YES Problem: U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) -> s(plus(activate(N),activate(M))) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(N,0()) -> U31(isNat(N),N) plus(N,s(M)) -> U41(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X Proof: DP Processor: DPs: U11#(tt(),V2) -> activate#(V2) U11#(tt(),V2) -> isNat#(activate(V2)) U11#(tt(),V2) -> U12#(isNat(activate(V2))) U31#(tt(),N) -> activate#(N) U41#(tt(),M,N) -> activate#(M) U41#(tt(),M,N) -> activate#(N) U41#(tt(),M,N) -> isNat#(activate(N)) U41#(tt(),M,N) -> U42#(isNat(activate(N)),activate(M),activate(N)) U42#(tt(),M,N) -> activate#(M) U42#(tt(),M,N) -> activate#(N) U42#(tt(),M,N) -> plus#(activate(N),activate(M)) U42#(tt(),M,N) -> s#(plus(activate(N),activate(M))) isNat#(n__plus(V1,V2)) -> activate#(V2) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)) isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2)) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNat#(activate(V1)) isNat#(n__s(V1)) -> U21#(isNat(activate(V1))) plus#(N,0()) -> isNat#(N) plus#(N,0()) -> U31#(isNat(N),N) plus#(N,s(M)) -> isNat#(M) plus#(N,s(M)) -> U41#(isNat(M),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__s(X)) -> activate#(X) activate#(n__s(X)) -> s#(activate(X)) TRS: U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) -> s(plus(activate(N),activate(M))) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(N,0()) -> U31(isNat(N),N) plus(N,s(M)) -> U41(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X TDG Processor: DPs: U11#(tt(),V2) -> activate#(V2) U11#(tt(),V2) -> isNat#(activate(V2)) U11#(tt(),V2) -> U12#(isNat(activate(V2))) U31#(tt(),N) -> activate#(N) U41#(tt(),M,N) -> activate#(M) U41#(tt(),M,N) -> activate#(N) U41#(tt(),M,N) -> isNat#(activate(N)) U41#(tt(),M,N) -> U42#(isNat(activate(N)),activate(M),activate(N)) U42#(tt(),M,N) -> activate#(M) U42#(tt(),M,N) -> activate#(N) U42#(tt(),M,N) -> plus#(activate(N),activate(M)) U42#(tt(),M,N) -> s#(plus(activate(N),activate(M))) isNat#(n__plus(V1,V2)) -> activate#(V2) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)) isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2)) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNat#(activate(V1)) isNat#(n__s(V1)) -> U21#(isNat(activate(V1))) plus#(N,0()) -> isNat#(N) plus#(N,0()) -> U31#(isNat(N),N) plus#(N,s(M)) -> isNat#(M) plus#(N,s(M)) -> U41#(isNat(M),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__s(X)) -> activate#(X) activate#(n__s(X)) -> s#(activate(X)) TRS: U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) -> s(plus(activate(N),activate(M))) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(N,0()) -> U31(isNat(N),N) plus(N,s(M)) -> U41(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X graph: plus#(N,0()) -> U31#(isNat(N),N) -> U31#(tt(),N) -> activate#(N) plus#(N,0()) -> isNat#(N) -> isNat#(n__s(V1)) -> U21#(isNat(activate(V1))) plus#(N,0()) -> isNat#(N) -> isNat#(n__s(V1)) -> isNat#(activate(V1)) plus#(N,0()) -> isNat#(N) -> isNat#(n__s(V1)) -> activate#(V1) plus#(N,0()) -> isNat#(N) -> isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2)) plus#(N,0()) -> isNat#(N) -> isNat#(n__plus(V1,V2)) -> isNat#(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)) -> U41#(isNat(M),M,N) -> U41#(tt(),M,N) -> U42#(isNat(activate(N)),activate(M),activate(N)) plus#(N,s(M)) -> U41#(isNat(M),M,N) -> U41#(tt(),M,N) -> isNat#(activate(N)) plus#(N,s(M)) -> U41#(isNat(M),M,N) -> U41#(tt(),M,N) -> activate#(N) plus#(N,s(M)) -> U41#(isNat(M),M,N) -> U41#(tt(),M,N) -> activate#(M) plus#(N,s(M)) -> isNat#(M) -> isNat#(n__s(V1)) -> U21#(isNat(activate(V1))) plus#(N,s(M)) -> isNat#(M) -> isNat#(n__s(V1)) -> isNat#(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#(isNat(activate(V1)),activate(V2)) 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)) -> activate#(V1) plus#(N,s(M)) -> isNat#(M) -> isNat#(n__plus(V1,V2)) -> activate#(V2) U42#(tt(),M,N) -> plus#(activate(N),activate(M)) -> plus#(N,s(M)) -> U41#(isNat(M),M,N) U42#(tt(),M,N) -> plus#(activate(N),activate(M)) -> plus#(N,s(M)) -> isNat#(M) U42#(tt(),M,N) -> plus#(activate(N),activate(M)) -> plus#(N,0()) -> U31#(isNat(N),N) U42#(tt(),M,N) -> plus#(activate(N),activate(M)) -> plus#(N,0()) -> isNat#(N) U42#(tt(),M,N) -> activate#(N) -> activate#(n__s(X)) -> s#(activate(X)) U42#(tt(),M,N) -> activate#(N) -> activate#(n__s(X)) -> activate#(X) U42#(tt(),M,N) -> activate#(N) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) U42#(tt(),M,N) -> activate#(N) -> activate#(n__plus(X1,X2)) -> activate#(X1) U42#(tt(),M,N) -> activate#(N) -> activate#(n__plus(X1,X2)) -> activate#(X2) U42#(tt(),M,N) -> activate#(N) -> activate#(n__0()) -> 0#() U42#(tt(),M,N) -> activate#(M) -> activate#(n__s(X)) -> s#(activate(X)) U42#(tt(),M,N) -> activate#(M) -> activate#(n__s(X)) -> activate#(X) U42#(tt(),M,N) -> activate#(M) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) U42#(tt(),M,N) -> activate#(M) -> activate#(n__plus(X1,X2)) -> activate#(X1) U42#(tt(),M,N) -> activate#(M) -> activate#(n__plus(X1,X2)) -> activate#(X2) U42#(tt(),M,N) -> activate#(M) -> activate#(n__0()) -> 0#() U41#(tt(),M,N) -> U42#(isNat(activate(N)),activate(M),activate(N)) -> U42#(tt(),M,N) -> s#(plus(activate(N),activate(M))) U41#(tt(),M,N) -> U42#(isNat(activate(N)),activate(M),activate(N)) -> U42#(tt(),M,N) -> plus#(activate(N),activate(M)) U41#(tt(),M,N) -> U42#(isNat(activate(N)),activate(M),activate(N)) -> U42#(tt(),M,N) -> activate#(N) U41#(tt(),M,N) -> U42#(isNat(activate(N)),activate(M),activate(N)) -> U42#(tt(),M,N) -> activate#(M) U41#(tt(),M,N) -> isNat#(activate(N)) -> isNat#(n__s(V1)) -> U21#(isNat(activate(V1))) U41#(tt(),M,N) -> isNat#(activate(N)) -> isNat#(n__s(V1)) -> isNat#(activate(V1)) U41#(tt(),M,N) -> isNat#(activate(N)) -> isNat#(n__s(V1)) -> activate#(V1) U41#(tt(),M,N) -> isNat#(activate(N)) -> isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2)) U41#(tt(),M,N) -> isNat#(activate(N)) -> isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)) U41#(tt(),M,N) -> isNat#(activate(N)) -> isNat#(n__plus(V1,V2)) -> activate#(V1) U41#(tt(),M,N) -> isNat#(activate(N)) -> isNat#(n__plus(V1,V2)) -> activate#(V2) 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__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#() 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__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#() 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__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#() isNat#(n__s(V1)) -> isNat#(activate(V1)) -> isNat#(n__s(V1)) -> U21#(isNat(activate(V1))) isNat#(n__s(V1)) -> isNat#(activate(V1)) -> isNat#(n__s(V1)) -> isNat#(activate(V1)) isNat#(n__s(V1)) -> isNat#(activate(V1)) -> isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNat#(activate(V1)) -> isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2)) 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)) -> activate#(V1) isNat#(n__s(V1)) -> isNat#(activate(V1)) -> isNat#(n__plus(V1,V2)) -> activate#(V2) 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__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)) -> isNat#(activate(V1)) -> isNat#(n__s(V1)) -> U21#(isNat(activate(V1))) isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)) -> isNat#(n__s(V1)) -> isNat#(activate(V1)) isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)) -> isNat#(n__s(V1)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)) -> isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2)) 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)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)) -> isNat#(n__plus(V1,V2)) -> activate#(V2) 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__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__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__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#(isNat(activate(V1)),activate(V2)) -> U11#(tt(),V2) -> U12#(isNat(activate(V2))) isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2)) -> U11#(tt(),V2) -> isNat#(activate(V2)) isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2)) -> U11#(tt(),V2) -> activate#(V2) 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__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__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) -> plus#(N,s(M)) -> U41#(isNat(M),M,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,0()) -> U31#(isNat(N),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__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__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__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__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(),V2) -> isNat#(activate(V2)) -> isNat#(n__s(V1)) -> U21#(isNat(activate(V1))) U11#(tt(),V2) -> isNat#(activate(V2)) -> isNat#(n__s(V1)) -> isNat#(activate(V1)) U11#(tt(),V2) -> isNat#(activate(V2)) -> isNat#(n__s(V1)) -> activate#(V1) U11#(tt(),V2) -> isNat#(activate(V2)) -> isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2)) U11#(tt(),V2) -> isNat#(activate(V2)) -> isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)) U11#(tt(),V2) -> isNat#(activate(V2)) -> isNat#(n__plus(V1,V2)) -> activate#(V1) U11#(tt(),V2) -> isNat#(activate(V2)) -> isNat#(n__plus(V1,V2)) -> activate#(V2) U11#(tt(),V2) -> activate#(V2) -> activate#(n__s(X)) -> s#(activate(X)) U11#(tt(),V2) -> activate#(V2) -> activate#(n__s(X)) -> activate#(X) U11#(tt(),V2) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) U11#(tt(),V2) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> activate#(X1) U11#(tt(),V2) -> activate#(V2) -> activate#(n__plus(X1,X2)) -> activate#(X2) U11#(tt(),V2) -> activate#(V2) -> activate#(n__0()) -> 0#() SCC Processor: #sccs: 1 #rules: 24 #arcs: 134/841 DPs: plus#(N,0()) -> U31#(isNat(N),N) U31#(tt(),N) -> activate#(N) 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__s(X)) -> activate#(X) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)) isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2)) U11#(tt(),V2) -> activate#(V2) U11#(tt(),V2) -> isNat#(activate(V2)) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNat#(activate(V1)) plus#(N,s(M)) -> isNat#(M) plus#(N,s(M)) -> U41#(isNat(M),M,N) U41#(tt(),M,N) -> activate#(M) U41#(tt(),M,N) -> activate#(N) U41#(tt(),M,N) -> isNat#(activate(N)) U41#(tt(),M,N) -> U42#(isNat(activate(N)),activate(M),activate(N)) U42#(tt(),M,N) -> activate#(M) U42#(tt(),M,N) -> activate#(N) U42#(tt(),M,N) -> plus#(activate(N),activate(M)) TRS: U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) -> s(plus(activate(N),activate(M))) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(N,0()) -> U31(isNat(N),N) plus(N,s(M)) -> U41(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X Matrix Interpretation Processor: dim=1 usable rules: U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) -> s(plus(activate(N),activate(M))) plus(N,0()) -> U31(isNat(N),N) plus(N,s(M)) -> U41(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X interpretation: [plus#](x0, x1) = x0 + 2x1, [U42#](x0, x1, x2) = 2x1 + x2, [U41#](x0, x1, x2) = 2x1 + x2, [U31#](x0, x1) = x1 + 1, [isNat#](x0) = x0, [activate#](x0) = x0, [U11#](x0, x1) = 2x1, [0] = 1/2, [n__s](x0) = x0, [n__plus](x0, x1) = 2x0 + 2x1, [n__0] = 1/2, [s](x0) = x0, [plus](x0, x1) = 2x0 + 2x1, [U42](x0, x1, x2) = 2x1 + 2x2, [U41](x0, x1, x2) = 2x1 + 2x2, [U31](x0, x1) = x1, [U21](x0) = 0, [U12](x0) = 5/2, [isNat](x0) = 0, [activate](x0) = x0, [U11](x0, x1) = 0, [tt] = 0 orientation: plus#(N,0()) = N + 1 >= N + 1 = U31#(isNat(N),N) U31#(tt(),N) = N + 1 >= N = activate#(N) activate#(n__plus(X1,X2)) = 2X1 + 2X2 >= X2 = activate#(X2) activate#(n__plus(X1,X2)) = 2X1 + 2X2 >= X1 = activate#(X1) activate#(n__plus(X1,X2)) = 2X1 + 2X2 >= X1 + 2X2 = plus#(activate(X1),activate(X2)) plus#(N,0()) = N + 1 >= N = isNat#(N) isNat#(n__plus(V1,V2)) = 2V1 + 2V2 >= V2 = activate#(V2) activate#(n__s(X)) = X >= X = activate#(X) isNat#(n__plus(V1,V2)) = 2V1 + 2V2 >= V1 = activate#(V1) isNat#(n__plus(V1,V2)) = 2V1 + 2V2 >= V1 = isNat#(activate(V1)) isNat#(n__plus(V1,V2)) = 2V1 + 2V2 >= 2V2 = U11#(isNat(activate(V1)),activate(V2)) U11#(tt(),V2) = 2V2 >= V2 = activate#(V2) U11#(tt(),V2) = 2V2 >= V2 = isNat#(activate(V2)) isNat#(n__s(V1)) = V1 >= V1 = activate#(V1) isNat#(n__s(V1)) = V1 >= V1 = isNat#(activate(V1)) plus#(N,s(M)) = 2M + N >= M = isNat#(M) plus#(N,s(M)) = 2M + N >= 2M + N = U41#(isNat(M),M,N) U41#(tt(),M,N) = 2M + N >= M = activate#(M) U41#(tt(),M,N) = 2M + N >= N = activate#(N) U41#(tt(),M,N) = 2M + N >= N = isNat#(activate(N)) U41#(tt(),M,N) = 2M + N >= 2M + N = U42#(isNat(activate(N)),activate(M),activate(N)) U42#(tt(),M,N) = 2M + N >= M = activate#(M) U42#(tt(),M,N) = 2M + N >= N = activate#(N) U42#(tt(),M,N) = 2M + N >= 2M + N = plus#(activate(N),activate(M)) U11(tt(),V2) = 0 >= 5/2 = U12(isNat(activate(V2))) U12(tt()) = 5/2 >= 0 = tt() U21(tt()) = 0 >= 0 = tt() U31(tt(),N) = N >= N = activate(N) U41(tt(),M,N) = 2M + 2N >= 2M + 2N = U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) = 2M + 2N >= 2M + 2N = s(plus(activate(N),activate(M))) isNat(n__0()) = 0 >= 0 = tt() isNat(n__plus(V1,V2)) = 0 >= 0 = U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) = 0 >= 0 = U21(isNat(activate(V1))) plus(N,0()) = 2N + 1 >= N = U31(isNat(N),N) plus(N,s(M)) = 2M + 2N >= 2M + 2N = U41(isNat(M),M,N) 0() = 1/2 >= 1/2 = n__0() plus(X1,X2) = 2X1 + 2X2 >= 2X1 + 2X2 = n__plus(X1,X2) s(X) = X >= X = n__s(X) activate(n__0()) = 1/2 >= 1/2 = 0() activate(n__plus(X1,X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = plus(activate(X1),activate(X2)) activate(n__s(X)) = X >= X = s(activate(X)) activate(X) = X >= X = X problem: DPs: plus#(N,0()) -> U31#(isNat(N),N) 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__s(X)) -> activate#(X) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)) isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2)) U11#(tt(),V2) -> activate#(V2) U11#(tt(),V2) -> isNat#(activate(V2)) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNat#(activate(V1)) plus#(N,s(M)) -> isNat#(M) plus#(N,s(M)) -> U41#(isNat(M),M,N) U41#(tt(),M,N) -> activate#(M) U41#(tt(),M,N) -> activate#(N) U41#(tt(),M,N) -> isNat#(activate(N)) U41#(tt(),M,N) -> U42#(isNat(activate(N)),activate(M),activate(N)) U42#(tt(),M,N) -> activate#(M) U42#(tt(),M,N) -> activate#(N) U42#(tt(),M,N) -> plus#(activate(N),activate(M)) TRS: U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) -> s(plus(activate(N),activate(M))) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(N,0()) -> U31(isNat(N),N) plus(N,s(M)) -> U41(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X Restore Modifier: DPs: plus#(N,0()) -> U31#(isNat(N),N) 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__s(X)) -> activate#(X) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)) isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2)) U11#(tt(),V2) -> activate#(V2) U11#(tt(),V2) -> isNat#(activate(V2)) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNat#(activate(V1)) plus#(N,s(M)) -> isNat#(M) plus#(N,s(M)) -> U41#(isNat(M),M,N) U41#(tt(),M,N) -> activate#(M) U41#(tt(),M,N) -> activate#(N) U41#(tt(),M,N) -> isNat#(activate(N)) U41#(tt(),M,N) -> U42#(isNat(activate(N)),activate(M),activate(N)) U42#(tt(),M,N) -> activate#(M) U42#(tt(),M,N) -> activate#(N) U42#(tt(),M,N) -> plus#(activate(N),activate(M)) TRS: U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) -> s(plus(activate(N),activate(M))) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(N,0()) -> U31(isNat(N),N) plus(N,s(M)) -> U41(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X SCC Processor: #sccs: 1 #rules: 21 #arcs: 102/484 DPs: plus#(N,s(M)) -> U41#(isNat(M),M,N) U41#(tt(),M,N) -> activate#(M) 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)) -> isNat#(M) isNat#(n__plus(V1,V2)) -> activate#(V2) activate#(n__s(X)) -> activate#(X) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)) isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2)) U11#(tt(),V2) -> activate#(V2) U11#(tt(),V2) -> isNat#(activate(V2)) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNat#(activate(V1)) U41#(tt(),M,N) -> activate#(N) U41#(tt(),M,N) -> isNat#(activate(N)) U41#(tt(),M,N) -> U42#(isNat(activate(N)),activate(M),activate(N)) U42#(tt(),M,N) -> activate#(M) U42#(tt(),M,N) -> activate#(N) U42#(tt(),M,N) -> plus#(activate(N),activate(M)) TRS: U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) -> s(plus(activate(N),activate(M))) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(N,0()) -> U31(isNat(N),N) plus(N,s(M)) -> U41(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X Arctic Interpretation Processor: dimension: 1 usable rules: U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) -> s(plus(activate(N),activate(M))) plus(N,0()) -> U31(isNat(N),N) plus(N,s(M)) -> U41(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X interpretation: [plus#](x0, x1) = x0 + x1 + 0, [U42#](x0, x1, x2) = x1 + x2 + 0, [U41#](x0, x1, x2) = x1 + x2 + 0, [isNat#](x0) = x0 + 0, [activate#](x0) = x0 + 0, [U11#](x0, x1) = 1x1 + 0, [0] = 0, [n__s](x0) = x0, [n__plus](x0, x1) = x0 + 1x1 + 1, [n__0] = 0, [s](x0) = x0 + 0, [plus](x0, x1) = x0 + 1x1 + 1, [U42](x0, x1, x2) = 1x1 + x2 + 1, [U41](x0, x1, x2) = 1x1 + x2 + 1, [U31](x0, x1) = x1 + 1, [U21](x0) = 1x0 + 0, [U12](x0) = 2x0 + 2, [isNat](x0) = x0 + 0, [activate](x0) = x0 + 0, [U11](x0, x1) = 1x1 + 0, [tt] = 4 orientation: plus#(N,s(M)) = M + N + 0 >= M + N + 0 = U41#(isNat(M),M,N) U41#(tt(),M,N) = M + N + 0 >= M + 0 = activate#(M) activate#(n__plus(X1,X2)) = X1 + 1X2 + 1 >= X2 + 0 = activate#(X2) activate#(n__plus(X1,X2)) = X1 + 1X2 + 1 >= X1 + 0 = activate#(X1) activate#(n__plus(X1,X2)) = X1 + 1X2 + 1 >= X1 + X2 + 0 = plus#(activate(X1),activate(X2)) plus#(N,s(M)) = M + N + 0 >= M + 0 = isNat#(M) isNat#(n__plus(V1,V2)) = V1 + 1V2 + 1 >= V2 + 0 = activate#(V2) activate#(n__s(X)) = X + 0 >= X + 0 = activate#(X) isNat#(n__plus(V1,V2)) = V1 + 1V2 + 1 >= V1 + 0 = activate#(V1) isNat#(n__plus(V1,V2)) = V1 + 1V2 + 1 >= V1 + 0 = isNat#(activate(V1)) isNat#(n__plus(V1,V2)) = V1 + 1V2 + 1 >= 1V2 + 1 = U11#(isNat(activate(V1)),activate(V2)) U11#(tt(),V2) = 1V2 + 0 >= V2 + 0 = activate#(V2) U11#(tt(),V2) = 1V2 + 0 >= V2 + 0 = isNat#(activate(V2)) isNat#(n__s(V1)) = V1 + 0 >= V1 + 0 = activate#(V1) isNat#(n__s(V1)) = V1 + 0 >= V1 + 0 = isNat#(activate(V1)) U41#(tt(),M,N) = M + N + 0 >= N + 0 = activate#(N) U41#(tt(),M,N) = M + N + 0 >= N + 0 = isNat#(activate(N)) U41#(tt(),M,N) = M + N + 0 >= M + N + 0 = U42#(isNat(activate(N)),activate(M),activate(N)) U42#(tt(),M,N) = M + N + 0 >= M + 0 = activate#(M) U42#(tt(),M,N) = M + N + 0 >= N + 0 = activate#(N) U42#(tt(),M,N) = M + N + 0 >= M + N + 0 = plus#(activate(N),activate(M)) U11(tt(),V2) = 1V2 + 0 >= 2V2 + 2 = U12(isNat(activate(V2))) U12(tt()) = 6 >= 4 = tt() U21(tt()) = 5 >= 4 = tt() U31(tt(),N) = N + 1 >= N + 0 = activate(N) U41(tt(),M,N) = 1M + N + 1 >= 1M + N + 1 = U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) = 1M + N + 1 >= 1M + N + 1 = s(plus(activate(N),activate(M))) isNat(n__0()) = 0 >= 4 = tt() isNat(n__plus(V1,V2)) = V1 + 1V2 + 1 >= 1V2 + 1 = U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) = V1 + 0 >= 1V1 + 1 = U21(isNat(activate(V1))) plus(N,0()) = N + 1 >= N + 1 = U31(isNat(N),N) plus(N,s(M)) = 1M + N + 1 >= 1M + N + 1 = U41(isNat(M),M,N) 0() = 0 >= 0 = n__0() plus(X1,X2) = X1 + 1X2 + 1 >= X1 + 1X2 + 1 = n__plus(X1,X2) s(X) = X + 0 >= X = n__s(X) activate(n__0()) = 0 >= 0 = 0() activate(n__plus(X1,X2)) = X1 + 1X2 + 1 >= X1 + 1X2 + 1 = plus(activate(X1),activate(X2)) activate(n__s(X)) = X + 0 >= X + 0 = s(activate(X)) activate(X) = X + 0 >= X = X problem: DPs: plus#(N,s(M)) -> U41#(isNat(M),M,N) U41#(tt(),M,N) -> activate#(M) activate#(n__plus(X1,X2)) -> activate#(X1) activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) plus#(N,s(M)) -> isNat#(M) activate#(n__s(X)) -> activate#(X) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)) isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2)) U11#(tt(),V2) -> activate#(V2) U11#(tt(),V2) -> isNat#(activate(V2)) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNat#(activate(V1)) U41#(tt(),M,N) -> activate#(N) U41#(tt(),M,N) -> isNat#(activate(N)) U41#(tt(),M,N) -> U42#(isNat(activate(N)),activate(M),activate(N)) U42#(tt(),M,N) -> activate#(M) U42#(tt(),M,N) -> activate#(N) U42#(tt(),M,N) -> plus#(activate(N),activate(M)) TRS: U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) -> s(plus(activate(N),activate(M))) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(N,0()) -> U31(isNat(N),N) plus(N,s(M)) -> U41(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X Restore Modifier: DPs: plus#(N,s(M)) -> U41#(isNat(M),M,N) U41#(tt(),M,N) -> activate#(M) activate#(n__plus(X1,X2)) -> activate#(X1) activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2)) plus#(N,s(M)) -> isNat#(M) activate#(n__s(X)) -> activate#(X) isNat#(n__plus(V1,V2)) -> activate#(V1) isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)) isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2)) U11#(tt(),V2) -> activate#(V2) U11#(tt(),V2) -> isNat#(activate(V2)) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNat#(activate(V1)) U41#(tt(),M,N) -> activate#(N) U41#(tt(),M,N) -> isNat#(activate(N)) U41#(tt(),M,N) -> U42#(isNat(activate(N)),activate(M),activate(N)) U42#(tt(),M,N) -> activate#(M) U42#(tt(),M,N) -> activate#(N) U42#(tt(),M,N) -> plus#(activate(N),activate(M)) TRS: U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) -> s(plus(activate(N),activate(M))) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(N,0()) -> U31(isNat(N),N) plus(N,s(M)) -> U41(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X Matrix Interpretation Processor: dim=1 usable rules: U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) -> s(plus(activate(N),activate(M))) plus(N,0()) -> U31(isNat(N),N) plus(N,s(M)) -> U41(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X interpretation: [plus#](x0, x1) = 2x0 + 2x1, [U42#](x0, x1, x2) = 2x1 + 2x2, [U41#](x0, x1, x2) = 2x1 + 2x2, [isNat#](x0) = 2x0, [activate#](x0) = 2x0, [U11#](x0, x1) = 2x1 + 7, [0] = 2, [n__s](x0) = x0, [n__plus](x0, x1) = x0 + 4x1 + 5, [n__0] = 2, [s](x0) = x0, [plus](x0, x1) = x0 + 4x1 + 5, [U42](x0, x1, x2) = 4x1 + x2 + 5, [U41](x0, x1, x2) = 4x1 + x2 + 5, [U31](x0, x1) = x1 + 7, [U21](x0) = 0, [U12](x0) = 5, [isNat](x0) = 0, [activate](x0) = x0, [U11](x0, x1) = 4x1 + 4, [tt] = 0 orientation: plus#(N,s(M)) = 2M + 2N >= 2M + 2N = U41#(isNat(M),M,N) U41#(tt(),M,N) = 2M + 2N >= 2M = activate#(M) activate#(n__plus(X1,X2)) = 2X1 + 8X2 + 10 >= 2X1 = activate#(X1) activate#(n__plus(X1,X2)) = 2X1 + 8X2 + 10 >= 2X1 + 2X2 = plus#(activate(X1),activate(X2)) plus#(N,s(M)) = 2M + 2N >= 2M = isNat#(M) activate#(n__s(X)) = 2X >= 2X = activate#(X) isNat#(n__plus(V1,V2)) = 2V1 + 8V2 + 10 >= 2V1 = activate#(V1) isNat#(n__plus(V1,V2)) = 2V1 + 8V2 + 10 >= 2V1 = isNat#(activate(V1)) isNat#(n__plus(V1,V2)) = 2V1 + 8V2 + 10 >= 2V2 + 7 = U11#(isNat(activate(V1)),activate(V2)) U11#(tt(),V2) = 2V2 + 7 >= 2V2 = activate#(V2) U11#(tt(),V2) = 2V2 + 7 >= 2V2 = isNat#(activate(V2)) isNat#(n__s(V1)) = 2V1 >= 2V1 = activate#(V1) isNat#(n__s(V1)) = 2V1 >= 2V1 = isNat#(activate(V1)) U41#(tt(),M,N) = 2M + 2N >= 2N = activate#(N) U41#(tt(),M,N) = 2M + 2N >= 2N = isNat#(activate(N)) U41#(tt(),M,N) = 2M + 2N >= 2M + 2N = U42#(isNat(activate(N)),activate(M),activate(N)) U42#(tt(),M,N) = 2M + 2N >= 2M = activate#(M) U42#(tt(),M,N) = 2M + 2N >= 2N = activate#(N) U42#(tt(),M,N) = 2M + 2N >= 2M + 2N = plus#(activate(N),activate(M)) U11(tt(),V2) = 4V2 + 4 >= 5 = U12(isNat(activate(V2))) U12(tt()) = 5 >= 0 = tt() U21(tt()) = 0 >= 0 = tt() U31(tt(),N) = N + 7 >= N = activate(N) U41(tt(),M,N) = 4M + N + 5 >= 4M + N + 5 = U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) = 4M + N + 5 >= 4M + N + 5 = s(plus(activate(N),activate(M))) isNat(n__0()) = 0 >= 0 = tt() isNat(n__plus(V1,V2)) = 0 >= 4V2 + 4 = U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) = 0 >= 0 = U21(isNat(activate(V1))) plus(N,0()) = N + 13 >= N + 7 = U31(isNat(N),N) plus(N,s(M)) = 4M + N + 5 >= 4M + N + 5 = U41(isNat(M),M,N) 0() = 2 >= 2 = n__0() plus(X1,X2) = X1 + 4X2 + 5 >= X1 + 4X2 + 5 = n__plus(X1,X2) s(X) = X >= X = n__s(X) activate(n__0()) = 2 >= 2 = 0() activate(n__plus(X1,X2)) = X1 + 4X2 + 5 >= X1 + 4X2 + 5 = plus(activate(X1),activate(X2)) activate(n__s(X)) = X >= X = s(activate(X)) activate(X) = X >= X = X problem: DPs: plus#(N,s(M)) -> U41#(isNat(M),M,N) U41#(tt(),M,N) -> activate#(M) plus#(N,s(M)) -> isNat#(M) activate#(n__s(X)) -> activate#(X) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNat#(activate(V1)) U41#(tt(),M,N) -> activate#(N) U41#(tt(),M,N) -> isNat#(activate(N)) U41#(tt(),M,N) -> U42#(isNat(activate(N)),activate(M),activate(N)) U42#(tt(),M,N) -> activate#(M) U42#(tt(),M,N) -> activate#(N) U42#(tt(),M,N) -> plus#(activate(N),activate(M)) TRS: U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) -> s(plus(activate(N),activate(M))) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(N,0()) -> U31(isNat(N),N) plus(N,s(M)) -> U41(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X Restore Modifier: DPs: plus#(N,s(M)) -> U41#(isNat(M),M,N) U41#(tt(),M,N) -> activate#(M) plus#(N,s(M)) -> isNat#(M) activate#(n__s(X)) -> activate#(X) isNat#(n__s(V1)) -> activate#(V1) isNat#(n__s(V1)) -> isNat#(activate(V1)) U41#(tt(),M,N) -> activate#(N) U41#(tt(),M,N) -> isNat#(activate(N)) U41#(tt(),M,N) -> U42#(isNat(activate(N)),activate(M),activate(N)) U42#(tt(),M,N) -> activate#(M) U42#(tt(),M,N) -> activate#(N) U42#(tt(),M,N) -> plus#(activate(N),activate(M)) TRS: U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) -> s(plus(activate(N),activate(M))) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(N,0()) -> U31(isNat(N),N) plus(N,s(M)) -> U41(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X SCC Processor: #sccs: 3 #rules: 5 #arcs: 87/144 DPs: plus#(N,s(M)) -> U41#(isNat(M),M,N) U41#(tt(),M,N) -> U42#(isNat(activate(N)),activate(M),activate(N)) U42#(tt(),M,N) -> plus#(activate(N),activate(M)) TRS: U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) -> s(plus(activate(N),activate(M))) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(N,0()) -> U31(isNat(N),N) plus(N,s(M)) -> U41(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X Matrix Interpretation Processor: dim=1 usable rules: U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) -> s(plus(activate(N),activate(M))) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(N,0()) -> U31(isNat(N),N) plus(N,s(M)) -> U41(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X interpretation: [plus#](x0, x1) = 4x1, [U42#](x0, x1, x2) = 4x0 + 4x1 + 6, [U41#](x0, x1, x2) = 7x0 + 4x1 + 7, [0] = 0, [n__s](x0) = x0 + 6, [n__plus](x0, x1) = x0 + 3x1, [n__0] = 0, [s](x0) = x0 + 6, [plus](x0, x1) = x0 + 3x1, [U42](x0, x1, x2) = 6x0 + 3x1 + x2, [U41](x0, x1, x2) = 6x0 + 3x1 + x2 + 6, [U31](x0, x1) = x1, [U21](x0) = 1, [U12](x0) = 1, [isNat](x0) = 2, [activate](x0) = x0, [U11](x0, x1) = 1, [tt] = 1 orientation: plus#(N,s(M)) = 4M + 24 >= 4M + 21 = U41#(isNat(M),M,N) U41#(tt(),M,N) = 4M + 14 >= 4M + 14 = U42#(isNat(activate(N)),activate(M),activate(N)) U42#(tt(),M,N) = 4M + 10 >= 4M = plus#(activate(N),activate(M)) U11(tt(),V2) = 1 >= 1 = U12(isNat(activate(V2))) U12(tt()) = 1 >= 1 = tt() U21(tt()) = 1 >= 1 = tt() U31(tt(),N) = N >= N = activate(N) U41(tt(),M,N) = 3M + N + 12 >= 3M + N + 12 = U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) = 3M + N + 6 >= 3M + N + 6 = s(plus(activate(N),activate(M))) isNat(n__0()) = 2 >= 1 = tt() isNat(n__plus(V1,V2)) = 2 >= 1 = U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) = 2 >= 1 = U21(isNat(activate(V1))) plus(N,0()) = N >= N = U31(isNat(N),N) plus(N,s(M)) = 3M + N + 18 >= 3M + N + 18 = U41(isNat(M),M,N) 0() = 0 >= 0 = n__0() plus(X1,X2) = X1 + 3X2 >= X1 + 3X2 = n__plus(X1,X2) s(X) = X + 6 >= X + 6 = n__s(X) activate(n__0()) = 0 >= 0 = 0() activate(n__plus(X1,X2)) = X1 + 3X2 >= X1 + 3X2 = plus(activate(X1),activate(X2)) activate(n__s(X)) = X + 6 >= X + 6 = s(activate(X)) activate(X) = X >= X = X problem: DPs: U41#(tt(),M,N) -> U42#(isNat(activate(N)),activate(M),activate(N)) TRS: U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) -> s(plus(activate(N),activate(M))) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(N,0()) -> U31(isNat(N),N) plus(N,s(M)) -> U41(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X Restore Modifier: DPs: U41#(tt(),M,N) -> U42#(isNat(activate(N)),activate(M),activate(N)) TRS: U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) -> s(plus(activate(N),activate(M))) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(N,0()) -> U31(isNat(N),N) plus(N,s(M)) -> U41(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X SCC Processor: #sccs: 0 #rules: 0 #arcs: 3/1 DPs: isNat#(n__s(V1)) -> isNat#(activate(V1)) TRS: U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) -> s(plus(activate(N),activate(M))) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(N,0()) -> U31(isNat(N),N) plus(N,s(M)) -> U41(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X Matrix Interpretation Processor: dim=1 usable rules: U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) -> s(plus(activate(N),activate(M))) plus(N,0()) -> U31(isNat(N),N) plus(N,s(M)) -> U41(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X interpretation: [isNat#](x0) = 2x0 + 5, [0] = 0, [n__s](x0) = x0 + 1, [n__plus](x0, x1) = x0 + x1, [n__0] = 0, [s](x0) = x0 + 1, [plus](x0, x1) = x0 + x1, [U42](x0, x1, x2) = x1 + x2 + 1, [U41](x0, x1, x2) = x1 + x2 + 1, [U31](x0, x1) = x1, [U21](x0) = 2, [U12](x0) = 1, [isNat](x0) = 0, [activate](x0) = x0, [U11](x0, x1) = 1, [tt] = 0 orientation: isNat#(n__s(V1)) = 2V1 + 7 >= 2V1 + 5 = isNat#(activate(V1)) U11(tt(),V2) = 1 >= 1 = U12(isNat(activate(V2))) U12(tt()) = 1 >= 0 = tt() U21(tt()) = 2 >= 0 = tt() U31(tt(),N) = N >= N = activate(N) U41(tt(),M,N) = M + N + 1 >= M + N + 1 = U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) = M + N + 1 >= M + N + 1 = s(plus(activate(N),activate(M))) isNat(n__0()) = 0 >= 0 = tt() isNat(n__plus(V1,V2)) = 0 >= 1 = U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) = 0 >= 2 = U21(isNat(activate(V1))) plus(N,0()) = N >= N = U31(isNat(N),N) plus(N,s(M)) = M + N + 1 >= M + N + 1 = U41(isNat(M),M,N) 0() = 0 >= 0 = n__0() plus(X1,X2) = X1 + X2 >= X1 + X2 = n__plus(X1,X2) s(X) = X + 1 >= X + 1 = n__s(X) activate(n__0()) = 0 >= 0 = 0() activate(n__plus(X1,X2)) = X1 + X2 >= X1 + X2 = plus(activate(X1),activate(X2)) activate(n__s(X)) = X + 1 >= X + 1 = s(activate(X)) activate(X) = X >= X = X problem: DPs: TRS: U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) -> s(plus(activate(N),activate(M))) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(N,0()) -> U31(isNat(N),N) plus(N,s(M)) -> U41(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X Qed DPs: activate#(n__s(X)) -> activate#(X) TRS: U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) -> s(plus(activate(N),activate(M))) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(N,0()) -> U31(isNat(N),N) plus(N,s(M)) -> U41(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X Size-Change Termination Processor: DPs: TRS: U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) -> s(plus(activate(N),activate(M))) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(N,0()) -> U31(isNat(N),N) plus(N,s(M)) -> U41(isNat(M),M,N) 0() -> n__0() plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X The DP: activate#(n__s(X)) -> activate#(X) has the edges: 0 > 0 Qed