1) = x1, [U21#](x0, x1) = x0 + x1, [U12#](x0, x1) = 7x1 + 0, [isNat#](x0) = x0, [activate#](x0) = x0, [U11#](x0, x1, x2) = x0 + x1 + 7x2, [n__and](x0, x1) = x1, [0] = 0, [n__s](x0) = x0, [n__isNatKind](x0) = x0, [isNatKind](x0) = x0, [n__plus](x0, x1) = x0 + 7x1, [n__0] = 0, [and](x0, x1) = x1, [s](x0) = x0, [plus](x0, x1) = x0 + 7x1, [U41](x0, x1, x2) = x0 + 7x1 + x2, [U31](x0, x1) = x0 + x1 + 0, [U22](x0) = 0, [U21](x0, x1) = 5x0 + 1x1 + 8, [U13](x0) = 2x0 + 0, [U12](x0, x1) = x0 + 0, [isNat](x0) = x0 + 0, [activate](x0) = x0, [U11](x0, x1, x2) = 1x0 + x1 + 1x2, [tt] = 0 orientation: isNatKind#(n__s(V1)) = V1 >= V1 = isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) = V1 + 7V2 >= V2 = activate#(V2) activate#(n__plus(X1,X2)) = X1 + 7X2 >= X1 + X2 = plus#(X1,X2) plus#(N,0()) = N + 0 >= N = isNat#(N) isNat#(n__plus(V1,V2)) = V1 + 7V2 >= V2 = activate#(V2) activate#(n__isNatKind(X)) = X >= X = isNatKind#(X) isNatKind#(n__plus(V1,V2)) = V1 + 7V2 >= V1 = activate#(V1) activate#(n__and(X1,X2)) = X2 >= X2 = and#(X1,X2) and#(tt(),X) = X >= X = activate#(X) isNatKind#(n__plus(V1,V2)) = V1 + 7V2 >= V1 = isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) = V1 + 7V2 >= V2 = and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind#(n__s(V1)) = V1 >= V1 = activate#(V1) isNat#(n__plus(V1,V2)) = V1 + 7V2 >= V1 = activate#(V1) isNat#(n__plus(V1,V2)) = V1 + 7V2 >= V1 = isNatKind#(activate(V1)) isNat#(n__plus(V1,V2)) = V1 + 7V2 >= V2 = and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNat#(n__plus(V1,V2)) = V1 + 7V2 >= V1 + 7V2 = U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1), activate(V2)) U11#(tt(),V1,V2) = V1 + 7V2 + 0 >= V2 = activate#(V2) U11#(tt(),V1,V2) = V1 + 7V2 + 0 >= V1 = activate#(V1) U11#(tt(),V1,V2) = V1 + 7V2 + 0 >= V1 = isNat#(activate(V1)) isNat#(n__s(V1)) = V1 >= V1 = activate#(V1) isNat#(n__s(V1)) = V1 >= V1 = isNatKind#(activate(V1)) isNat#(n__s(V1)) = V1 >= V1 = U21#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) = V1 + 0 >= V1 = activate#(V1) U21#(tt(),V1) = V1 + 0 >= V1 = isNat#(activate(V1)) U11#(tt(),V1,V2) = V1 + 7V2 + 0 >= 7V2 + 0 = U12#(isNat(activate(V1)),activate(V2)) U12#(tt(),V2) = 7V2 + 0 >= V2 = activate#(V2) U12#(tt(),V2) = 7V2 + 0 >= V2 = isNat#(activate(V2)) plus#(N,0()) = N + 0 >= N = and#(isNat(N),n__isNatKind(N)) plus#(N,0()) = N + 0 >= N = U31#(and(isNat(N),n__isNatKind(N)),N) U31#(tt(),N) = N >= N = activate#(N) plus#(N,s(M)) = M + N >= N = isNat#(N) plus#(N,s(M)) = M + N >= M = isNat#(M) plus#(N,s(M)) = M + N >= M = and#(isNat(M),n__isNatKind(M)) plus#(N,s(M)) = M + N >= N = and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))) plus#(N,s(M)) = M + N >= M + N = U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) U41#(tt(),M,N) = M + N + 0 >= M = activate#(M) U41#(tt(),M,N) = M + N + 0 >= N = activate#(N) U41#(tt(),M,N) = M + N + 0 >= M + N = plus#(activate(N),activate(M)) U11(tt(),V1,V2) = V1 + 1V2 + 1 >= V1 + 0 = U12(isNat(activate(V1)),activate(V2)) U12(tt(),V2) = 0 >= 2V2 + 2 = U13(isNat(activate(V2))) U13(tt()) = 2 >= 0 = tt() U21(tt(),V1) = 1V1 + 8 >= 0 = U22(isNat(activate(V1))) U22(tt()) = 0 >= 0 = tt() U31(tt(),N) = N + 0 >= N = activate(N) U41(tt(),M,N) = 7M + N + 0 >= 7M + N = s(plus(activate(N),activate(M))) and(tt(),X) = X >= X = activate(X) isNat(n__0()) = 0 >= 0 = tt() isNat(n__plus(V1,V2)) = V1 + 7V2 + 0 >= V1 + 1V2 = U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1), activate(V2)) isNat(n__s(V1)) = V1 + 0 >= 5V1 + 8 = U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) = 0 >= 0 = tt() isNatKind(n__plus(V1,V2)) = V1 + 7V2 >= V2 = and(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind(n__s(V1)) = V1 >= V1 = isNatKind(activate(V1)) plus(N,0()) = N + 7 >= N + 0 = U31(and(isNat(N),n__isNatKind(N)),N) plus(N,s(M)) = 7M + N >= 7M + N = U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) 0() = 0 >= 0 = n__0() plus(X1,X2) = X1 + 7X2 >= X1 + 7X2 = n__plus(X1,X2) isNatKind(X) = X >= X = n__isNatKind(X) s(X) = X >= X = n__s(X) and(X1,X2) = X2 >= X2 = n__and(X1,X2) activate(n__0()) = 0 >= 0 = 0() activate(n__plus(X1,X2)) = X1 + 7X2 >= X1 + 7X2 = plus(X1,X2) activate(n__isNatKind(X)) = X >= X = isNatKind(X) activate(n__s(X)) = X >= X = s(X) activate(n__and(X1,X2)) = X2 >= X2 = and(X1,X2) activate(X) = X >= X = X problem: DPs: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) activate#(n__plus(X1,X2)) -> plus#(X1,X2) plus#(N,0()) -> isNat#(N) activate#(n__isNatKind(X)) -> isNatKind#(X) isNatKind#(n__plus(V1,V2)) -> activate#(V1) activate#(n__and(X1,X2)) -> and#(X1,X2) and#(tt(),X) -> activate#(X) isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNatKind#(n__s(V1)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> isNatKind#(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) 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)) plus#(N,0()) -> and#(isNat(N),n__isNatKind(N)) plus#(N,0()) -> U31#(and(isNat(N),n__isNatKind(N)),N) U31#(tt(),N) -> activate#(N) plus#(N,s(M)) -> isNat#(N) plus#(N,s(M)) -> isNat#(M) plus#(N,s(M)) -> and#(isNat(M),n__isNatKind(M)) plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))) plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) U41#(tt(),M,N) -> activate#(M) U41#(tt(),M,N) -> activate#(N) U41#(tt(),M,N) -> plus#(activate(N),activate(M)) TRS: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2)) U12(tt(),V2) -> U13(isNat(activate(V2))) U13(tt()) -> tt() U21(tt(),V1) -> U22(isNat(activate(V1))) U22(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> s(plus(activate(N),activate(M))) and(tt(),X) -> activate(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind(n__s(V1)) -> isNatKind(activate(V1)) plus(N,0()) -> U31(and(isNat(N),n__isNatKind(N)),N) plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) isNatKind(X) -> n__isNatKind(X) s(X) -> n__s(X) and(X1,X2) -> n__and(X1,X2) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(X) activate(n__and(X1,X2)) -> and(X1,X2) activate(X) -> X Restore Modifier: DPs: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) activate#(n__plus(X1,X2)) -> plus#(X1,X2) plus#(N,0()) -> isNat#(N) activate#(n__isNatKind(X)) -> isNatKind#(X) isNatKind#(n__plus(V1,V2)) -> activate#(V1) activate#(n__and(X1,X2)) -> and#(X1,X2) and#(tt(),X) -> activate#(X) isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNatKind#(n__s(V1)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> isNatKind#(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) 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)) plus#(N,0()) -> and#(isNat(N),n__isNatKind(N)) plus#(N,0()) -> U31#(and(isNat(N),n__isNatKind(N)),N) U31#(tt(),N) -> activate#(N) plus#(N,s(M)) -> isNat#(N) plus#(N,s(M)) -> isNat#(M) plus#(N,s(M)) -> and#(isNat(M),n__isNatKind(M)) plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))) plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) U41#(tt(),M,N) -> activate#(M) U41#(tt(),M,N) -> activate#(N) U41#(tt(),M,N) -> plus#(activate(N),activate(M)) TRS: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2)) U12(tt(),V2) -> U13(isNat(activate(V2))) U13(tt()) -> tt() U21(tt(),V1) -> U22(isNat(activate(V1))) U22(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> s(plus(activate(N),activate(M))) and(tt(),X) -> activate(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind(n__s(V1)) -> isNatKind(activate(V1)) plus(N,0()) -> U31(and(isNat(N),n__isNatKind(N)),N) plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) isNatKind(X) -> n__isNatKind(X) s(X) -> n__s(X) and(X1,X2) -> n__and(X1,X2) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(X) activate(n__and(X1,X2)) -> and(X1,X2) activate(X) -> X SCC Processor: #sccs: 1 #rules: 30 #arcs: 154/961 DPs: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) -> activate#(V1) activate#(n__plus(X1,X2)) -> plus#(X1,X2) plus#(N,0()) -> isNat#(N) isNat#(n__plus(V1,V2)) -> activate#(V1) activate#(n__isNatKind(X)) -> isNatKind#(X) isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)) isNatKind#(n__s(V1)) -> activate#(V1) activate#(n__and(X1,X2)) -> and#(X1,X2) and#(tt(),X) -> activate#(X) isNat#(n__plus(V1,V2)) -> isNatKind#(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) 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)) plus#(N,0()) -> and#(isNat(N),n__isNatKind(N)) plus#(N,0()) -> U31#(and(isNat(N),n__isNatKind(N)),N) U31#(tt(),N) -> activate#(N) plus#(N,s(M)) -> isNat#(N) plus#(N,s(M)) -> isNat#(M) plus#(N,s(M)) -> and#(isNat(M),n__isNatKind(M)) plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))) plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) U41#(tt(),M,N) -> activate#(M) U41#(tt(),M,N) -> activate#(N) U41#(tt(),M,N) -> plus#(activate(N),activate(M)) TRS: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2)) U12(tt(),V2) -> U13(isNat(activate(V2))) U13(tt()) -> tt() U21(tt(),V1) -> U22(isNat(activate(V1))) U22(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> s(plus(activate(N),activate(M))) and(tt(),X) -> activate(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind(n__s(V1)) -> isNatKind(activate(V1)) plus(N,0()) -> U31(and(isNat(N),n__isNatKind(N)),N) plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) isNatKind(X) -> n__isNatKind(X) s(X) -> n__s(X) and(X1,X2) -> n__and(X1,X2) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(X) activate(n__and(X1,X2)) -> and(X1,X2) activate(X) -> X Arctic Interpretation Processor: dimension: 1 usable rules: U31(tt(),N) -> activate(N) U41(tt(),M,N) -> s(plus(activate(N),activate(M))) and(tt(),X) -> activate(X) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind(n__s(V1)) -> isNatKind(activate(V1)) plus(N,0()) -> U31(and(isNat(N),n__isNatKind(N)),N) plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) isNatKind(X) -> n__isNatKind(X) s(X) -> n__s(X) and(X1,X2) -> n__and(X1,X2) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(X) activate(n__and(X1,X2)) -> and(X1,X2) activate(X) -> X interpretation: [isNatKind#](x0) = 3x0, [and#](x0, x1) = 3x1, [plus#](x0, x1) = 4x0 + 4x1 + 0, [U41#](x0, x1, x2) = 4x1 + 4x2 + 1, [U31#](x0, x1) = 2x0 + 4x1, [U21#](x0, x1) = 2x0 + 4x1 + 4, [isNat#](x0) = 4x0 + 1, [activate#](x0) = 3x0, [U11#](x0, x1, x2) = 5x1 + 2, [n__and](x0, x1) = x1, [0] = 2, [n__s](x0) = x0 + 0, [n__isNatKind](x0) = x0, [isNatKind](x0) = x0, [n__plus](x0, x1) = 1x0 + 2x1 + 0, [n__0] = 2, [and](x0, x1) = x1, [s](x0) = x0 + 0, [plus](x0, x1) = 1x0 + 2x1 + 0, [U41](x0, x1, x2) = x0 + 2x1 + 1x2 + 0, [U31](x0, x1) = 1x1, [U22](x0) = 0, [U21](x0, x1) = x0 + 4x1, [U13](x0) = 6, [U12](x0, x1) = 2x0 + 2x1 + 4, [isNat](x0) = 2x0 + 3, [activate](x0) = x0, [U11](x0, x1, x2) = x1 + x2, [tt] = 0 orientation: isNatKind#(n__s(V1)) = 3V1 + 3 >= 3V1 = isNatKind#(activate(V1)) isNatKind#(n__plus(V1,V2)) = 4V1 + 5V2 + 3 >= 3V1 = activate#(V1) activate#(n__plus(X1,X2)) = 4X1 + 5X2 + 3 >= 4X1 + 4X2 + 0 = plus#(X1,X2) plus#(N,0()) = 4N + 6 >= 4N + 1 = isNat#(N) isNat#(n__plus(V1,V2)) = 5V1 + 6V2 + 4 >= 3V1 = activate#(V1) activate#(n__isNatKind(X)) = 3X >= 3X = isNatKind#(X) isNatKind#(n__plus(V1,V2)) = 4V1 + 5V2 + 3 >= 3V1 = isNatKind#(activate(V1)) isNatKind#(n__s(V1)) = 3V1 + 3 >= 3V1 = activate#(V1) activate#(n__and(X1,X2)) = 3X2 >= 3X2 = and#(X1,X2) and#(tt(),X) = 3X >= 3X = activate#(X) isNat#(n__plus(V1,V2)) = 5V1 + 6V2 + 4 >= 3V1 = isNatKind#(activate(V1)) isNat#(n__plus(V1,V2)) = 5V1 + 6V2 + 4 >= 5V1 + 2 = U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1), activate(V2)) U11#(tt(),V1,V2) = 5V1 + 2 >= 3V1 = activate#(V1) U11#(tt(),V1,V2) = 5V1 + 2 >= 4V1 + 1 = isNat#(activate(V1)) isNat#(n__s(V1)) = 4V1 + 4 >= 3V1 = activate#(V1) isNat#(n__s(V1)) = 4V1 + 4 >= 3V1 = isNatKind#(activate(V1)) isNat#(n__s(V1)) = 4V1 + 4 >= 4V1 + 4 = U21#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) = 4V1 + 4 >= 3V1 = activate#(V1) U21#(tt(),V1) = 4V1 + 4 >= 4V1 + 1 = isNat#(activate(V1)) plus#(N,0()) = 4N + 6 >= 3N = and#(isNat(N),n__isNatKind(N)) plus#(N,0()) = 4N + 6 >= 4N = U31#(and(isNat(N),n__isNatKind(N)),N) U31#(tt(),N) = 4N + 2 >= 3N = activate#(N) plus#(N,s(M)) = 4M + 4N + 4 >= 4N + 1 = isNat#(N) plus#(N,s(M)) = 4M + 4N + 4 >= 4M + 1 = isNat#(M) plus#(N,s(M)) = 4M + 4N + 4 >= 3M = and#(isNat(M),n__isNatKind(M)) plus#(N,s(M)) = 4M + 4N + 4 >= 3N = and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))) plus#(N,s(M)) = 4M + 4N + 4 >= 4M + 4N + 1 = U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) U41#(tt(),M,N) = 4M + 4N + 1 >= 3M = activate#(M) U41#(tt(),M,N) = 4M + 4N + 1 >= 3N = activate#(N) U41#(tt(),M,N) = 4M + 4N + 1 >= 4M + 4N + 0 = plus#(activate(N),activate(M)) U11(tt(),V1,V2) = V1 + V2 >= 4V1 + 2V2 + 5 = U12(isNat(activate(V1)),activate(V2)) U12(tt(),V2) = 2V2 + 4 >= 6 = U13(isNat(activate(V2))) U13(tt()) = 6 >= 0 = tt() U21(tt(),V1) = 4V1 + 0 >= 0 = U22(isNat(activate(V1))) U22(tt()) = 0 >= 0 = tt() U31(tt(),N) = 1N >= N = activate(N) U41(tt(),M,N) = 2M + 1N + 0 >= 2M + 1N + 0 = s(plus(activate(N),activate(M))) and(tt(),X) = X >= X = activate(X) isNat(n__0()) = 4 >= 0 = tt() isNat(n__plus(V1,V2)) = 3V1 + 4V2 + 3 >= V1 + V2 = U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1), activate(V2)) isNat(n__s(V1)) = 2V1 + 3 >= 4V1 = U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) = 2 >= 0 = tt() isNatKind(n__plus(V1,V2)) = 1V1 + 2V2 + 0 >= V2 = and(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind(n__s(V1)) = V1 + 0 >= V1 = isNatKind(activate(V1)) plus(N,0()) = 1N + 4 >= 1N = U31(and(isNat(N),n__isNatKind(N)),N) plus(N,s(M)) = 2M + 1N + 2 >= 2M + 1N + 0 = U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) 0() = 2 >= 2 = n__0() plus(X1,X2) = 1X1 + 2X2 + 0 >= 1X1 + 2X2 + 0 = n__plus(X1,X2) isNatKind(X) = X >= X = n__isNatKind(X) s(X) = X + 0 >= X + 0 = n__s(X) and(X1,X2) = X2 >= X2 = n__and(X1,X2) activate(n__0()) = 2 >= 2 = 0() activate(n__plus(X1,X2)) = 1X1 + 2X2 + 0 >= 1X1 + 2X2 + 0 = plus(X1,X2) activate(n__isNatKind(X)) = X >= X = isNatKind(X) activate(n__s(X)) = X + 0 >= X + 0 = s(X) activate(n__and(X1,X2)) = X2 >= X2 = and(X1,X2) activate(X) = X >= X = X problem: DPs: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) activate#(n__plus(X1,X2)) -> plus#(X1,X2) plus#(N,0()) -> isNat#(N) activate#(n__isNatKind(X)) -> isNatKind#(X) isNatKind#(n__s(V1)) -> activate#(V1) activate#(n__and(X1,X2)) -> and#(X1,X2) and#(tt(),X) -> activate#(X) 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)) U21#(tt(),V1) -> isNat#(activate(V1)) plus#(N,0()) -> U31#(and(isNat(N),n__isNatKind(N)),N) plus#(N,s(M)) -> isNat#(N) plus#(N,s(M)) -> isNat#(M) plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) U41#(tt(),M,N) -> plus#(activate(N),activate(M)) TRS: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2)) U12(tt(),V2) -> U13(isNat(activate(V2))) U13(tt()) -> tt() U21(tt(),V1) -> U22(isNat(activate(V1))) U22(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> s(plus(activate(N),activate(M))) and(tt(),X) -> activate(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind(n__s(V1)) -> isNatKind(activate(V1)) plus(N,0()) -> U31(and(isNat(N),n__isNatKind(N)),N) plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) isNatKind(X) -> n__isNatKind(X) s(X) -> n__s(X) and(X1,X2) -> n__and(X1,X2) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(X) activate(n__and(X1,X2)) -> and(X1,X2) activate(X) -> X Restore Modifier: DPs: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) activate#(n__plus(X1,X2)) -> plus#(X1,X2) plus#(N,0()) -> isNat#(N) activate#(n__isNatKind(X)) -> isNatKind#(X) isNatKind#(n__s(V1)) -> activate#(V1) activate#(n__and(X1,X2)) -> and#(X1,X2) and#(tt(),X) -> activate#(X) 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)) U21#(tt(),V1) -> isNat#(activate(V1)) plus#(N,0()) -> U31#(and(isNat(N),n__isNatKind(N)),N) plus#(N,s(M)) -> isNat#(N) plus#(N,s(M)) -> isNat#(M) plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) U41#(tt(),M,N) -> plus#(activate(N),activate(M)) TRS: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2)) U12(tt(),V2) -> U13(isNat(activate(V2))) U13(tt()) -> tt() U21(tt(),V1) -> U22(isNat(activate(V1))) U22(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> s(plus(activate(N),activate(M))) and(tt(),X) -> activate(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind(n__s(V1)) -> isNatKind(activate(V1)) plus(N,0()) -> U31(and(isNat(N),n__isNatKind(N)),N) plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) isNatKind(X) -> n__isNatKind(X) s(X) -> n__s(X) and(X1,X2) -> n__and(X1,X2) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(X) activate(n__and(X1,X2)) -> and(X1,X2) activate(X) -> X SCC Processor: #sccs: 3 #rules: 9 #arcs: 108/225 DPs: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) isNatKind#(n__s(V1)) -> activate#(V1) activate#(n__isNatKind(X)) -> isNatKind#(X) activate#(n__and(X1,X2)) -> and#(X1,X2) and#(tt(),X) -> activate#(X) TRS: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2)) U12(tt(),V2) -> U13(isNat(activate(V2))) U13(tt()) -> tt() U21(tt(),V1) -> U22(isNat(activate(V1))) U22(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> s(plus(activate(N),activate(M))) and(tt(),X) -> activate(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind(n__s(V1)) -> isNatKind(activate(V1)) plus(N,0()) -> U31(and(isNat(N),n__isNatKind(N)),N) plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) isNatKind(X) -> n__isNatKind(X) s(X) -> n__s(X) and(X1,X2) -> n__and(X1,X2) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(X) activate(n__and(X1,X2)) -> and(X1,X2) activate(X) -> X Arctic Interpretation Processor: dimension: 1 usable rules: U31(tt(),N) -> activate(N) U41(tt(),M,N) -> s(plus(activate(N),activate(M))) and(tt(),X) -> activate(X) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind(n__s(V1)) -> isNatKind(activate(V1)) plus(N,0()) -> U31(and(isNat(N),n__isNatKind(N)),N) plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) isNatKind(X) -> n__isNatKind(X) s(X) -> n__s(X) and(X1,X2) -> n__and(X1,X2) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(X) activate(n__and(X1,X2)) -> and(X1,X2) activate(X) -> X interpretation: [isNatKind#](x0) = x0, [and#](x0, x1) = 1x1, [activate#](x0) = x0, [n__and](x0, x1) = 1x1, [0] = 1, [n__s](x0) = x0 + 0, [n__isNatKind](x0) = 3x0, [isNatKind](x0) = 3x0, [n__plus](x0, x1) = 4x0 + 4x1 + 4, [n__0] = 1, [and](x0, x1) = 1x1 + 0, [s](x0) = x0 + 0, [plus](x0, x1) = 4x0 + 4x1 + 4, [U41](x0, x1, x2) = 4x1 + 4x2 + 4, [U31](x0, x1) = 1x1 + 0, [U22](x0) = 7x0 + 0, [U21](x0, x1) = x0 + 3x1 + 4, [U13](x0) = 1x0 + 1, [U12](x0, x1) = 2x1, [isNat](x0) = x0 + 0, [activate](x0) = x0 + 0, [U11](x0, x1, x2) = x0 + 4x1 + 4x2 + 0, [tt] = 0 orientation: isNatKind#(n__s(V1)) = V1 + 0 >= V1 + 0 = isNatKind#(activate(V1)) isNatKind#(n__s(V1)) = V1 + 0 >= V1 = activate#(V1) activate#(n__isNatKind(X)) = 3X >= X = isNatKind#(X) activate#(n__and(X1,X2)) = 1X2 >= 1X2 = and#(X1,X2) and#(tt(),X) = 1X >= X = activate#(X) U11(tt(),V1,V2) = 4V1 + 4V2 + 0 >= 2V2 + 2 = U12(isNat(activate(V1)),activate(V2)) U12(tt(),V2) = 2V2 >= 1V2 + 1 = U13(isNat(activate(V2))) U13(tt()) = 1 >= 0 = tt() U21(tt(),V1) = 3V1 + 4 >= 7V1 + 7 = U22(isNat(activate(V1))) U22(tt()) = 7 >= 0 = tt() U31(tt(),N) = 1N + 0 >= N + 0 = activate(N) U41(tt(),M,N) = 4M + 4N + 4 >= 4M + 4N + 4 = s(plus(activate(N),activate(M))) and(tt(),X) = 1X + 0 >= X + 0 = activate(X) isNat(n__0()) = 1 >= 0 = tt() isNat(n__plus(V1,V2)) = 4V1 + 4V2 + 4 >= 4V1 + 4V2 + 4 = U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1), activate(V2)) isNat(n__s(V1)) = V1 + 0 >= 3V1 + 4 = U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) = 4 >= 0 = tt() isNatKind(n__plus(V1,V2)) = 7V1 + 7V2 + 7 >= 4V2 + 4 = and(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind(n__s(V1)) = 3V1 + 3 >= 3V1 + 3 = isNatKind(activate(V1)) plus(N,0()) = 4N + 5 >= 1N + 0 = U31(and(isNat(N),n__isNatKind(N)),N) plus(N,s(M)) = 4M + 4N + 4 >= 4M + 4N + 4 = U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) 0() = 1 >= 1 = n__0() plus(X1,X2) = 4X1 + 4X2 + 4 >= 4X1 + 4X2 + 4 = n__plus(X1,X2) isNatKind(X) = 3X >= 3X = n__isNatKind(X) s(X) = X + 0 >= X + 0 = n__s(X) and(X1,X2) = 1X2 + 0 >= 1X2 = n__and(X1,X2) activate(n__0()) = 1 >= 1 = 0() activate(n__plus(X1,X2)) = 4X1 + 4X2 + 4 >= 4X1 + 4X2 + 4 = plus(X1,X2) activate(n__isNatKind(X)) = 3X + 0 >= 3X = isNatKind(X) activate(n__s(X)) = X + 0 >= X + 0 = s(X) activate(n__and(X1,X2)) = 1X2 + 0 >= 1X2 + 0 = and(X1,X2) activate(X) = X + 0 >= X = X problem: DPs: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) isNatKind#(n__s(V1)) -> activate#(V1) activate#(n__and(X1,X2)) -> and#(X1,X2) TRS: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2)) U12(tt(),V2) -> U13(isNat(activate(V2))) U13(tt()) -> tt() U21(tt(),V1) -> U22(isNat(activate(V1))) U22(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> s(plus(activate(N),activate(M))) and(tt(),X) -> activate(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind(n__s(V1)) -> isNatKind(activate(V1)) plus(N,0()) -> U31(and(isNat(N),n__isNatKind(N)),N) plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) isNatKind(X) -> n__isNatKind(X) s(X) -> n__s(X) and(X1,X2) -> n__and(X1,X2) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(X) activate(n__and(X1,X2)) -> and(X1,X2) activate(X) -> X Restore Modifier: DPs: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)) isNatKind#(n__s(V1)) -> activate#(V1) activate#(n__and(X1,X2)) -> and#(X1,X2) TRS: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2)) U12(tt(),V2) -> U13(isNat(activate(V2))) U13(tt()) -> tt() U21(tt(),V1) -> U22(isNat(activate(V1))) U22(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> s(plus(activate(N),activate(M))) and(tt(),X) -> activate(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind(n__s(V1)) -> isNatKind(activate(V1)) plus(N,0()) -> U31(and(isNat(N),n__isNatKind(N)),N) plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) isNatKind(X) -> n__isNatKind(X) s(X) -> n__s(X) and(X1,X2) -> n__and(X1,X2) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(X) activate(n__and(X1,X2)) -> and(X1,X2) activate(X) -> X SCC Processor: #sccs: 1 #rules: 1 #arcs: 9/9 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(isNat(N),n__isNatKind(N))),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) isNatKind(X) -> n__isNatKind(X) s(X) -> n__s(X) and(X1,X2) -> n__and(X1,X2) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(X) activate(n__and(X1,X2)) -> and(X1,X2) activate(X) -> X 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(isNat(N),n__isNatKind(N))),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) isNatKind(X) -> n__isNatKind(X) s(X) -> n__s(X) and(X1,X2) -> n__and(X1,X2) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(X) activate(n__and(X1,X2)) -> and(X1,X2) activate(X) -> X interpretation: [isNatKind#](x0) = 4x0, [n__and](x0, x1) = 4x1, [0] = 0, [n__s](x0) = x0 + 1, [n__isNatKind](x0) = 0, [isNatKind](x0) = 0, [n__plus](x0, x1) = x0 + 5x1 + 3, [n__0] = 0, [and](x0, x1) = 4x1, [s](x0) = x0 + 1, [plus](x0, x1) = x0 + 5x1 + 3, [U41](x0, x1, x2) = 5x1 + x2 + 5, [U31](x0, x1) = x1, [U22](x0) = 1, [U21](x0, x1) = 2, [U13](x0) = 0, [U12](x0, x1) = 0, [isNat](x0) = 2, [activate](x0) = x0, [U11](x0, x1, x2) = 0, [tt] = 0 orientation: isNatKind#(n__s(V1)) = 4V1 + 4 >= 4V1 = 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) = 2 >= 1 = U22(isNat(activate(V1))) U22(tt()) = 1 >= 0 = tt() U31(tt(),N) = N >= N = activate(N) U41(tt(),M,N) = 5M + N + 5 >= 5M + N + 4 = s(plus(activate(N),activate(M))) and(tt(),X) = 4X >= X = activate(X) isNat(n__0()) = 2 >= 0 = tt() isNat(n__plus(V1,V2)) = 2 >= 0 = U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1), activate(V2)) isNat(n__s(V1)) = 2 >= 2 = 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 + 3 >= N = U31(and(isNat(N),n__isNatKind(N)),N) plus(N,s(M)) = 5M + N + 8 >= 5M + N + 5 = U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) 0() = 0 >= 0 = n__0() plus(X1,X2) = X1 + 5X2 + 3 >= X1 + 5X2 + 3 = n__plus(X1,X2) isNatKind(X) = 0 >= 0 = n__isNatKind(X) s(X) = X + 1 >= X + 1 = n__s(X) and(X1,X2) = 4X2 >= 4X2 = n__and(X1,X2) activate(n__0()) = 0 >= 0 = 0() activate(n__plus(X1,X2)) = X1 + 5X2 + 3 >= X1 + 5X2 + 3 = plus(X1,X2) activate(n__isNatKind(X)) = 0 >= 0 = isNatKind(X) activate(n__s(X)) = X + 1 >= X + 1 = s(X) activate(n__and(X1,X2)) = 4X2 >= 4X2 = and(X1,X2) 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(isNat(N),n__isNatKind(N))),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) isNatKind(X) -> n__isNatKind(X) s(X) -> n__s(X) and(X1,X2) -> n__and(X1,X2) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(X) activate(n__and(X1,X2)) -> and(X1,X2) activate(X) -> X Qed DPs: plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) U41#(tt(),M,N) -> plus#(activate(N),activate(M)) TRS: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2)) U12(tt(),V2) -> U13(isNat(activate(V2))) U13(tt()) -> tt() U21(tt(),V1) -> U22(isNat(activate(V1))) U22(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> s(plus(activate(N),activate(M))) and(tt(),X) -> activate(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind(n__s(V1)) -> isNatKind(activate(V1)) plus(N,0()) -> U31(and(isNat(N),n__isNatKind(N)),N) plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) isNatKind(X) -> n__isNatKind(X) s(X) -> n__s(X) and(X1,X2) -> n__and(X1,X2) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(X) activate(n__and(X1,X2)) -> and(X1,X2) activate(X) -> X Matrix Interpretation Processor: dim=1 usable rules: U31(tt(),N) -> activate(N) U41(tt(),M,N) -> s(plus(activate(N),activate(M))) and(tt(),X) -> activate(X) isNatKind(n__0()) -> tt() isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind(n__s(V1)) -> isNatKind(activate(V1)) plus(N,0()) -> U31(and(isNat(N),n__isNatKind(N)),N) plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) isNatKind(X) -> n__isNatKind(X) s(X) -> n__s(X) and(X1,X2) -> n__and(X1,X2) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(X) activate(n__and(X1,X2)) -> and(X1,X2) activate(X) -> X interpretation: [plus#](x0, x1) = x1, [U41#](x0, x1, x2) = x1 + 1/2, [n__and](x0, x1) = 3/2x1 + 2, [0] = 0, [n__s](x0) = x0 + 1/2, [n__isNatKind](x0) = 2x0 + 2, [isNatKind](x0) = 2x0 + 2, [n__plus](x0, x1) = 3/2x0 + 2x1 + 3/2, [n__0] = 0, [and](x0, x1) = 3/2x1 + 2, [s](x0) = x0 + 1/2, [plus](x0, x1) = 3/2x0 + 2x1 + 3/2, [U41](x0, x1, x2) = 2x1 + 3/2x2 + 5/2, [U31](x0, x1) = x1, [U22](x0) = 2x0, [U21](x0, x1) = 2x1 + 1, [U13](x0) = 1/2x0 + 3, [U12](x0, x1) = x1 + 1, [isNat](x0) = 2, [activate](x0) = x0, [U11](x0, x1, x2) = 2x0 + 2x1 + 5/2x2, [tt] = 2 orientation: plus#(N,s(M)) = M + 1/2 >= M + 1/2 = U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) U41#(tt(),M,N) = M + 1/2 >= M = plus#(activate(N),activate(M)) U11(tt(),V1,V2) = 2V1 + 5/2V2 + 4 >= V2 + 1 = U12(isNat(activate(V1)),activate(V2)) U12(tt(),V2) = V2 + 1 >= 4 = U13(isNat(activate(V2))) U13(tt()) = 4 >= 2 = tt() U21(tt(),V1) = 2V1 + 1 >= 4 = U22(isNat(activate(V1))) U22(tt()) = 4 >= 2 = tt() U31(tt(),N) = N >= N = activate(N) U41(tt(),M,N) = 2M + 3/2N + 5/2 >= 2M + 3/2N + 2 = s(plus(activate(N),activate(M))) and(tt(),X) = 3/2X + 2 >= X = activate(X) isNat(n__0()) = 2 >= 2 = tt() isNat(n__plus(V1,V2)) = 2 >= 2V1 + 17/2V2 + 10 = U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1), activate(V2)) isNat(n__s(V1)) = 2 >= 2V1 + 1 = U21(isNatKind(activate(V1)),activate(V1)) isNatKind(n__0()) = 2 >= 2 = tt() isNatKind(n__plus(V1,V2)) = 3V1 + 4V2 + 5 >= 3V2 + 5 = and(isNatKind(activate(V1)),n__isNatKind(activate(V2))) isNatKind(n__s(V1)) = 2V1 + 3 >= 2V1 + 2 = isNatKind(activate(V1)) plus(N,0()) = 3/2N + 3/2 >= N = U31(and(isNat(N),n__isNatKind(N)),N) plus(N,s(M)) = 2M + 3/2N + 5/2 >= 2M + 3/2N + 5/2 = U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) 0() = 0 >= 0 = n__0() plus(X1,X2) = 3/2X1 + 2X2 + 3/2 >= 3/2X1 + 2X2 + 3/2 = n__plus(X1,X2) isNatKind(X) = 2X + 2 >= 2X + 2 = n__isNatKind(X) s(X) = X + 1/2 >= X + 1/2 = n__s(X) and(X1,X2) = 3/2X2 + 2 >= 3/2X2 + 2 = n__and(X1,X2) activate(n__0()) = 0 >= 0 = 0() activate(n__plus(X1,X2)) = 3/2X1 + 2X2 + 3/2 >= 3/2X1 + 2X2 + 3/2 = plus(X1,X2) activate(n__isNatKind(X)) = 2X + 2 >= 2X + 2 = isNatKind(X) activate(n__s(X)) = X + 1/2 >= X + 1/2 = s(X) activate(n__and(X1,X2)) = 3/2X2 + 2 >= 3/2X2 + 2 = and(X1,X2) activate(X) = X >= X = X problem: DPs: plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(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(isNat(N),n__isNatKind(N))),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) isNatKind(X) -> n__isNatKind(X) s(X) -> n__s(X) and(X1,X2) -> n__and(X1,X2) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(X) activate(n__and(X1,X2)) -> and(X1,X2) activate(X) -> X Restore Modifier: DPs: plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(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(isNat(N),n__isNatKind(N))),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) isNatKind(X) -> n__isNatKind(X) s(X) -> n__s(X) and(X1,X2) -> n__and(X1,X2) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(X) activate(n__and(X1,X2)) -> and(X1,X2) activate(X) -> X SCC Processor: #sccs: 0 #rules: 0 #arcs: 2/1 DPs: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)) 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(isNat(N),n__isNatKind(N))),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) isNatKind(X) -> n__isNatKind(X) s(X) -> n__s(X) and(X1,X2) -> n__and(X1,X2) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(X) activate(n__and(X1,X2)) -> and(X1,X2) activate(X) -> X 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(isNat(N),n__isNatKind(N))),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) isNatKind(X) -> n__isNatKind(X) s(X) -> n__s(X) and(X1,X2) -> n__and(X1,X2) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(X) activate(n__and(X1,X2)) -> and(X1,X2) activate(X) -> X interpretation: [U21#](x0, x1) = x1, [isNat#](x0) = x0, [n__and](x0, x1) = 2x0 + x1, [0] = 0, [n__s](x0) = x0 + 1, [n__isNatKind](x0) = 0, [isNatKind](x0) = 0, [n__plus](x0, x1) = x0 + x1, [n__0] = 0, [and](x0, x1) = 2x0 + x1, [s](x0) = x0 + 1, [plus](x0, x1) = x0 + x1, [U41](x0, x1, x2) = x1 + x2 + 1, [U31](x0, x1) = x1, [U22](x0) = x0, [U21](x0, x1) = x1 + 4, [U13](x0) = 1, [U12](x0, x1) = 1, [isNat](x0) = x0 + 3, [activate](x0) = x0, [U11](x0, x1, x2) = x1 + 1, [tt] = 0 orientation: isNat#(n__s(V1)) = V1 + 1 >= V1 = U21#(isNatKind(activate(V1)),activate(V1)) U21#(tt(),V1) = V1 >= V1 = isNat#(activate(V1)) U11(tt(),V1,V2) = V1 + 1 >= 1 = U12(isNat(activate(V1)),activate(V2)) U12(tt(),V2) = 1 >= 1 = U13(isNat(activate(V2))) U13(tt()) = 1 >= 0 = tt() U21(tt(),V1) = V1 + 4 >= V1 + 3 = 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) = X >= X = activate(X) isNat(n__0()) = 3 >= 0 = tt() isNat(n__plus(V1,V2)) = V1 + V2 + 3 >= V1 + 1 = U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1), activate(V2)) isNat(n__s(V1)) = V1 + 4 >= V1 + 4 = 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(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) = 2X1 + X2 >= 2X1 + X2 = n__and(X1,X2) activate(n__0()) = 0 >= 0 = 0() activate(n__plus(X1,X2)) = X1 + X2 >= X1 + X2 = plus(X1,X2) activate(n__isNatKind(X)) = 0 >= 0 = isNatKind(X) activate(n__s(X)) = X + 1 >= X + 1 = s(X) activate(n__and(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = and(X1,X2) 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(isNat(N),n__isNatKind(N))),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) isNatKind(X) -> n__isNatKind(X) s(X) -> n__s(X) and(X1,X2) -> n__and(X1,X2) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(X) activate(n__and(X1,X2)) -> and(X1,X2) activate(X) -> X Restore Modifier: DPs: 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(isNat(N),n__isNatKind(N))),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) isNatKind(X) -> n__isNatKind(X) s(X) -> n__s(X) and(X1,X2) -> n__and(X1,X2) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(X) activate(n__and(X1,X2)) -> and(X1,X2) activate(X) -> X SCC Processor: #sccs: 0 #rules: 0 #arcs: 2/1