WORST_CASE(?,O(n^2)) * Step 1: DependencyPairs WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: U11(tt(),M,N) -> U12(tt(),activate(M),activate(N)) U12(tt(),M,N) -> s(plus(activate(N),activate(M))) U21(tt(),M,N) -> U22(tt(),activate(M),activate(N)) U22(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(X) -> X plus(N,0()) -> N plus(N,s(M)) -> U11(tt(),M,N) x(N,0()) -> 0() x(N,s(M)) -> U21(tt(),M,N) - Signature: {U11/3,U12/3,U21/3,U22/3,activate/1,plus/2,x/2} / {0/0,s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {U11,U12,U21,U22,activate,plus,x} 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)) U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N)),activate#(M),activate#(N)) U22#(tt(),M,N) -> c_4(plus#(x(activate(N),activate(M)),activate(N)) ,x#(activate(N),activate(M)) ,activate#(N) ,activate#(M) ,activate#(N)) activate#(X) -> c_5() plus#(N,0()) -> c_6() plus#(N,s(M)) -> c_7(U11#(tt(),M,N)) x#(N,0()) -> c_8() x#(N,s(M)) -> c_9(U21#(tt(),M,N)) Weak DPs and mark the set of starting terms. * Step 2: PredecessorEstimation WORST_CASE(?,O(n^2)) + 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)) U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N)),activate#(M),activate#(N)) U22#(tt(),M,N) -> c_4(plus#(x(activate(N),activate(M)),activate(N)) ,x#(activate(N),activate(M)) ,activate#(N) ,activate#(M) ,activate#(N)) activate#(X) -> c_5() plus#(N,0()) -> c_6() plus#(N,s(M)) -> c_7(U11#(tt(),M,N)) x#(N,0()) -> c_8() x#(N,s(M)) -> c_9(U21#(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))) U21(tt(),M,N) -> U22(tt(),activate(M),activate(N)) U22(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(X) -> X plus(N,0()) -> N plus(N,s(M)) -> U11(tt(),M,N) x(N,0()) -> 0() x(N,s(M)) -> U21(tt(),M,N) - Signature: {U11/3,U12/3,U21/3,U22/3,activate/1,plus/2,x/2,U11#/3,U12#/3,U21#/3,U22#/3,activate#/1,plus#/2,x#/2} / {0/0 ,s/1,tt/0,c_1/3,c_2/3,c_3/3,c_4/5,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,U21#,U22#,activate#,plus# ,x#} and constructors {0,s,tt} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {5,6,8} by application of Pre({5,6,8}) = {1,2,3,4}. 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: U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N)),activate#(M),activate#(N)) 4: U22#(tt(),M,N) -> c_4(plus#(x(activate(N),activate(M)),activate(N)) ,x#(activate(N),activate(M)) ,activate#(N) ,activate#(M) ,activate#(N)) 5: activate#(X) -> c_5() 6: plus#(N,0()) -> c_6() 7: plus#(N,s(M)) -> c_7(U11#(tt(),M,N)) 8: x#(N,0()) -> c_8() 9: x#(N,s(M)) -> c_9(U21#(tt(),M,N)) * Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + 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)) U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N)),activate#(M),activate#(N)) U22#(tt(),M,N) -> c_4(plus#(x(activate(N),activate(M)),activate(N)) ,x#(activate(N),activate(M)) ,activate#(N) ,activate#(M) ,activate#(N)) plus#(N,s(M)) -> c_7(U11#(tt(),M,N)) x#(N,s(M)) -> c_9(U21#(tt(),M,N)) - Weak DPs: activate#(X) -> c_5() plus#(N,0()) -> c_6() x#(N,0()) -> c_8() - Weak TRS: U11(tt(),M,N) -> U12(tt(),activate(M),activate(N)) U12(tt(),M,N) -> s(plus(activate(N),activate(M))) U21(tt(),M,N) -> U22(tt(),activate(M),activate(N)) U22(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(X) -> X plus(N,0()) -> N plus(N,s(M)) -> U11(tt(),M,N) x(N,0()) -> 0() x(N,s(M)) -> U21(tt(),M,N) - Signature: {U11/3,U12/3,U21/3,U22/3,activate/1,plus/2,x/2,U11#/3,U12#/3,U21#/3,U22#/3,activate#/1,plus#/2,x#/2} / {0/0 ,s/1,tt/0,c_1/3,c_2/3,c_3/3,c_4/5,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,U21#,U22#,activate#,plus# ,x#} 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_5():7 -->_2 activate#(X) -> c_5():7 2:S:U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M)),activate#(N),activate#(M)) -->_1 plus#(N,s(M)) -> c_7(U11#(tt(),M,N)):5 -->_1 plus#(N,0()) -> c_6():8 -->_3 activate#(X) -> c_5():7 -->_2 activate#(X) -> c_5():7 3:S:U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N)),activate#(M),activate#(N)) -->_1 U22#(tt(),M,N) -> c_4(plus#(x(activate(N),activate(M)),activate(N)) ,x#(activate(N),activate(M)) ,activate#(N) ,activate#(M) ,activate#(N)):4 -->_3 activate#(X) -> c_5():7 -->_2 activate#(X) -> c_5():7 4:S:U22#(tt(),M,N) -> c_4(plus#(x(activate(N),activate(M)),activate(N)) ,x#(activate(N),activate(M)) ,activate#(N) ,activate#(M) ,activate#(N)) -->_2 x#(N,s(M)) -> c_9(U21#(tt(),M,N)):6 -->_1 plus#(N,s(M)) -> c_7(U11#(tt(),M,N)):5 -->_2 x#(N,0()) -> c_8():9 -->_1 plus#(N,0()) -> c_6():8 -->_5 activate#(X) -> c_5():7 -->_4 activate#(X) -> c_5():7 -->_3 activate#(X) -> c_5():7 5:S:plus#(N,s(M)) -> c_7(U11#(tt(),M,N)) -->_1 U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N)),activate#(M),activate#(N)):1 6:S:x#(N,s(M)) -> c_9(U21#(tt(),M,N)) -->_1 U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N)),activate#(M),activate#(N)):3 7:W:activate#(X) -> c_5() 8:W:plus#(N,0()) -> c_6() 9:W:x#(N,0()) -> c_8() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 9: x#(N,0()) -> c_8() 7: activate#(X) -> c_5() 8: plus#(N,0()) -> c_6() * Step 4: SimplifyRHS WORST_CASE(?,O(n^2)) + 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)) U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N)),activate#(M),activate#(N)) U22#(tt(),M,N) -> c_4(plus#(x(activate(N),activate(M)),activate(N)) ,x#(activate(N),activate(M)) ,activate#(N) ,activate#(M) ,activate#(N)) plus#(N,s(M)) -> c_7(U11#(tt(),M,N)) x#(N,s(M)) -> c_9(U21#(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))) U21(tt(),M,N) -> U22(tt(),activate(M),activate(N)) U22(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(X) -> X plus(N,0()) -> N plus(N,s(M)) -> U11(tt(),M,N) x(N,0()) -> 0() x(N,s(M)) -> U21(tt(),M,N) - Signature: {U11/3,U12/3,U21/3,U22/3,activate/1,plus/2,x/2,U11#/3,U12#/3,U21#/3,U22#/3,activate#/1,plus#/2,x#/2} / {0/0 ,s/1,tt/0,c_1/3,c_2/3,c_3/3,c_4/5,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,U21#,U22#,activate#,plus# ,x#} 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_7(U11#(tt(),M,N)):5 3:S:U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N)),activate#(M),activate#(N)) -->_1 U22#(tt(),M,N) -> c_4(plus#(x(activate(N),activate(M)),activate(N)) ,x#(activate(N),activate(M)) ,activate#(N) ,activate#(M) ,activate#(N)):4 4:S:U22#(tt(),M,N) -> c_4(plus#(x(activate(N),activate(M)),activate(N)) ,x#(activate(N),activate(M)) ,activate#(N) ,activate#(M) ,activate#(N)) -->_2 x#(N,s(M)) -> c_9(U21#(tt(),M,N)):6 -->_1 plus#(N,s(M)) -> c_7(U11#(tt(),M,N)):5 5:S:plus#(N,s(M)) -> c_7(U11#(tt(),M,N)) -->_1 U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N)),activate#(M),activate#(N)):1 6:S:x#(N,s(M)) -> c_9(U21#(tt(),M,N)) -->_1 U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N)),activate#(M),activate#(N)):3 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))) U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N))) U22#(tt(),M,N) -> c_4(plus#(x(activate(N),activate(M)),activate(N)),x#(activate(N),activate(M))) * Step 5: DecomposeDG WORST_CASE(?,O(n^2)) + 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))) U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N))) U22#(tt(),M,N) -> c_4(plus#(x(activate(N),activate(M)),activate(N)),x#(activate(N),activate(M))) plus#(N,s(M)) -> c_7(U11#(tt(),M,N)) x#(N,s(M)) -> c_9(U21#(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))) U21(tt(),M,N) -> U22(tt(),activate(M),activate(N)) U22(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(X) -> X plus(N,0()) -> N plus(N,s(M)) -> U11(tt(),M,N) x(N,0()) -> 0() x(N,s(M)) -> U21(tt(),M,N) - Signature: {U11/3,U12/3,U21/3,U22/3,activate/1,plus/2,x/2,U11#/3,U12#/3,U21#/3,U22#/3,activate#/1,plus#/2,x#/2} / {0/0 ,s/1,tt/0,c_1/1,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,U21#,U22#,activate#,plus# ,x#} and constructors {0,s,tt} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N))) U22#(tt(),M,N) -> c_4(plus#(x(activate(N),activate(M)),activate(N)),x#(activate(N),activate(M))) x#(N,s(M)) -> c_9(U21#(tt(),M,N)) and a lower component 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_7(U11#(tt(),M,N)) Further, following extension rules are added to the lower component. U21#(tt(),M,N) -> U22#(tt(),activate(M),activate(N)) U22#(tt(),M,N) -> plus#(x(activate(N),activate(M)),activate(N)) U22#(tt(),M,N) -> x#(activate(N),activate(M)) x#(N,s(M)) -> U21#(tt(),M,N) ** Step 5.a:1: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N))) U22#(tt(),M,N) -> c_4(plus#(x(activate(N),activate(M)),activate(N)),x#(activate(N),activate(M))) x#(N,s(M)) -> c_9(U21#(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))) U21(tt(),M,N) -> U22(tt(),activate(M),activate(N)) U22(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(X) -> X plus(N,0()) -> N plus(N,s(M)) -> U11(tt(),M,N) x(N,0()) -> 0() x(N,s(M)) -> U21(tt(),M,N) - Signature: {U11/3,U12/3,U21/3,U22/3,activate/1,plus/2,x/2,U11#/3,U12#/3,U21#/3,U22#/3,activate#/1,plus#/2,x#/2} / {0/0 ,s/1,tt/0,c_1/1,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,U21#,U22#,activate#,plus# ,x#} and constructors {0,s,tt} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N))) -->_1 U22#(tt(),M,N) -> c_4(plus#(x(activate(N),activate(M)),activate(N)),x#(activate(N),activate(M))):2 2:S:U22#(tt(),M,N) -> c_4(plus#(x(activate(N),activate(M)),activate(N)),x#(activate(N),activate(M))) -->_2 x#(N,s(M)) -> c_9(U21#(tt(),M,N)):3 3:S:x#(N,s(M)) -> c_9(U21#(tt(),M,N)) -->_1 U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N))):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: U22#(tt(),M,N) -> c_4(x#(activate(N),activate(M))) ** Step 5.a:2: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N))) U22#(tt(),M,N) -> c_4(x#(activate(N),activate(M))) x#(N,s(M)) -> c_9(U21#(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))) U21(tt(),M,N) -> U22(tt(),activate(M),activate(N)) U22(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(X) -> X plus(N,0()) -> N plus(N,s(M)) -> U11(tt(),M,N) x(N,0()) -> 0() x(N,s(M)) -> U21(tt(),M,N) - Signature: {U11/3,U12/3,U21/3,U22/3,activate/1,plus/2,x/2,U11#/3,U12#/3,U21#/3,U22#/3,activate#/1,plus#/2,x#/2} / {0/0 ,s/1,tt/0,c_1/1,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,U21#,U22#,activate#,plus# ,x#} and constructors {0,s,tt} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: activate(X) -> X U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N))) U22#(tt(),M,N) -> c_4(x#(activate(N),activate(M))) x#(N,s(M)) -> c_9(U21#(tt(),M,N)) ** Step 5.a:3: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N))) U22#(tt(),M,N) -> c_4(x#(activate(N),activate(M))) x#(N,s(M)) -> c_9(U21#(tt(),M,N)) - Weak TRS: activate(X) -> X - Signature: {U11/3,U12/3,U21/3,U22/3,activate/1,plus/2,x/2,U11#/3,U12#/3,U21#/3,U22#/3,activate#/1,plus#/2,x#/2} / {0/0 ,s/1,tt/0,c_1/1,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,U21#,U22#,activate#,plus# ,x#} 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(U22#) = {2,3}, uargs(x#) = {1,2}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_9) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(U11) = [0] p(U12) = [2] x2 + [1] p(U21) = [0] p(U22) = [0] p(activate) = [1] x1 + [9] p(plus) = [0] p(s) = [1] x1 + [15] p(tt) = [1] p(x) = [2] x1 + [0] p(U11#) = [0] p(U12#) = [0] p(U21#) = [10] x1 + [1] x2 + [1] x3 + [0] p(U22#) = [4] x1 + [1] x2 + [1] x3 + [7] p(activate#) = [0] p(plus#) = [2] x1 + [0] p(x#) = [1] x1 + [1] x2 + [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [1] x1 + [0] Following rules are strictly oriented: x#(N,s(M)) = [1] M + [1] N + [15] > [1] M + [1] N + [10] = c_9(U21#(tt(),M,N)) Following rules are (at-least) weakly oriented: U21#(tt(),M,N) = [1] M + [1] N + [10] >= [1] M + [1] N + [29] = c_3(U22#(tt(),activate(M),activate(N))) U22#(tt(),M,N) = [1] M + [1] N + [11] >= [1] M + [1] N + [18] = c_4(x#(activate(N),activate(M))) activate(X) = [1] X + [9] >= [1] X + [0] = X Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 5.a:4: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N))) U22#(tt(),M,N) -> c_4(x#(activate(N),activate(M))) - Weak DPs: x#(N,s(M)) -> c_9(U21#(tt(),M,N)) - Weak TRS: activate(X) -> X - Signature: {U11/3,U12/3,U21/3,U22/3,activate/1,plus/2,x/2,U11#/3,U12#/3,U21#/3,U22#/3,activate#/1,plus#/2,x#/2} / {0/0 ,s/1,tt/0,c_1/1,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,U21#,U22#,activate#,plus# ,x#} 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(U22#) = {2,3}, uargs(x#) = {1,2}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_9) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(U11) = [0] p(U12) = [1] p(U21) = [1] p(U22) = [0] p(activate) = [1] x1 + [4] p(plus) = [8] x1 + [0] p(s) = [1] x1 + [9] p(tt) = [1] p(x) = [0] p(U11#) = [0] p(U12#) = [4] x2 + [1] x3 + [0] p(U21#) = [2] x1 + [1] x2 + [1] x3 + [2] p(U22#) = [4] x1 + [1] x2 + [1] x3 + [11] p(activate#) = [2] p(plus#) = [4] x1 + [1] x2 + [0] p(x#) = [1] x1 + [1] x2 + [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [1] x1 + [5] Following rules are strictly oriented: U22#(tt(),M,N) = [1] M + [1] N + [15] > [1] M + [1] N + [8] = c_4(x#(activate(N),activate(M))) Following rules are (at-least) weakly oriented: U21#(tt(),M,N) = [1] M + [1] N + [4] >= [1] M + [1] N + [23] = c_3(U22#(tt(),activate(M),activate(N))) x#(N,s(M)) = [1] M + [1] N + [9] >= [1] M + [1] N + [9] = c_9(U21#(tt(),M,N)) activate(X) = [1] X + [4] >= [1] X + [0] = X Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 5.a:5: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N))) - Weak DPs: U22#(tt(),M,N) -> c_4(x#(activate(N),activate(M))) x#(N,s(M)) -> c_9(U21#(tt(),M,N)) - Weak TRS: activate(X) -> X - Signature: {U11/3,U12/3,U21/3,U22/3,activate/1,plus/2,x/2,U11#/3,U12#/3,U21#/3,U22#/3,activate#/1,plus#/2,x#/2} / {0/0 ,s/1,tt/0,c_1/1,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,U21#,U22#,activate#,plus# ,x#} 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(U22#) = {2,3}, uargs(x#) = {1,2}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_9) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [8] p(U11) = [1] p(U12) = [0] p(U21) = [0] p(U22) = [0] p(activate) = [1] x1 + [3] p(plus) = [0] p(s) = [1] x1 + [13] p(tt) = [6] p(x) = [0] p(U11#) = [0] p(U12#) = [0] p(U21#) = [1] x2 + [1] x3 + [13] p(U22#) = [1] x1 + [1] x2 + [1] x3 + [0] p(activate#) = [1] x1 + [1] p(plus#) = [2] x1 + [0] p(x#) = [1] x1 + [1] x2 + [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [1] x1 + [0] Following rules are strictly oriented: U21#(tt(),M,N) = [1] M + [1] N + [13] > [1] M + [1] N + [12] = c_3(U22#(tt(),activate(M),activate(N))) Following rules are (at-least) weakly oriented: U22#(tt(),M,N) = [1] M + [1] N + [6] >= [1] M + [1] N + [6] = c_4(x#(activate(N),activate(M))) x#(N,s(M)) = [1] M + [1] N + [13] >= [1] M + [1] N + [13] = c_9(U21#(tt(),M,N)) activate(X) = [1] X + [3] >= [1] X + [0] = X Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 5.a:6: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: U21#(tt(),M,N) -> c_3(U22#(tt(),activate(M),activate(N))) U22#(tt(),M,N) -> c_4(x#(activate(N),activate(M))) x#(N,s(M)) -> c_9(U21#(tt(),M,N)) - Weak TRS: activate(X) -> X - Signature: {U11/3,U12/3,U21/3,U22/3,activate/1,plus/2,x/2,U11#/3,U12#/3,U21#/3,U22#/3,activate#/1,plus#/2,x#/2} / {0/0 ,s/1,tt/0,c_1/1,c_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,U21#,U22#,activate#,plus# ,x#} and constructors {0,s,tt} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 5.b:1: NaturalMI 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_7(U11#(tt(),M,N)) - Weak DPs: U21#(tt(),M,N) -> U22#(tt(),activate(M),activate(N)) U22#(tt(),M,N) -> plus#(x(activate(N),activate(M)),activate(N)) U22#(tt(),M,N) -> x#(activate(N),activate(M)) x#(N,s(M)) -> U21#(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))) U21(tt(),M,N) -> U22(tt(),activate(M),activate(N)) U22(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(X) -> X plus(N,0()) -> N plus(N,s(M)) -> U11(tt(),M,N) x(N,0()) -> 0() x(N,s(M)) -> U21(tt(),M,N) - Signature: {U11/3,U12/3,U21/3,U22/3,activate/1,plus/2,x/2,U11#/3,U12#/3,U21#/3,U22#/3,activate#/1,plus#/2,x#/2} / {0/0 ,s/1,tt/0,c_1/1,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,U21#,U22#,activate#,plus# ,x#} and constructors {0,s,tt} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_1) = {1}, uargs(c_2) = {1}, uargs(c_7) = {1} Following symbols are considered usable: {activate,U11#,U12#,U21#,U22#,activate#,plus#,x#} TcT has computed the following interpretation: p(0) = [1] p(U11) = [2] x1 + [1] x2 + [0] p(U12) = [2] x1 + [1] p(U21) = [4] x2 + [0] p(U22) = [9] x2 + [1] x3 + [0] p(activate) = [1] x1 + [0] p(plus) = [4] p(s) = [1] x1 + [4] p(tt) = [2] p(x) = [2] p(U11#) = [9] x1 + [4] x2 + [0] p(U12#) = [7] x1 + [4] x2 + [2] p(U21#) = [4] x3 + [4] p(U22#) = [2] x1 + [4] x3 + [0] p(activate#) = [2] x1 + [1] p(plus#) = [4] x2 + [2] p(x#) = [4] x1 + [4] p(c_1) = [1] x1 + [2] p(c_2) = [1] x1 + [12] p(c_3) = [1] x1 + [1] p(c_4) = [4] x2 + [4] p(c_5) = [1] p(c_6) = [8] p(c_7) = [1] x1 + [0] p(c_8) = [0] p(c_9) = [1] x1 + [1] Following rules are strictly oriented: U12#(tt(),M,N) = [4] M + [16] > [4] M + [14] = c_2(plus#(activate(N),activate(M))) Following rules are (at-least) weakly oriented: U11#(tt(),M,N) = [4] M + [18] >= [4] M + [18] = c_1(U12#(tt(),activate(M),activate(N))) U21#(tt(),M,N) = [4] N + [4] >= [4] N + [4] = U22#(tt(),activate(M),activate(N)) U22#(tt(),M,N) = [4] N + [4] >= [4] N + [2] = plus#(x(activate(N),activate(M)),activate(N)) U22#(tt(),M,N) = [4] N + [4] >= [4] N + [4] = x#(activate(N),activate(M)) plus#(N,s(M)) = [4] M + [18] >= [4] M + [18] = c_7(U11#(tt(),M,N)) x#(N,s(M)) = [4] N + [4] >= [4] N + [4] = U21#(tt(),M,N) activate(X) = [1] X + [0] >= [1] X + [0] = X ** Step 5.b:2: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N))) plus#(N,s(M)) -> c_7(U11#(tt(),M,N)) - Weak DPs: U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M))) U21#(tt(),M,N) -> U22#(tt(),activate(M),activate(N)) U22#(tt(),M,N) -> plus#(x(activate(N),activate(M)),activate(N)) U22#(tt(),M,N) -> x#(activate(N),activate(M)) x#(N,s(M)) -> U21#(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))) U21(tt(),M,N) -> U22(tt(),activate(M),activate(N)) U22(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(X) -> X plus(N,0()) -> N plus(N,s(M)) -> U11(tt(),M,N) x(N,0()) -> 0() x(N,s(M)) -> U21(tt(),M,N) - Signature: {U11/3,U12/3,U21/3,U22/3,activate/1,plus/2,x/2,U11#/3,U12#/3,U21#/3,U22#/3,activate#/1,plus#/2,x#/2} / {0/0 ,s/1,tt/0,c_1/1,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,U21#,U22#,activate#,plus# ,x#} and constructors {0,s,tt} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_1) = {1}, uargs(c_2) = {1}, uargs(c_7) = {1} Following symbols are considered usable: {activate,U11#,U12#,U21#,U22#,activate#,plus#,x#} TcT has computed the following interpretation: p(0) = [0] p(U11) = [1] x1 + [8] p(U12) = [4] x2 + [0] p(U21) = [2] x3 + [0] p(U22) = [1] x1 + [8] x2 + [7] p(activate) = [1] x1 + [0] p(plus) = [5] x1 + [8] x2 + [0] p(s) = [1] x1 + [2] p(tt) = [2] p(x) = [4] x1 + [2] x2 + [0] p(U11#) = [8] x1 + [8] x2 + [0] p(U12#) = [6] x1 + [8] x2 + [0] p(U21#) = [4] x1 + [10] x2 + [8] x3 + [14] p(U22#) = [2] x1 + [10] x2 + [8] x3 + [0] p(activate#) = [8] x1 + [0] p(plus#) = [8] x2 + [4] p(x#) = [8] x1 + [10] x2 + [2] p(c_1) = [1] x1 + [0] p(c_2) = [1] x1 + [8] p(c_3) = [0] p(c_4) = [1] x1 + [1] p(c_5) = [1] p(c_6) = [0] p(c_7) = [1] x1 + [4] p(c_8) = [1] p(c_9) = [1] x1 + [2] Following rules are strictly oriented: U11#(tt(),M,N) = [8] M + [16] > [8] M + [12] = c_1(U12#(tt(),activate(M),activate(N))) Following rules are (at-least) weakly oriented: U12#(tt(),M,N) = [8] M + [12] >= [8] M + [12] = c_2(plus#(activate(N),activate(M))) U21#(tt(),M,N) = [10] M + [8] N + [22] >= [10] M + [8] N + [4] = U22#(tt(),activate(M),activate(N)) U22#(tt(),M,N) = [10] M + [8] N + [4] >= [8] N + [4] = plus#(x(activate(N),activate(M)),activate(N)) U22#(tt(),M,N) = [10] M + [8] N + [4] >= [10] M + [8] N + [2] = x#(activate(N),activate(M)) plus#(N,s(M)) = [8] M + [20] >= [8] M + [20] = c_7(U11#(tt(),M,N)) x#(N,s(M)) = [10] M + [8] N + [22] >= [10] M + [8] N + [22] = U21#(tt(),M,N) activate(X) = [1] X + [0] >= [1] X + [0] = X ** Step 5.b:3: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: plus#(N,s(M)) -> c_7(U11#(tt(),M,N)) - 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))) U21#(tt(),M,N) -> U22#(tt(),activate(M),activate(N)) U22#(tt(),M,N) -> plus#(x(activate(N),activate(M)),activate(N)) U22#(tt(),M,N) -> x#(activate(N),activate(M)) x#(N,s(M)) -> U21#(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))) U21(tt(),M,N) -> U22(tt(),activate(M),activate(N)) U22(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(X) -> X plus(N,0()) -> N plus(N,s(M)) -> U11(tt(),M,N) x(N,0()) -> 0() x(N,s(M)) -> U21(tt(),M,N) - Signature: {U11/3,U12/3,U21/3,U22/3,activate/1,plus/2,x/2,U11#/3,U12#/3,U21#/3,U22#/3,activate#/1,plus#/2,x#/2} / {0/0 ,s/1,tt/0,c_1/1,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,U21#,U22#,activate#,plus# ,x#} and constructors {0,s,tt} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_1) = {1}, uargs(c_2) = {1}, uargs(c_7) = {1} Following symbols are considered usable: {activate,U11#,U12#,U21#,U22#,activate#,plus#,x#} TcT has computed the following interpretation: p(0) = [6] p(U11) = [1] x3 + [0] p(U12) = [2] x2 + [4] x3 + [0] p(U21) = [2] x2 + [13] p(U22) = [4] x2 + [0] p(activate) = [1] x1 + [0] p(plus) = [2] x2 + [0] p(s) = [1] x1 + [2] p(tt) = [4] p(x) = [1] x2 + [1] p(U11#) = [2] x1 + [14] x2 + [5] p(U12#) = [2] x1 + [14] x2 + [0] p(U21#) = [1] x1 + [14] x3 + [4] p(U22#) = [14] x3 + [8] p(activate#) = [0] p(plus#) = [14] x2 + [0] p(x#) = [14] x1 + [8] p(c_1) = [1] x1 + [4] p(c_2) = [1] x1 + [8] p(c_3) = [2] x1 + [2] p(c_4) = [0] p(c_5) = [0] p(c_6) = [1] p(c_7) = [1] x1 + [1] p(c_8) = [1] p(c_9) = [1] x1 + [0] Following rules are strictly oriented: plus#(N,s(M)) = [14] M + [28] > [14] M + [14] = c_7(U11#(tt(),M,N)) Following rules are (at-least) weakly oriented: U11#(tt(),M,N) = [14] M + [13] >= [14] M + [12] = c_1(U12#(tt(),activate(M),activate(N))) U12#(tt(),M,N) = [14] M + [8] >= [14] M + [8] = c_2(plus#(activate(N),activate(M))) U21#(tt(),M,N) = [14] N + [8] >= [14] N + [8] = U22#(tt(),activate(M),activate(N)) U22#(tt(),M,N) = [14] N + [8] >= [14] N + [0] = plus#(x(activate(N),activate(M)),activate(N)) U22#(tt(),M,N) = [14] N + [8] >= [14] N + [8] = x#(activate(N),activate(M)) x#(N,s(M)) = [14] N + [8] >= [14] N + [8] = U21#(tt(),M,N) activate(X) = [1] X + [0] >= [1] X + [0] = X ** Step 5.b:4: 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))) U21#(tt(),M,N) -> U22#(tt(),activate(M),activate(N)) U22#(tt(),M,N) -> plus#(x(activate(N),activate(M)),activate(N)) U22#(tt(),M,N) -> x#(activate(N),activate(M)) plus#(N,s(M)) -> c_7(U11#(tt(),M,N)) x#(N,s(M)) -> U21#(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))) U21(tt(),M,N) -> U22(tt(),activate(M),activate(N)) U22(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N)) activate(X) -> X plus(N,0()) -> N plus(N,s(M)) -> U11(tt(),M,N) x(N,0()) -> 0() x(N,s(M)) -> U21(tt(),M,N) - Signature: {U11/3,U12/3,U21/3,U22/3,activate/1,plus/2,x/2,U11#/3,U12#/3,U21#/3,U22#/3,activate#/1,plus#/2,x#/2} / {0/0 ,s/1,tt/0,c_1/1,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,U21#,U22#,activate#,plus# ,x#} and constructors {0,s,tt} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))