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 Matrix Interpretation Processor: dim=1 interpretation: [isNat#](x0) = x0 + 3, [and#](x0, x1) = x0 + x1, [plus#](x0, x1) = x0 + 4x1 + 1, [U21#](x0, x1, x2) = 4x1 + x2 + 4, [activate#](x0) = x0 + 3, [U11#](x0, x1) = x1 + 5, [0] = 4, [n__s](x0) = x0 + 1, [n__isNat](x0) = x0 + 1, [n__plus](x0, x1) = x0 + 6x1 + 1, [isNat](x0) = x0 + 1, [n__0] = 4, [and](x0, x1) = 2x1, [s](x0) = x0 + 1, [plus](x0, x1) = x0 + 6x1 + 1, [U21](x0, x1, x2) = 6x1 + x2 + 7, [activate](x0) = x0, [U11](x0, x1) = x1 + 1, [tt] = 5 orientation: isNat#(n__s(V1)) = V1 + 4 >= V1 + 3 = isNat#(activate(V1)) isNat#(n__plus(V1,V2)) = V1 + 6V2 + 4 >= V2 + 3 = activate#(V2) activate#(n__plus(X1,X2)) = X1 + 6X2 + 4 >= X1 + 4X2 + 1 = plus#(X1,X2) plus#(N,0()) = N + 17 >= N + 3 = isNat#(N) isNat#(n__plus(V1,V2)) = V1 + 6V2 + 4 >= V1 + 3 = activate#(V1) activate#(n__isNat(X)) = X + 4 >= X + 3 = isNat#(X) isNat#(n__plus(V1,V2)) = V1 + 6V2 + 4 >= V1 + 3 = isNat#(activate(V1)) isNat#(n__plus(V1,V2)) = V1 + 6V2 + 4 >= V1 + V2 + 2 = and#(isNat(activate(V1)),n__isNat(activate(V2))) and#(tt(),X) = X + 5 >= X + 3 = activate#(X) isNat#(n__s(V1)) = V1 + 4 >= V1 + 3 = activate#(V1) plus#(N,0()) = N + 17 >= N + 5 = U11#(isNat(N),N) U11#(tt(),N) = N + 5 >= N + 3 = activate#(N) plus#(N,s(M)) = 4M + N + 5 >= M + 3 = isNat#(M) plus#(N,s(M)) = 4M + N + 5 >= M + N + 2 = and#(isNat(M),n__isNat(N)) plus#(N,s(M)) = 4M + N + 5 >= 4M + N + 4 = U21#(and(isNat(M),n__isNat(N)),M,N) U21#(tt(),M,N) = 4M + N + 4 >= M + 3 = activate#(M) U21#(tt(),M,N) = 4M + N + 4 >= N + 3 = activate#(N) U21#(tt(),M,N) = 4M + N + 4 >= 4M + N + 1 = plus#(activate(N),activate(M)) U11(tt(),N) = N + 1 >= N = activate(N) U21(tt(),M,N) = 6M + N + 7 >= 6M + N + 2 = s(plus(activate(N),activate(M))) and(tt(),X) = 2X >= X = activate(X) isNat(n__0()) = 5 >= 5 = tt() isNat(n__plus(V1,V2)) = V1 + 6V2 + 2 >= 2V2 + 2 = and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) = V1 + 2 >= V1 + 1 = isNat(activate(V1)) plus(N,0()) = N + 25 >= N + 1 = U11(isNat(N),N) plus(N,s(M)) = 6M + N + 7 >= 6M + N + 7 = U21(and(isNat(M),n__isNat(N)),M,N) 0() = 4 >= 4 = n__0() plus(X1,X2) = X1 + 6X2 + 1 >= X1 + 6X2 + 1 = n__plus(X1,X2) isNat(X) = X + 1 >= X + 1 = n__isNat(X) s(X) = X + 1 >= X + 1 = n__s(X) activate(n__0()) = 4 >= 4 = 0() activate(n__plus(X1,X2)) = X1 + 6X2 + 1 >= X1 + 6X2 + 1 = plus(X1,X2) activate(n__isNat(X)) = X + 1 >= X + 1 = isNat(X) activate(n__s(X)) = X + 1 >= X + 1 = s(X) activate(X) = X >= X = X 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