MAYBE Problem: U11(tt(),N) -> activate(N) U21(tt(),M,N) -> s(plus(activate(N),activate(M))) and(tt(),X) -> activate(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) plus(N,0()) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) isNat(X) -> n__isNat(X) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__isNat(X)) -> isNat(X) activate(n__s(X)) -> s(activate(X)) activate(X) -> X Proof: DP Processor: DPs: U11#(tt(),N) -> activate#(N) U21#(tt(),M,N) -> activate#(M) U21#(tt(),M,N) -> activate#(N) U21#(tt(),M,N) -> plus#(activate(N),activate(M)) U21#(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)) -> isNat#(activate(V1)) isNat#(n__plus(V1,V2)) -> and#(isNat(activate(V1)),n__isNat(activate(V2))) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNat#(activate(V1)) plus#(N,0()) -> isNat#(N) plus#(N,0()) -> U11#(isNat(N),N) plus#(N,s(M)) -> isNat#(M) plus#(N,s(M)) -> and#(isNat(M),n__isNat(N)) plus#(N,s(M)) -> U21#(and(isNat(M),n__isNat(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__isNat(X)) -> isNat#(X) activate#(n__s(X)) -> activate#(X) activate#(n__s(X)) -> s#(activate(X)) TRS: U11(tt(),N) -> activate(N) U21(tt(),M,N) -> s(plus(activate(N),activate(M))) and(tt(),X) -> activate(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) plus(N,0()) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) isNat(X) -> n__isNat(X) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__isNat(X)) -> isNat(X) activate(n__s(X)) -> s(activate(X)) activate(X) -> X CDG Processor: DPs: U11#(tt(),N) -> activate#(N) U21#(tt(),M,N) -> activate#(M) U21#(tt(),M,N) -> activate#(N) U21#(tt(),M,N) -> plus#(activate(N),activate(M)) U21#(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)) -> isNat#(activate(V1)) isNat#(n__plus(V1,V2)) -> and#(isNat(activate(V1)),n__isNat(activate(V2))) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNat#(activate(V1)) plus#(N,0()) -> isNat#(N) plus#(N,0()) -> U11#(isNat(N),N) plus#(N,s(M)) -> isNat#(M) plus#(N,s(M)) -> and#(isNat(M),n__isNat(N)) plus#(N,s(M)) -> U21#(and(isNat(M),n__isNat(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__isNat(X)) -> isNat#(X) activate#(n__s(X)) -> activate#(X) activate#(n__s(X)) -> s#(activate(X)) TRS: U11(tt(),N) -> activate(N) U21(tt(),M,N) -> s(plus(activate(N),activate(M))) and(tt(),X) -> activate(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) plus(N,0()) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) isNat(X) -> n__isNat(X) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__isNat(X)) -> isNat(X) activate(n__s(X)) -> s(activate(X)) activate(X) -> X graph: isNat#(n__s(V1)) -> isNat#(activate(V1)) -> isNat#(n__plus(V1,V2)) -> activate#(V2) isNat#(n__s(V1)) -> isNat#(activate(V1)) -> isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__s(V1)) -> isNat#(activate(V1)) -> isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)) isNat#(n__s(V1)) -> isNat#(activate(V1)) -> isNat#(n__plus(V1,V2)) -> and#(isNat(activate(V1)),n__isNat(activate(V2))) isNat#(n__s(V1)) -> isNat#(activate(V1)) -> isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNat#(activate(V1)) -> isNat#(n__s(V1)) -> isNat#(activate(V1)) isNat#(n__s(V1)) -> activate#(V1) -> activate#(n__0()) -> 0#() isNat#(n__s(V1)) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> 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)) -> plus#(activate(X1),activate(X2)) isNat#(n__s(V1)) -> activate#(V1) -> activate#(n__isNat(X)) -> isNat#(X) isNat#(n__s(V1)) -> activate#(V1) -> activate#(n__s(X)) -> activate#(X) isNat#(n__s(V1)) -> activate#(V1) -> activate#(n__s(X)) -> s#(activate(X)) isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)) -> isNat#(n__plus(V1,V2)) -> activate#(V2) isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)) -> isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)) -> isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)) isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)) -> isNat#(n__plus(V1,V2)) -> and#(isNat(activate(V1)),n__isNat(activate(V2))) isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)) -> isNat#(n__s(V1)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)) -> isNat#(n__s(V1)) -> isNat#(activate(V1)) isNat#(n__plus(V1,V2)) -> and#(isNat(activate(V1)),n__isNat(activate(V2))) -> and#(tt(),X) -> activate#(X) isNat#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__0()) -> 0#() isNat#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> 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)) -> plus#(activate(X1),activate(X2)) isNat#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__isNat(X)) -> isNat#(X) isNat#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__s(X)) -> activate#(X) isNat#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__s(X)) -> s#(activate(X)) isNat#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__0()) -> 0#() isNat#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__plus(X1,X2)) -> 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)) -> plus#(activate(X1),activate(X2)) isNat#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__isNat(X)) -> isNat#(X) isNat#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__s(X)) -> activate#(X) isNat#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__s(X)) -> s#(activate(X)) and#(tt(),X) -> activate#(X) -> activate#(n__0()) -> 0#() and#(tt(),X) -> activate#(X) -> activate#(n__plus(X1,X2)) -> activate#(X2) and#(tt(),X) -> activate#(X) -> activate#(n__plus(X1,X2)) -> activate#(X1) and#(tt(),X) -> activate#(X) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) and#(tt(),X) -> activate#(X) -> activate#(n__isNat(X)) -> isNat#(X) and#(tt(),X) -> activate#(X) -> activate#(n__s(X)) -> activate#(X) and#(tt(),X) -> activate#(X) -> activate#(n__s(X)) -> s#(activate(X)) plus#(N,0()) -> isNat#(N) -> isNat#(n__plus(V1,V2)) -> activate#(V2) plus#(N,0()) -> isNat#(N) -> isNat#(n__plus(V1,V2)) -> activate#(V1) plus#(N,0()) -> isNat#(N) -> isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)) plus#(N,0()) -> isNat#(N) -> isNat#(n__plus(V1,V2)) -> and#(isNat(activate(V1)),n__isNat(activate(V2))) plus#(N,0()) -> isNat#(N) -> isNat#(n__s(V1)) -> activate#(V1) plus#(N,0()) -> isNat#(N) -> isNat#(n__s(V1)) -> isNat#(activate(V1)) plus#(N,0()) -> U11#(isNat(N),N) -> U11#(tt(),N) -> activate#(N) plus#(N,s(M)) -> isNat#(M) -> isNat#(n__plus(V1,V2)) -> activate#(V2) plus#(N,s(M)) -> isNat#(M) -> isNat#(n__plus(V1,V2)) -> activate#(V1) plus#(N,s(M)) -> isNat#(M) -> isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)) plus#(N,s(M)) -> isNat#(M) -> isNat#(n__plus(V1,V2)) -> and#(isNat(activate(V1)),n__isNat(activate(V2))) plus#(N,s(M)) -> isNat#(M) -> isNat#(n__s(V1)) -> activate#(V1) plus#(N,s(M)) -> isNat#(M) -> isNat#(n__s(V1)) -> isNat#(activate(V1)) plus#(N,s(M)) -> and#(isNat(M),n__isNat(N)) -> and#(tt(),X) -> activate#(X) plus#(N,s(M)) -> U21#(and(isNat(M),n__isNat(N)),M,N) -> U21#(tt(),M,N) -> activate#(M) plus#(N,s(M)) -> U21#(and(isNat(M),n__isNat(N)),M,N) -> U21#(tt(),M,N) -> activate#(N) plus#(N,s(M)) -> U21#(and(isNat(M),n__isNat(N)),M,N) -> U21#(tt(),M,N) -> plus#(activate(N),activate(M)) plus#(N,s(M)) -> U21#(and(isNat(M),n__isNat(N)),M,N) -> U21#(tt(),M,N) -> s#(plus(activate(N),activate(M))) U21#(tt(),M,N) -> plus#(activate(N),activate(M)) -> plus#(N,0()) -> isNat#(N) U21#(tt(),M,N) -> plus#(activate(N),activate(M)) -> plus#(N,0()) -> U11#(isNat(N),N) U21#(tt(),M,N) -> plus#(activate(N),activate(M)) -> plus#(N,s(M)) -> isNat#(M) U21#(tt(),M,N) -> plus#(activate(N),activate(M)) -> plus#(N,s(M)) -> and#(isNat(M),n__isNat(N)) U21#(tt(),M,N) -> plus#(activate(N),activate(M)) -> plus#(N,s(M)) -> U21#(and(isNat(M),n__isNat(N)),M,N) U21#(tt(),M,N) -> activate#(M) -> activate#(n__0()) -> 0#() U21#(tt(),M,N) -> activate#(M) -> activate#(n__plus(X1,X2)) -> activate#(X2) U21#(tt(),M,N) -> activate#(M) -> activate#(n__plus(X1,X2)) -> activate#(X1) U21#(tt(),M,N) -> activate#(M) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) U21#(tt(),M,N) -> activate#(M) -> activate#(n__isNat(X)) -> isNat#(X) U21#(tt(),M,N) -> activate#(M) -> activate#(n__s(X)) -> activate#(X) U21#(tt(),M,N) -> activate#(M) -> activate#(n__s(X)) -> s#(activate(X)) U21#(tt(),M,N) -> activate#(N) -> activate#(n__0()) -> 0#() U21#(tt(),M,N) -> activate#(N) -> activate#(n__plus(X1,X2)) -> activate#(X2) U21#(tt(),M,N) -> activate#(N) -> activate#(n__plus(X1,X2)) -> activate#(X1) U21#(tt(),M,N) -> activate#(N) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) U21#(tt(),M,N) -> activate#(N) -> activate#(n__isNat(X)) -> isNat#(X) U21#(tt(),M,N) -> activate#(N) -> activate#(n__s(X)) -> activate#(X) U21#(tt(),M,N) -> activate#(N) -> activate#(n__s(X)) -> s#(activate(X)) activate#(n__s(X)) -> activate#(X) -> activate#(n__0()) -> 0#() activate#(n__s(X)) -> activate#(X) -> activate#(n__plus(X1,X2)) -> 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)) -> plus#(activate(X1),activate(X2)) activate#(n__s(X)) -> activate#(X) -> activate#(n__isNat(X)) -> isNat#(X) activate#(n__s(X)) -> activate#(X) -> activate#(n__s(X)) -> activate#(X) activate#(n__s(X)) -> activate#(X) -> activate#(n__s(X)) -> s#(activate(X)) activate#(n__isNat(X)) -> isNat#(X) -> isNat#(n__plus(V1,V2)) -> activate#(V2) activate#(n__isNat(X)) -> isNat#(X) -> isNat#(n__plus(V1,V2)) -> activate#(V1) activate#(n__isNat(X)) -> isNat#(X) -> isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)) activate#(n__isNat(X)) -> isNat#(X) -> isNat#(n__plus(V1,V2)) -> and#(isNat(activate(V1)),n__isNat(activate(V2))) activate#(n__isNat(X)) -> isNat#(X) -> isNat#(n__s(V1)) -> activate#(V1) activate#(n__isNat(X)) -> isNat#(X) -> isNat#(n__s(V1)) -> isNat#(activate(V1)) activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) -> plus#(N,0()) -> isNat#(N) activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) -> plus#(N,0()) -> U11#(isNat(N),N) 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,s(M)) -> and#(isNat(M),n__isNat(N)) activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) -> plus#(N,s(M)) -> U21#(and(isNat(M),n__isNat(N)),M,N) activate#(n__plus(X1,X2)) -> activate#(X2) -> activate#(n__0()) -> 0#() activate#(n__plus(X1,X2)) -> activate#(X2) -> activate#(n__plus(X1,X2)) -> 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)) -> plus#(activate(X1),activate(X2)) activate#(n__plus(X1,X2)) -> activate#(X2) -> activate#(n__isNat(X)) -> isNat#(X) activate#(n__plus(X1,X2)) -> activate#(X2) -> activate#(n__s(X)) -> activate#(X) activate#(n__plus(X1,X2)) -> activate#(X2) -> activate#(n__s(X)) -> s#(activate(X)) activate#(n__plus(X1,X2)) -> activate#(X1) -> activate#(n__0()) -> 0#() activate#(n__plus(X1,X2)) -> activate#(X1) -> activate#(n__plus(X1,X2)) -> 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)) -> plus#(activate(X1),activate(X2)) activate#(n__plus(X1,X2)) -> activate#(X1) -> activate#(n__isNat(X)) -> isNat#(X) activate#(n__plus(X1,X2)) -> activate#(X1) -> activate#(n__s(X)) -> activate#(X) activate#(n__plus(X1,X2)) -> activate#(X1) -> activate#(n__s(X)) -> s#(activate(X)) U11#(tt(),N) -> activate#(N) -> activate#(n__0()) -> 0#() U11#(tt(),N) -> activate#(N) -> activate#(n__plus(X1,X2)) -> activate#(X2) U11#(tt(),N) -> activate#(N) -> activate#(n__plus(X1,X2)) -> activate#(X1) U11#(tt(),N) -> activate#(N) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) U11#(tt(),N) -> activate#(N) -> activate#(n__isNat(X)) -> isNat#(X) U11#(tt(),N) -> activate#(N) -> activate#(n__s(X)) -> activate#(X) U11#(tt(),N) -> activate#(N) -> activate#(n__s(X)) -> s#(activate(X)) SCC Processor: #sccs: 1 #rules: 21 #arcs: 117/576 DPs: isNat#(n__s(V1)) -> isNat#(activate(V1)) isNat#(n__s(V1)) -> activate#(V1) activate#(n__s(X)) -> activate#(X) activate#(n__isNat(X)) -> isNat#(X) isNat#(n__plus(V1,V2)) -> and#(isNat(activate(V1)),n__isNat(activate(V2))) and#(tt(),X) -> activate#(X) activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) plus#(N,s(M)) -> U21#(and(isNat(M),n__isNat(N)),M,N) U21#(tt(),M,N) -> plus#(activate(N),activate(M)) plus#(N,s(M)) -> and#(isNat(M),n__isNat(N)) plus#(N,s(M)) -> isNat#(M) isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)) isNat#(n__plus(V1,V2)) -> activate#(V1) activate#(n__plus(X1,X2)) -> activate#(X1) activate#(n__plus(X1,X2)) -> activate#(X2) isNat#(n__plus(V1,V2)) -> activate#(V2) plus#(N,0()) -> U11#(isNat(N),N) U11#(tt(),N) -> activate#(N) plus#(N,0()) -> isNat#(N) U21#(tt(),M,N) -> activate#(N) U21#(tt(),M,N) -> activate#(M) TRS: U11(tt(),N) -> activate(N) U21(tt(),M,N) -> s(plus(activate(N),activate(M))) and(tt(),X) -> activate(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) plus(N,0()) -> U11(isNat(N),N) plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) isNat(X) -> n__isNat(X) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__isNat(X)) -> isNat(X) activate(n__s(X)) -> s(activate(X)) activate(X) -> X Open