YES 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(X1,X2) activate(n__isNat(X)) -> isNat(X) activate(n__s(X)) -> s(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)) -> plus#(X1,X2) activate#(n__isNat(X)) -> isNat#(X) activate#(n__s(X)) -> s#(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(X1,X2) activate(n__isNat(X)) -> isNat(X) activate(n__s(X)) -> s(X) activate(X) -> X TDG 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)) -> plus#(X1,X2) activate#(n__isNat(X)) -> isNat#(X) activate#(n__s(X)) -> s#(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(X1,X2) activate(n__isNat(X)) -> isNat(X) activate(n__s(X)) -> s(X) activate(X) -> X graph: 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)) -> and#(isNat(activate(V1)),n__isNat(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#(X) isNat#(n__s(V1)) -> activate#(V1) -> activate#(n__isNat(X)) -> isNat#(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)) -> 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)) -> and#(isNat(activate(V1)),n__isNat(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)) -> and#(isNat(activate(V1)),n__isNat(activate(V2))) -> and#(tt(),X) -> activate#(X) isNat#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__s(X)) -> s#(X) isNat#(n__plus(V1,V2)) -> activate#(V2) -> activate#(n__isNat(X)) -> isNat#(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__s(X)) -> s#(X) isNat#(n__plus(V1,V2)) -> activate#(V1) -> activate#(n__isNat(X)) -> isNat#(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#() and#(tt(),X) -> activate#(X) -> activate#(n__s(X)) -> s#(X) and#(tt(),X) -> activate#(X) -> activate#(n__isNat(X)) -> isNat#(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()) -> 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)) -> and#(isNat(activate(V1)),n__isNat(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,0()) -> U11#(isNat(N),N) -> U11#(tt(),N) -> activate#(N) 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)) -> and#(isNat(activate(V1)),n__isNat(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) 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) -> s#(plus(activate(N),activate(M))) 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) -> activate#(N) plus#(N,s(M)) -> U21#(and(isNat(M),n__isNat(N)),M,N) -> U21#(tt(),M,N) -> activate#(M) 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) -> 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)) -> isNat#(M) 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,0()) -> isNat#(N) U21#(tt(),M,N) -> activate#(N) -> activate#(n__s(X)) -> s#(X) U21#(tt(),M,N) -> activate#(N) -> activate#(n__isNat(X)) -> isNat#(X) U21#(tt(),M,N) -> activate#(N) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) U21#(tt(),M,N) -> activate#(N) -> activate#(n__0()) -> 0#() U21#(tt(),M,N) -> activate#(M) -> activate#(n__s(X)) -> s#(X) U21#(tt(),M,N) -> activate#(M) -> activate#(n__isNat(X)) -> isNat#(X) U21#(tt(),M,N) -> activate#(M) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) U21#(tt(),M,N) -> activate#(M) -> activate#(n__0()) -> 0#() activate#(n__isNat(X)) -> isNat#(X) -> isNat#(n__s(V1)) -> isNat#(activate(V1)) activate#(n__isNat(X)) -> isNat#(X) -> isNat#(n__s(V1)) -> activate#(V1) activate#(n__isNat(X)) -> isNat#(X) -> isNat#(n__plus(V1,V2)) -> and#(isNat(activate(V1)),n__isNat(activate(V2))) 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)) -> activate#(V1) activate#(n__isNat(X)) -> isNat#(X) -> isNat#(n__plus(V1,V2)) -> activate#(V2) activate#(n__plus(X1,X2)) -> plus#(X1,X2) -> plus#(N,s(M)) -> U21#(and(isNat(M),n__isNat(N)),M,N) activate#(n__plus(X1,X2)) -> plus#(X1,X2) -> plus#(N,s(M)) -> and#(isNat(M),n__isNat(N)) activate#(n__plus(X1,X2)) -> plus#(X1,X2) -> plus#(N,s(M)) -> isNat#(M) activate#(n__plus(X1,X2)) -> plus#(X1,X2) -> plus#(N,0()) -> U11#(isNat(N),N) activate#(n__plus(X1,X2)) -> plus#(X1,X2) -> plus#(N,0()) -> isNat#(N) U11#(tt(),N) -> activate#(N) -> activate#(n__s(X)) -> s#(X) U11#(tt(),N) -> activate#(N) -> activate#(n__isNat(X)) -> isNat#(X) U11#(tt(),N) -> activate#(N) -> activate#(n__plus(X1,X2)) -> plus#(X1,X2) U11#(tt(),N) -> activate#(N) -> activate#(n__0()) -> 0#() SCC Processor: #sccs: 1 #rules: 18 #arcs: 75/441 DPs: isNat#(n__s(V1)) -> isNat#(activate(V1)) isNat#(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#(V1) activate#(n__isNat(X)) -> isNat#(X) isNat#(n__plus(V1,V2)) -> isNat#(activate(V1)) isNat#(n__plus(V1,V2)) -> and#(isNat(activate(V1)),n__isNat(activate(V2))) and#(tt(),X) -> activate#(X) isNat#(n__s(V1)) -> activate#(V1) plus#(N,0()) -> U11#(isNat(N),N) U11#(tt(),N) -> activate#(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) U21#(tt(),M,N) -> activate#(M) U21#(tt(),M,N) -> activate#(N) U21#(tt(),M,N) -> plus#(activate(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(X1,X2) activate(n__isNat(X)) -> isNat(X) activate(n__s(X)) -> s(X) activate(X) -> X Arctic Interpretation Processor: dimension: 1 interpretation: [isNat#](x0) = x0, [and#](x0, x1) = x0 + x1, [plus#](x0, x1) = 4x0 + x1, [U21#](x0, x1, x2) = x1 + 4x2, [activate#](x0) = x0, [U11#](x0, x1) = 4x1 + 0, [0] = 5, [n__s](x0) = x0, [n__isNat](x0) = x0, [n__plus](x0, x1) = 6x0 + x1 + 5, [isNat](x0) = x0, [n__0] = 5, [and](x0, x1) = x1 + 0, [s](x0) = x0, [plus](x0, x1) = 6x0 + x1 + 5, [U21](x0, x1, x2) = x0 + x1 + 6x2 + 0, [activate](x0) = x0, [U11](x0, x1) = 6x1 + 0, [tt] = 5 orientation: isNat#(n__s(V1)) = V1 >= V1 = isNat#(activate(V1)) isNat#(n__plus(V1,V2)) = 6V1 + V2 + 5 >= V2 = activate#(V2) activate#(n__plus(X1,X2)) = 6X1 + X2 + 5 >= 4X1 + X2 = plus#(X1,X2) plus#(N,0()) = 4N + 5 >= N = isNat#(N) isNat#(n__plus(V1,V2)) = 6V1 + V2 + 5 >= V1 = activate#(V1) activate#(n__isNat(X)) = X >= X = isNat#(X) isNat#(n__plus(V1,V2)) = 6V1 + V2 + 5 >= V1 = isNat#(activate(V1)) isNat#(n__plus(V1,V2)) = 6V1 + V2 + 5 >= V1 + V2 = and#(isNat(activate(V1)),n__isNat(activate(V2))) and#(tt(),X) = X + 5 >= X = activate#(X) isNat#(n__s(V1)) = V1 >= V1 = activate#(V1) plus#(N,0()) = 4N + 5 >= 4N + 0 = U11#(isNat(N),N) U11#(tt(),N) = 4N + 0 >= N = activate#(N) plus#(N,s(M)) = M + 4N >= M = isNat#(M) plus#(N,s(M)) = M + 4N >= M + N = and#(isNat(M),n__isNat(N)) plus#(N,s(M)) = M + 4N >= M + 4N = U21#(and(isNat(M),n__isNat(N)),M,N) U21#(tt(),M,N) = M + 4N >= M = activate#(M) U21#(tt(),M,N) = M + 4N >= N = activate#(N) U21#(tt(),M,N) = M + 4N >= M + 4N = plus#(activate(N),activate(M)) U11(tt(),N) = 6N + 0 >= N = activate(N) U21(tt(),M,N) = M + 6N + 5 >= M + 6N + 5 = s(plus(activate(N),activate(M))) and(tt(),X) = X + 0 >= X = activate(X) isNat(n__0()) = 5 >= 5 = tt() isNat(n__plus(V1,V2)) = 6V1 + V2 + 5 >= V2 + 0 = and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) = V1 >= V1 = isNat(activate(V1)) plus(N,0()) = 6N + 5 >= 6N + 0 = U11(isNat(N),N) plus(N,s(M)) = M + 6N + 5 >= M + 6N + 0 = U21(and(isNat(M),n__isNat(N)),M,N) 0() = 5 >= 5 = n__0() plus(X1,X2) = 6X1 + X2 + 5 >= 6X1 + X2 + 5 = n__plus(X1,X2) isNat(X) = X >= X = n__isNat(X) s(X) = X >= X = n__s(X) activate(n__0()) = 5 >= 5 = 0() activate(n__plus(X1,X2)) = 6X1 + X2 + 5 >= 6X1 + X2 + 5 = plus(X1,X2) activate(n__isNat(X)) = X >= X = isNat(X) activate(n__s(X)) = X >= X = s(X) activate(X) = X >= X = X problem: DPs: isNat#(n__s(V1)) -> isNat#(activate(V1)) isNat#(n__plus(V1,V2)) -> activate#(V2) activate#(n__plus(X1,X2)) -> plus#(X1,X2) activate#(n__isNat(X)) -> isNat#(X) isNat#(n__plus(V1,V2)) -> and#(isNat(activate(V1)),n__isNat(activate(V2))) and#(tt(),X) -> activate#(X) isNat#(n__s(V1)) -> activate#(V1) 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) U21#(tt(),M,N) -> activate#(M) U21#(tt(),M,N) -> plus#(activate(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(X1,X2) activate(n__isNat(X)) -> isNat(X) activate(n__s(X)) -> s(X) activate(X) -> X SCC Processor: #sccs: 1 #rules: 12 #arcs: 60/169 DPs: isNat#(n__s(V1)) -> isNat#(activate(V1)) isNat#(n__plus(V1,V2)) -> activate#(V2) activate#(n__plus(X1,X2)) -> plus#(X1,X2) plus#(N,s(M)) -> isNat#(M) isNat#(n__plus(V1,V2)) -> and#(isNat(activate(V1)),n__isNat(activate(V2))) and#(tt(),X) -> activate#(X) activate#(n__isNat(X)) -> isNat#(X) isNat#(n__s(V1)) -> activate#(V1) plus#(N,s(M)) -> and#(isNat(M),n__isNat(N)) plus#(N,s(M)) -> U21#(and(isNat(M),n__isNat(N)),M,N) U21#(tt(),M,N) -> activate#(M) U21#(tt(),M,N) -> plus#(activate(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(X1,X2) activate(n__isNat(X)) -> isNat(X) activate(n__s(X)) -> s(X) activate(X) -> X Arctic Interpretation Processor: dimension: 1 interpretation: [isNat#](x0) = x0, [and#](x0, x1) = x1 + 0, [plus#](x0, x1) = 1x0 + x1, [U21#](x0, x1, x2) = x0 + x1 + 1x2 + 0, [activate#](x0) = x0 + 0, [0] = 6, [n__s](x0) = x0 + 1, [n__isNat](x0) = x0 + 0, [n__plus](x0, x1) = 1x0 + x1 + 4, [isNat](x0) = x0 + 0, [n__0] = 6, [and](x0, x1) = x0 + x1 + 0, [s](x0) = x0 + 1, [plus](x0, x1) = 1x0 + x1 + 4, [U21](x0, x1, x2) = x0 + x1 + 1x2 + 4, [activate](x0) = x0, [U11](x0, x1) = x1 + 6, [tt] = 2 orientation: isNat#(n__s(V1)) = V1 + 1 >= V1 = isNat#(activate(V1)) isNat#(n__plus(V1,V2)) = 1V1 + V2 + 4 >= V2 + 0 = activate#(V2) activate#(n__plus(X1,X2)) = 1X1 + X2 + 4 >= 1X1 + X2 = plus#(X1,X2) plus#(N,s(M)) = M + 1N + 1 >= M = isNat#(M) isNat#(n__plus(V1,V2)) = 1V1 + V2 + 4 >= V2 + 0 = and#(isNat(activate(V1)),n__isNat(activate(V2))) and#(tt(),X) = X + 0 >= X + 0 = activate#(X) activate#(n__isNat(X)) = X + 0 >= X = isNat#(X) isNat#(n__s(V1)) = V1 + 1 >= V1 + 0 = activate#(V1) plus#(N,s(M)) = M + 1N + 1 >= N + 0 = and#(isNat(M),n__isNat(N)) plus#(N,s(M)) = M + 1N + 1 >= M + 1N + 0 = U21#(and(isNat(M),n__isNat(N)),M,N) U21#(tt(),M,N) = M + 1N + 2 >= M + 0 = activate#(M) U21#(tt(),M,N) = M + 1N + 2 >= M + 1N = plus#(activate(N),activate(M)) U11(tt(),N) = N + 6 >= N = activate(N) U21(tt(),M,N) = M + 1N + 4 >= M + 1N + 4 = s(plus(activate(N),activate(M))) and(tt(),X) = X + 2 >= X = activate(X) isNat(n__0()) = 6 >= 2 = tt() isNat(n__plus(V1,V2)) = 1V1 + V2 + 4 >= V1 + V2 + 0 = and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) = V1 + 1 >= V1 + 0 = isNat(activate(V1)) plus(N,0()) = 1N + 6 >= N + 6 = U11(isNat(N),N) plus(N,s(M)) = M + 1N + 4 >= M + 1N + 4 = U21(and(isNat(M),n__isNat(N)),M,N) 0() = 6 >= 6 = n__0() plus(X1,X2) = 1X1 + X2 + 4 >= 1X1 + X2 + 4 = n__plus(X1,X2) isNat(X) = X + 0 >= X + 0 = n__isNat(X) s(X) = X + 1 >= X + 1 = n__s(X) activate(n__0()) = 6 >= 6 = 0() activate(n__plus(X1,X2)) = 1X1 + X2 + 4 >= 1X1 + X2 + 4 = plus(X1,X2) activate(n__isNat(X)) = X + 0 >= X + 0 = isNat(X) activate(n__s(X)) = X + 1 >= X + 1 = s(X) activate(X) = X >= X = X problem: DPs: isNat#(n__s(V1)) -> isNat#(activate(V1)) isNat#(n__plus(V1,V2)) -> activate#(V2) activate#(n__plus(X1,X2)) -> plus#(X1,X2) plus#(N,s(M)) -> isNat#(M) isNat#(n__plus(V1,V2)) -> and#(isNat(activate(V1)),n__isNat(activate(V2))) and#(tt(),X) -> activate#(X) activate#(n__isNat(X)) -> isNat#(X) isNat#(n__s(V1)) -> activate#(V1) plus#(N,s(M)) -> U21#(and(isNat(M),n__isNat(N)),M,N) U21#(tt(),M,N) -> activate#(M) U21#(tt(),M,N) -> plus#(activate(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(X1,X2) activate(n__isNat(X)) -> isNat(X) activate(n__s(X)) -> s(X) activate(X) -> X Arctic Interpretation Processor: dimension: 1 interpretation: [isNat#](x0) = x0 + 1, [and#](x0, x1) = x0 + 3x1 + 0, [plus#](x0, x1) = 1x1 + 0, [U21#](x0, x1, x2) = 1x1 + 1, [activate#](x0) = x0 + 1, [0] = 4, [n__s](x0) = x0, [n__isNat](x0) = x0 + 0, [n__plus](x0, x1) = 1x0 + 3x1 + 4, [isNat](x0) = x0 + 0, [n__0] = 4, [and](x0, x1) = x0 + 3x1 + 0, [s](x0) = x0 + 0, [plus](x0, x1) = 1x0 + 3x1 + 4, [U21](x0, x1, x2) = 3x1 + 1x2 + 4, [activate](x0) = x0 + 0, [U11](x0, x1) = 1x1 + 7, [tt] = 4 orientation: isNat#(n__s(V1)) = V1 + 1 >= V1 + 1 = isNat#(activate(V1)) isNat#(n__plus(V1,V2)) = 1V1 + 3V2 + 4 >= V2 + 1 = activate#(V2) activate#(n__plus(X1,X2)) = 1X1 + 3X2 + 4 >= 1X2 + 0 = plus#(X1,X2) plus#(N,s(M)) = 1M + 1 >= M + 1 = isNat#(M) isNat#(n__plus(V1,V2)) = 1V1 + 3V2 + 4 >= V1 + 3V2 + 3 = and#(isNat(activate(V1)),n__isNat(activate(V2))) and#(tt(),X) = 3X + 4 >= X + 1 = activate#(X) activate#(n__isNat(X)) = X + 1 >= X + 1 = isNat#(X) isNat#(n__s(V1)) = V1 + 1 >= V1 + 1 = activate#(V1) plus#(N,s(M)) = 1M + 1 >= 1M + 1 = U21#(and(isNat(M),n__isNat(N)),M,N) U21#(tt(),M,N) = 1M + 1 >= M + 1 = activate#(M) U21#(tt(),M,N) = 1M + 1 >= 1M + 1 = plus#(activate(N),activate(M)) U11(tt(),N) = 1N + 7 >= N + 0 = activate(N) U21(tt(),M,N) = 3M + 1N + 4 >= 3M + 1N + 4 = s(plus(activate(N),activate(M))) and(tt(),X) = 3X + 4 >= X + 0 = activate(X) isNat(n__0()) = 4 >= 4 = tt() isNat(n__plus(V1,V2)) = 1V1 + 3V2 + 4 >= V1 + 3V2 + 3 = and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) = V1 + 0 >= V1 + 0 = isNat(activate(V1)) plus(N,0()) = 1N + 7 >= 1N + 7 = U11(isNat(N),N) plus(N,s(M)) = 3M + 1N + 4 >= 3M + 1N + 4 = U21(and(isNat(M),n__isNat(N)),M,N) 0() = 4 >= 4 = n__0() plus(X1,X2) = 1X1 + 3X2 + 4 >= 1X1 + 3X2 + 4 = n__plus(X1,X2) isNat(X) = X + 0 >= X + 0 = n__isNat(X) s(X) = X + 0 >= X = n__s(X) activate(n__0()) = 4 >= 4 = 0() activate(n__plus(X1,X2)) = 1X1 + 3X2 + 4 >= 1X1 + 3X2 + 4 = plus(X1,X2) activate(n__isNat(X)) = X + 0 >= X + 0 = isNat(X) activate(n__s(X)) = X + 0 >= X + 0 = s(X) activate(X) = X + 0 >= X = X problem: DPs: isNat#(n__s(V1)) -> isNat#(activate(V1)) plus#(N,s(M)) -> isNat#(M) 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) plus#(N,s(M)) -> U21#(and(isNat(M),n__isNat(N)),M,N) U21#(tt(),M,N) -> activate#(M) U21#(tt(),M,N) -> plus#(activate(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(X1,X2) activate(n__isNat(X)) -> isNat(X) activate(n__s(X)) -> s(X) activate(X) -> X SCC Processor: #sccs: 2 #rules: 5 #arcs: 30/64 DPs: plus#(N,s(M)) -> U21#(and(isNat(M),n__isNat(N)),M,N) U21#(tt(),M,N) -> plus#(activate(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(X1,X2) activate(n__isNat(X)) -> isNat(X) activate(n__s(X)) -> s(X) activate(X) -> X KBO Processor: argument filtering: pi(tt) = [] pi(U11) = [1] pi(activate) = [0] pi(U21) = [1,2] pi(plus) = [0,1] pi(s) = [0] pi(and) = [1] pi(n__0) = [] pi(isNat) = 0 pi(n__plus) = [0,1] pi(n__isNat) = 0 pi(n__s) = [0] pi(0) = [] pi(U21#) = [1] pi(plus#) = 1 weight function: w0 = 1 w(U21#) = w(0) = w(n__s) = w(n__0) = w(and) = w(s) = w(U21) = w( U11) = w(tt) = 1 w(plus#) = w(n__isNat) = w(n__plus) = w(isNat) = w(plus) = w( activate) = 0 precedence: activate > 0 > plus > n__plus ~ U21 > n__0 ~ s > plus# ~ U21# ~ n__s ~ n__isNat ~ isNat ~ and ~ U11 ~ tt problem: DPs: 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(X1,X2) activate(n__isNat(X)) -> isNat(X) activate(n__s(X)) -> s(X) activate(X) -> X Qed DPs: isNat#(n__s(V1)) -> isNat#(activate(V1)) isNat#(n__s(V1)) -> activate#(V1) activate#(n__isNat(X)) -> isNat#(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(X1,X2) activate(n__isNat(X)) -> isNat(X) activate(n__s(X)) -> s(X) activate(X) -> X Arctic Interpretation Processor: dimension: 1 interpretation: [isNat#](x0) = 1x0 + 0, [activate#](x0) = x0, [0] = 0, [n__s](x0) = x0 + 0, [n__isNat](x0) = 2x0 + 0, [n__plus](x0, x1) = x0 + x1 + 0, [isNat](x0) = 2x0 + 0, [n__0] = 0, [and](x0, x1) = x1 + 0, [s](x0) = x0 + 0, [plus](x0, x1) = x0 + x1 + 0, [U21](x0, x1, x2) = x1 + x2 + 0, [activate](x0) = x0 + 0, [U11](x0, x1) = x1 + 0, [tt] = 0 orientation: isNat#(n__s(V1)) = 1V1 + 1 >= 1V1 + 1 = isNat#(activate(V1)) isNat#(n__s(V1)) = 1V1 + 1 >= V1 = activate#(V1) activate#(n__isNat(X)) = 2X + 0 >= 1X + 0 = isNat#(X) U11(tt(),N) = N + 0 >= N + 0 = activate(N) U21(tt(),M,N) = M + N + 0 >= M + N + 0 = s(plus(activate(N),activate(M))) and(tt(),X) = X + 0 >= X + 0 = activate(X) isNat(n__0()) = 2 >= 0 = tt() isNat(n__plus(V1,V2)) = 2V1 + 2V2 + 2 >= 2V2 + 2 = and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) = 2V1 + 2 >= 2V1 + 2 = isNat(activate(V1)) plus(N,0()) = N + 0 >= N + 0 = U11(isNat(N),N) plus(N,s(M)) = M + N + 0 >= M + N + 0 = U21(and(isNat(M),n__isNat(N)),M,N) 0() = 0 >= 0 = n__0() plus(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = n__plus(X1,X2) isNat(X) = 2X + 0 >= 2X + 0 = n__isNat(X) s(X) = X + 0 >= X + 0 = n__s(X) activate(n__0()) = 0 >= 0 = 0() activate(n__plus(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = plus(X1,X2) activate(n__isNat(X)) = 2X + 0 >= 2X + 0 = isNat(X) activate(n__s(X)) = X + 0 >= X + 0 = s(X) activate(X) = X + 0 >= X = X problem: DPs: isNat#(n__s(V1)) -> isNat#(activate(V1)) activate#(n__isNat(X)) -> isNat#(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(X1,X2) activate(n__isNat(X)) -> isNat(X) activate(n__s(X)) -> s(X) activate(X) -> X SCC Processor: #sccs: 1 #rules: 1 #arcs: 5/4 DPs: isNat#(n__s(V1)) -> isNat#(activate(V1)) 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(X1,X2) activate(n__isNat(X)) -> isNat(X) activate(n__s(X)) -> s(X) activate(X) -> X KBO Processor: argument filtering: pi(tt) = [] pi(U11) = [1] pi(activate) = [0] pi(U21) = [1,2] pi(plus) = [0,1] pi(s) = [0] pi(and) = [1] pi(n__0) = [] pi(isNat) = 0 pi(n__plus) = [0,1] pi(n__isNat) = 0 pi(n__s) = [0] pi(0) = [] pi(isNat#) = 0 weight function: w0 = 1 w(isNat#) = w(0) = w(n__s) = w(n__0) = w(and) = w(s) = w( U21) = w(U11) = w(tt) = 1 w(n__isNat) = w(n__plus) = w(isNat) = w(plus) = w(activate) = 0 precedence: activate > plus > n__plus ~ U21 > 0 > n__0 > s > isNat# ~ n__s ~ n__isNat ~ isNat ~ and ~ U11 ~ tt problem: DPs: 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(X1,X2) activate(n__isNat(X)) -> isNat(X) activate(n__s(X)) -> s(X) activate(X) -> X Qed