WORST_CASE(?,O(n^1)) * Step 1: DependencyPairs WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: U11(tt(),M,N) -> U12(tt(),activate(M),activate(N)) U12(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X plus(N,0()) -> N plus(N,s(M)) -> U11(tt(),M,N) - Signature: {U11/3,U12/3,activate/1,plus/2} / {0/0,s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {U11,U12,activate,plus} and constructors {0,s,tt} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N)),activate#(M),activate#(N)) U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M)),activate#(N),activate#(M)) activate#(X) -> c_3() plus#(N,0()) -> c_4() plus#(N,s(M)) -> c_5(U11#(tt(),M,N)) Weak DPs and mark the set of starting terms. * Step 2: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N)),activate#(M),activate#(N)) U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M)),activate#(N),activate#(M)) activate#(X) -> c_3() plus#(N,0()) -> c_4() plus#(N,s(M)) -> c_5(U11#(tt(),M,N)) - Weak TRS: U11(tt(),M,N) -> U12(tt(),activate(M),activate(N)) U12(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X plus(N,0()) -> N plus(N,s(M)) -> U11(tt(),M,N) - Signature: {U11/3,U12/3,activate/1,plus/2,U11#/3,U12#/3,activate#/1,plus#/2} / {0/0,s/1,tt/0,c_1/3,c_2/3,c_3/0,c_4/0 ,c_5/1} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,plus#} and constructors {0,s,tt} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {3,4} by application of Pre({3,4}) = {1,2}. Here rules are labelled as follows: 1: U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N)),activate#(M),activate#(N)) 2: U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M)),activate#(N),activate#(M)) 3: activate#(X) -> c_3() 4: plus#(N,0()) -> c_4() 5: plus#(N,s(M)) -> c_5(U11#(tt(),M,N)) * Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N)),activate#(M),activate#(N)) U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M)),activate#(N),activate#(M)) plus#(N,s(M)) -> c_5(U11#(tt(),M,N)) - Weak DPs: activate#(X) -> c_3() plus#(N,0()) -> c_4() - Weak TRS: U11(tt(),M,N) -> U12(tt(),activate(M),activate(N)) U12(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X plus(N,0()) -> N plus(N,s(M)) -> U11(tt(),M,N) - Signature: {U11/3,U12/3,activate/1,plus/2,U11#/3,U12#/3,activate#/1,plus#/2} / {0/0,s/1,tt/0,c_1/3,c_2/3,c_3/0,c_4/0 ,c_5/1} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,plus#} and constructors {0,s,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N)),activate#(M),activate#(N)) -->_1 U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M)),activate#(N),activate#(M)):2 -->_3 activate#(X) -> c_3():4 -->_2 activate#(X) -> c_3():4 2:S:U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M)),activate#(N),activate#(M)) -->_1 plus#(N,s(M)) -> c_5(U11#(tt(),M,N)):3 -->_1 plus#(N,0()) -> c_4():5 -->_3 activate#(X) -> c_3():4 -->_2 activate#(X) -> c_3():4 3:S:plus#(N,s(M)) -> c_5(U11#(tt(),M,N)) -->_1 U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N)),activate#(M),activate#(N)):1 4:W:activate#(X) -> c_3() 5:W:plus#(N,0()) -> c_4() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 4: activate#(X) -> c_3() 5: plus#(N,0()) -> c_4() * Step 4: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N)),activate#(M),activate#(N)) U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M)),activate#(N),activate#(M)) plus#(N,s(M)) -> c_5(U11#(tt(),M,N)) - Weak TRS: U11(tt(),M,N) -> U12(tt(),activate(M),activate(N)) U12(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X plus(N,0()) -> N plus(N,s(M)) -> U11(tt(),M,N) - Signature: {U11/3,U12/3,activate/1,plus/2,U11#/3,U12#/3,activate#/1,plus#/2} / {0/0,s/1,tt/0,c_1/3,c_2/3,c_3/0,c_4/0 ,c_5/1} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,plus#} and constructors {0,s,tt} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N)),activate#(M),activate#(N)) -->_1 U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M)),activate#(N),activate#(M)):2 2:S:U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M)),activate#(N),activate#(M)) -->_1 plus#(N,s(M)) -> c_5(U11#(tt(),M,N)):3 3:S:plus#(N,s(M)) -> c_5(U11#(tt(),M,N)) -->_1 U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N)),activate#(M),activate#(N)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N))) U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M))) * Step 5: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N))) U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M))) plus#(N,s(M)) -> c_5(U11#(tt(),M,N)) - Weak TRS: U11(tt(),M,N) -> U12(tt(),activate(M),activate(N)) U12(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X plus(N,0()) -> N plus(N,s(M)) -> U11(tt(),M,N) - Signature: {U11/3,U12/3,activate/1,plus/2,U11#/3,U12#/3,activate#/1,plus#/2} / {0/0,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/0 ,c_5/1} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,plus#} and constructors {0,s,tt} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: activate(X) -> X U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N))) U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M))) plus#(N,s(M)) -> c_5(U11#(tt(),M,N)) * Step 6: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N))) U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M))) plus#(N,s(M)) -> c_5(U11#(tt(),M,N)) - Weak TRS: activate(X) -> X - Signature: {U11/3,U12/3,activate/1,plus/2,U11#/3,U12#/3,activate#/1,plus#/2} / {0/0,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/0 ,c_5/1} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,plus#} and constructors {0,s,tt} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(U12#) = {2,3}, uargs(plus#) = {1,2}, uargs(c_1) = {1}, uargs(c_2) = {1}, uargs(c_5) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(U11) = [0] p(U12) = [0] p(activate) = [1] x1 + [1] p(plus) = [0] p(s) = [1] x1 + [3] p(tt) = [1] p(U11#) = [2] x1 + [1] x2 + [1] x3 + [0] p(U12#) = [1] x2 + [1] x3 + [3] p(activate#) = [0] p(plus#) = [1] x1 + [1] x2 + [0] p(c_1) = [1] x1 + [0] p(c_2) = [1] x1 + [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [1] x1 + [0] Following rules are strictly oriented: U12#(tt(),M,N) = [1] M + [1] N + [3] > [1] M + [1] N + [2] = c_2(plus#(activate(N),activate(M))) plus#(N,s(M)) = [1] M + [1] N + [3] > [1] M + [1] N + [2] = c_5(U11#(tt(),M,N)) Following rules are (at-least) weakly oriented: U11#(tt(),M,N) = [1] M + [1] N + [2] >= [1] M + [1] N + [5] = c_1(U12#(tt(),activate(M),activate(N))) activate(X) = [1] X + [1] >= [1] X + [0] = X Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 7: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N))) - Weak DPs: U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M))) plus#(N,s(M)) -> c_5(U11#(tt(),M,N)) - Weak TRS: activate(X) -> X - Signature: {U11/3,U12/3,activate/1,plus/2,U11#/3,U12#/3,activate#/1,plus#/2} / {0/0,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/0 ,c_5/1} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,plus#} and constructors {0,s,tt} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(U12#) = {2,3}, uargs(plus#) = {1,2}, uargs(c_1) = {1}, uargs(c_2) = {1}, uargs(c_5) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(U11) = [0] p(U12) = [2] x1 + [0] p(activate) = [1] x1 + [1] p(plus) = [0] p(s) = [1] x1 + [12] p(tt) = [8] p(U11#) = [2] x1 + [1] x2 + [1] x3 + [7] p(U12#) = [1] x1 + [1] x2 + [1] x3 + [9] p(activate#) = [8] x1 + [2] p(plus#) = [1] x1 + [1] x2 + [11] p(c_1) = [1] x1 + [1] p(c_2) = [1] x1 + [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [1] x1 + [0] Following rules are strictly oriented: U11#(tt(),M,N) = [1] M + [1] N + [23] > [1] M + [1] N + [20] = c_1(U12#(tt(),activate(M),activate(N))) Following rules are (at-least) weakly oriented: U12#(tt(),M,N) = [1] M + [1] N + [17] >= [1] M + [1] N + [13] = c_2(plus#(activate(N),activate(M))) plus#(N,s(M)) = [1] M + [1] N + [23] >= [1] M + [1] N + [23] = c_5(U11#(tt(),M,N)) activate(X) = [1] X + [1] >= [1] X + [0] = X Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 8: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N))) U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M))) plus#(N,s(M)) -> c_5(U11#(tt(),M,N)) - Weak TRS: activate(X) -> X - Signature: {U11/3,U12/3,activate/1,plus/2,U11#/3,U12#/3,activate#/1,plus#/2} / {0/0,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/0 ,c_5/1} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,plus#} and constructors {0,s,tt} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))