/export/starexec/sandbox2/solver/bin/starexec_run_ttt2 /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- 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#(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#() 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#() 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 usable rules: 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 interpretation: [and#](x0, x1) = x0 + 3x1, [plus](x0, x1) = 2x0 + 2x1 + 0, [U21#](x0, x1, x2) = 5x1 + 4x2, [isNat](x0) = x0, [U11](x0, x1) = x0 + 2x1 + 2, [U21](x0, x1, x2) = 2x1 + 2x2 + 3, [plus#](x0, x1) = 4x0 + 5x1, [isNat#](x0) = 3x0, [s](x0) = x0 + 1, [n__isNat](x0) = x0, [tt] = 0, [n__0] = 0, [activate#](x0) = 3x0, [0] = 0, [n__s](x0) = x0 + 1, [n__plus](x0, x1) = 2x0 + 2x1 + 0, [U11#](x0, x1) = 4x1 + 0, [and](x0, x1) = 1x0 + 2x1, [activate](x0) = x0 orientation: isNat#(n__s(V1)) = 3V1 + 4 >= 3V1 = isNat#(activate(V1)) isNat#(n__plus(V1,V2)) = 5V1 + 5V2 + 3 >= 3V2 = activate#(V2) activate#(n__plus(X1,X2)) = 5X1 + 5X2 + 3 >= 4X1 + 5X2 = plus#(X1,X2) plus#(N,0()) = 4N + 5 >= 3N = isNat#(N) isNat#(n__plus(V1,V2)) = 5V1 + 5V2 + 3 >= 3V1 = activate#(V1) activate#(n__isNat(X)) = 3X >= 3X = isNat#(X) isNat#(n__plus(V1,V2)) = 5V1 + 5V2 + 3 >= 3V1 = isNat#(activate(V1)) isNat#(n__plus(V1,V2)) = 5V1 + 5V2 + 3 >= V1 + 3V2 = and#(isNat(activate(V1)),n__isNat(activate(V2))) and#(tt(),X) = 3X + 0 >= 3X = activate#(X) isNat#(n__s(V1)) = 3V1 + 4 >= 3V1 = activate#(V1) plus#(N,0()) = 4N + 5 >= 4N + 0 = U11#(isNat(N),N) U11#(tt(),N) = 4N + 0 >= 3N = activate#(N) plus#(N,s(M)) = 5M + 4N + 6 >= 3M = isNat#(M) plus#(N,s(M)) = 5M + 4N + 6 >= M + 3N = and#(isNat(M),n__isNat(N)) plus#(N,s(M)) = 5M + 4N + 6 >= 5M + 4N = U21#(and(isNat(M),n__isNat(N)),M,N) U21#(tt(),M,N) = 5M + 4N >= 3M = activate#(M) U21#(tt(),M,N) = 5M + 4N >= 3N = activate#(N) U21#(tt(),M,N) = 5M + 4N >= 5M + 4N = plus#(activate(N),activate(M)) U11(tt(),N) = 2N + 2 >= N = activate(N) U21(tt(),M,N) = 2M + 2N + 3 >= 2M + 2N + 1 = s(plus(activate(N),activate(M))) and(tt(),X) = 2X + 1 >= X = activate(X) isNat(n__0()) = 0 >= 0 = tt() isNat(n__plus(V1,V2)) = 2V1 + 2V2 + 0 >= 1V1 + 2V2 = and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) = V1 + 1 >= V1 = isNat(activate(V1)) plus(N,0()) = 2N + 2 >= 2N + 2 = U11(isNat(N),N) plus(N,s(M)) = 2M + 2N + 3 >= 2M + 2N + 3 = U21(and(isNat(M),n__isNat(N)),M,N) 0() = 0 >= 0 = n__0() plus(X1,X2) = 2X1 + 2X2 + 0 >= 2X1 + 2X2 + 0 = n__plus(X1,X2) isNat(X) = X >= X = n__isNat(X) s(X) = X + 1 >= X + 1 = n__s(X) activate(n__0()) = 0 >= 0 = 0() activate(n__plus(X1,X2)) = 2X1 + 2X2 + 0 >= 2X1 + 2X2 + 0 = plus(X1,X2) activate(n__isNat(X)) = X >= X = 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)) activate#(n__plus(X1,X2)) -> plus#(X1,X2) activate#(n__isNat(X)) -> isNat#(X) and#(tt(),X) -> activate#(X) isNat#(n__s(V1)) -> activate#(V1) plus#(N,0()) -> U11#(isNat(N),N) 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 Restore Modifier: DPs: isNat#(n__s(V1)) -> isNat#(activate(V1)) activate#(n__plus(X1,X2)) -> plus#(X1,X2) activate#(n__isNat(X)) -> isNat#(X) and#(tt(),X) -> activate#(X) isNat#(n__s(V1)) -> activate#(V1) plus#(N,0()) -> U11#(isNat(N),N) 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 SCC Processor: #sccs: 2 #rules: 5 #arcs: 60/64 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 usable rules: 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 interpretation: [plus](x0, x1) = 4x0 + 5x1 + 5, [isNat](x0) = 2x0 + 2, [U11](x0, x1) = 4x1 + 7, [U21](x0, x1, x2) = 5x1 + 4x2 + 5, [isNat#](x0) = 2x0 + 2, [s](x0) = x0 + 0, [n__isNat](x0) = 2x0 + 2, [tt] = 2, [n__0] = 2, [activate#](x0) = 2x0, [0] = 2, [n__s](x0) = x0 + 0, [n__plus](x0, x1) = 4x0 + 5x1 + 5, [and](x0, x1) = 2x0 + 5x1 + 0, [activate](x0) = x0 orientation: isNat#(n__s(V1)) = 2V1 + 2 >= 2V1 + 2 = isNat#(activate(V1)) isNat#(n__s(V1)) = 2V1 + 2 >= 2V1 = activate#(V1) activate#(n__isNat(X)) = 4X + 4 >= 2X + 2 = isNat#(X) U11(tt(),N) = 4N + 7 >= N = activate(N) U21(tt(),M,N) = 5M + 4N + 5 >= 5M + 4N + 5 = s(plus(activate(N),activate(M))) and(tt(),X) = 5X + 4 >= X = activate(X) isNat(n__0()) = 4 >= 2 = tt() isNat(n__plus(V1,V2)) = 6V1 + 7V2 + 7 >= 4V1 + 7V2 + 7 = and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) = 2V1 + 2 >= 2V1 + 2 = isNat(activate(V1)) plus(N,0()) = 4N + 7 >= 4N + 7 = U11(isNat(N),N) plus(N,s(M)) = 5M + 4N + 5 >= 5M + 4N + 5 = U21(and(isNat(M),n__isNat(N)),M,N) 0() = 2 >= 2 = n__0() plus(X1,X2) = 4X1 + 5X2 + 5 >= 4X1 + 5X2 + 5 = n__plus(X1,X2) isNat(X) = 2X + 2 >= 2X + 2 = n__isNat(X) s(X) = X + 0 >= X + 0 = n__s(X) activate(n__0()) = 2 >= 2 = 0() activate(n__plus(X1,X2)) = 4X1 + 5X2 + 5 >= 4X1 + 5X2 + 5 = plus(X1,X2) activate(n__isNat(X)) = 2X + 2 >= 2X + 2 = isNat(X) activate(n__s(X)) = X + 0 >= X + 0 = s(X) activate(X) = X >= X = X problem: DPs: isNat#(n__s(V1)) -> isNat#(activate(V1)) isNat#(n__s(V1)) -> 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 Restore Modifier: DPs: isNat#(n__s(V1)) -> isNat#(activate(V1)) isNat#(n__s(V1)) -> 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 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 Matrix Interpretation Processor: dim=1 usable rules: 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 interpretation: [plus](x0, x1) = 2x0 + 3x1, [isNat](x0) = 2x0, [U11](x0, x1) = 2x1, [U21](x0, x1, x2) = 3x1 + 2x2 + 1, [isNat#](x0) = x0 + 6, [s](x0) = x0 + 1, [n__isNat](x0) = 2x0, [tt] = 1, [n__0] = 4, [0] = 4, [n__s](x0) = x0 + 1, [n__plus](x0, x1) = 2x0 + 3x1, [and](x0, x1) = x0 + 2x1, [activate](x0) = x0 orientation: isNat#(n__s(V1)) = V1 + 7 >= V1 + 6 = isNat#(activate(V1)) U11(tt(),N) = 2N >= N = activate(N) U21(tt(),M,N) = 3M + 2N + 1 >= 3M + 2N + 1 = s(plus(activate(N),activate(M))) and(tt(),X) = 2X + 1 >= X = activate(X) isNat(n__0()) = 8 >= 1 = tt() isNat(n__plus(V1,V2)) = 4V1 + 6V2 >= 2V1 + 4V2 = and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) = 2V1 + 2 >= 2V1 = isNat(activate(V1)) plus(N,0()) = 2N + 12 >= 2N = U11(isNat(N),N) plus(N,s(M)) = 3M + 2N + 3 >= 3M + 2N + 1 = U21(and(isNat(M),n__isNat(N)),M,N) 0() = 4 >= 4 = n__0() plus(X1,X2) = 2X1 + 3X2 >= 2X1 + 3X2 = n__plus(X1,X2) isNat(X) = 2X >= 2X = n__isNat(X) s(X) = X + 1 >= X + 1 = n__s(X) activate(n__0()) = 4 >= 4 = 0() activate(n__plus(X1,X2)) = 2X1 + 3X2 >= 2X1 + 3X2 = plus(X1,X2) activate(n__isNat(X)) = 2X >= 2X = 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 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 Matrix Interpretation Processor: dim=1 usable rules: 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 interpretation: [plus](x0, x1) = 2x0 + x1 + 2, [U21#](x0, x1, x2) = x1 + 2, [isNat](x0) = 0, [U11](x0, x1) = 2x1 + 1, [U21](x0, x1, x2) = x1 + 2x2 + 6, [plus#](x0, x1) = x1 + 2, [s](x0) = x0 + 4, [n__isNat](x0) = 0, [tt] = 0, [n__0] = 0, [0] = 0, [n__s](x0) = x0 + 4, [n__plus](x0, x1) = 2x0 + x1 + 2, [and](x0, x1) = 4x1, [activate](x0) = x0 orientation: plus#(N,s(M)) = M + 6 >= M + 2 = U21#(and(isNat(M),n__isNat(N)),M,N) U21#(tt(),M,N) = M + 2 >= M + 2 = plus#(activate(N),activate(M)) U11(tt(),N) = 2N + 1 >= N = activate(N) U21(tt(),M,N) = M + 2N + 6 >= M + 2N + 6 = s(plus(activate(N),activate(M))) and(tt(),X) = 4X >= X = activate(X) isNat(n__0()) = 0 >= 0 = tt() isNat(n__plus(V1,V2)) = 0 >= 0 = and(isNat(activate(V1)),n__isNat(activate(V2))) isNat(n__s(V1)) = 0 >= 0 = isNat(activate(V1)) plus(N,0()) = 2N + 2 >= 2N + 1 = U11(isNat(N),N) plus(N,s(M)) = M + 2N + 6 >= M + 2N + 6 = U21(and(isNat(M),n__isNat(N)),M,N) 0() = 0 >= 0 = n__0() plus(X1,X2) = 2X1 + X2 + 2 >= 2X1 + X2 + 2 = n__plus(X1,X2) isNat(X) = 0 >= 0 = n__isNat(X) s(X) = X + 4 >= X + 4 = n__s(X) activate(n__0()) = 0 >= 0 = 0() activate(n__plus(X1,X2)) = 2X1 + X2 + 2 >= 2X1 + X2 + 2 = plus(X1,X2) activate(n__isNat(X)) = 0 >= 0 = isNat(X) activate(n__s(X)) = X + 4 >= X + 4 = s(X) activate(X) = X >= X = X problem: DPs: 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 Restore Modifier: DPs: 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: 0 #rules: 0 #arcs: 2/1