WORST_CASE(?,O(n^1)) * Step 1: DependencyPairs WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: 2nd(cons(X,n__cons(Y,Z))) -> activate(Y) activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) - Signature: {2nd/1,activate/1,cons/2,from/1,s/1} / {n__cons/2,n__from/1,n__s/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd,activate,cons,from,s} and constructors {n__cons ,n__from,n__s} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs 2nd#(cons(X,n__cons(Y,Z))) -> c_1(activate#(Y)) activate#(X) -> c_2() activate#(n__cons(X1,X2)) -> c_3(cons#(activate(X1),X2),activate#(X1)) activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) cons#(X1,X2) -> c_6() from#(X) -> c_7(cons#(X,n__from(n__s(X)))) from#(X) -> c_8() s#(X) -> c_9() Weak DPs and mark the set of starting terms. * Step 2: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: 2nd#(cons(X,n__cons(Y,Z))) -> c_1(activate#(Y)) activate#(X) -> c_2() activate#(n__cons(X1,X2)) -> c_3(cons#(activate(X1),X2),activate#(X1)) activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) cons#(X1,X2) -> c_6() from#(X) -> c_7(cons#(X,n__from(n__s(X)))) from#(X) -> c_8() s#(X) -> c_9() - Weak TRS: 2nd(cons(X,n__cons(Y,Z))) -> activate(Y) activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) - Signature: {2nd/1,activate/1,cons/2,from/1,s/1,2nd#/1,activate#/1,cons#/2,from#/1,s#/1} / {n__cons/2,n__from/1,n__s/1 ,c_1/1,c_2/0,c_3/2,c_4/2,c_5/2,c_6/0,c_7/1,c_8/0,c_9/0} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,cons#,from#,s#} and constructors {n__cons ,n__from,n__s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2,6,8,9} by application of Pre({1,2,6,8,9}) = {3,4,5,7}. Here rules are labelled as follows: 1: 2nd#(cons(X,n__cons(Y,Z))) -> c_1(activate#(Y)) 2: activate#(X) -> c_2() 3: activate#(n__cons(X1,X2)) -> c_3(cons#(activate(X1),X2),activate#(X1)) 4: activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)) 5: activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) 6: cons#(X1,X2) -> c_6() 7: from#(X) -> c_7(cons#(X,n__from(n__s(X)))) 8: from#(X) -> c_8() 9: s#(X) -> c_9() * Step 3: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__cons(X1,X2)) -> c_3(cons#(activate(X1),X2),activate#(X1)) activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) from#(X) -> c_7(cons#(X,n__from(n__s(X)))) - Weak DPs: 2nd#(cons(X,n__cons(Y,Z))) -> c_1(activate#(Y)) activate#(X) -> c_2() cons#(X1,X2) -> c_6() from#(X) -> c_8() s#(X) -> c_9() - Weak TRS: 2nd(cons(X,n__cons(Y,Z))) -> activate(Y) activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) - Signature: {2nd/1,activate/1,cons/2,from/1,s/1,2nd#/1,activate#/1,cons#/2,from#/1,s#/1} / {n__cons/2,n__from/1,n__s/1 ,c_1/1,c_2/0,c_3/2,c_4/2,c_5/2,c_6/0,c_7/1,c_8/0,c_9/0} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,cons#,from#,s#} and constructors {n__cons ,n__from,n__s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {4} by application of Pre({4}) = {2}. Here rules are labelled as follows: 1: activate#(n__cons(X1,X2)) -> c_3(cons#(activate(X1),X2),activate#(X1)) 2: activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)) 3: activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) 4: from#(X) -> c_7(cons#(X,n__from(n__s(X)))) 5: 2nd#(cons(X,n__cons(Y,Z))) -> c_1(activate#(Y)) 6: activate#(X) -> c_2() 7: cons#(X1,X2) -> c_6() 8: from#(X) -> c_8() 9: s#(X) -> c_9() * Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__cons(X1,X2)) -> c_3(cons#(activate(X1),X2),activate#(X1)) activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) - Weak DPs: 2nd#(cons(X,n__cons(Y,Z))) -> c_1(activate#(Y)) activate#(X) -> c_2() cons#(X1,X2) -> c_6() from#(X) -> c_7(cons#(X,n__from(n__s(X)))) from#(X) -> c_8() s#(X) -> c_9() - Weak TRS: 2nd(cons(X,n__cons(Y,Z))) -> activate(Y) activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) - Signature: {2nd/1,activate/1,cons/2,from/1,s/1,2nd#/1,activate#/1,cons#/2,from#/1,s#/1} / {n__cons/2,n__from/1,n__s/1 ,c_1/1,c_2/0,c_3/2,c_4/2,c_5/2,c_6/0,c_7/1,c_8/0,c_9/0} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,cons#,from#,s#} and constructors {n__cons ,n__from,n__s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:activate#(n__cons(X1,X2)) -> c_3(cons#(activate(X1),X2),activate#(X1)) -->_2 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):3 -->_2 activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)):2 -->_1 cons#(X1,X2) -> c_6():6 -->_2 activate#(X) -> c_2():5 -->_2 activate#(n__cons(X1,X2)) -> c_3(cons#(activate(X1),X2),activate#(X1)):1 2:S:activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)) -->_1 from#(X) -> c_7(cons#(X,n__from(n__s(X)))):7 -->_2 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):3 -->_1 from#(X) -> c_8():8 -->_2 activate#(X) -> c_2():5 -->_2 activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)):2 -->_2 activate#(n__cons(X1,X2)) -> c_3(cons#(activate(X1),X2),activate#(X1)):1 3:S:activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) -->_1 s#(X) -> c_9():9 -->_2 activate#(X) -> c_2():5 -->_2 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):3 -->_2 activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)):2 -->_2 activate#(n__cons(X1,X2)) -> c_3(cons#(activate(X1),X2),activate#(X1)):1 4:W:2nd#(cons(X,n__cons(Y,Z))) -> c_1(activate#(Y)) 5:W:activate#(X) -> c_2() 6:W:cons#(X1,X2) -> c_6() 7:W:from#(X) -> c_7(cons#(X,n__from(n__s(X)))) -->_1 cons#(X1,X2) -> c_6():6 8:W:from#(X) -> c_8() 9:W:s#(X) -> c_9() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 4: 2nd#(cons(X,n__cons(Y,Z))) -> c_1(activate#(Y)) 8: from#(X) -> c_8() 7: from#(X) -> c_7(cons#(X,n__from(n__s(X)))) 6: cons#(X1,X2) -> c_6() 5: activate#(X) -> c_2() 9: s#(X) -> c_9() * Step 5: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__cons(X1,X2)) -> c_3(cons#(activate(X1),X2),activate#(X1)) activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) - Weak TRS: 2nd(cons(X,n__cons(Y,Z))) -> activate(Y) activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) - Signature: {2nd/1,activate/1,cons/2,from/1,s/1,2nd#/1,activate#/1,cons#/2,from#/1,s#/1} / {n__cons/2,n__from/1,n__s/1 ,c_1/1,c_2/0,c_3/2,c_4/2,c_5/2,c_6/0,c_7/1,c_8/0,c_9/0} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,cons#,from#,s#} and constructors {n__cons ,n__from,n__s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:activate#(n__cons(X1,X2)) -> c_3(cons#(activate(X1),X2),activate#(X1)) -->_2 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):3 -->_2 activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)):2 -->_2 activate#(n__cons(X1,X2)) -> c_3(cons#(activate(X1),X2),activate#(X1)):1 2:S:activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)) -->_2 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):3 -->_2 activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)):2 -->_2 activate#(n__cons(X1,X2)) -> c_3(cons#(activate(X1),X2),activate#(X1)):1 3:S:activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) -->_2 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):3 -->_2 activate#(n__from(X)) -> c_4(from#(activate(X)),activate#(X)):2 -->_2 activate#(n__cons(X1,X2)) -> c_3(cons#(activate(X1),X2),activate#(X1)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: activate#(n__cons(X1,X2)) -> c_3(activate#(X1)) activate#(n__from(X)) -> c_4(activate#(X)) activate#(n__s(X)) -> c_5(activate#(X)) * Step 6: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__cons(X1,X2)) -> c_3(activate#(X1)) activate#(n__from(X)) -> c_4(activate#(X)) activate#(n__s(X)) -> c_5(activate#(X)) - Weak TRS: 2nd(cons(X,n__cons(Y,Z))) -> activate(Y) activate(X) -> X activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) cons(X1,X2) -> n__cons(X1,X2) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) - Signature: {2nd/1,activate/1,cons/2,from/1,s/1,2nd#/1,activate#/1,cons#/2,from#/1,s#/1} / {n__cons/2,n__from/1,n__s/1 ,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,cons#,from#,s#} and constructors {n__cons ,n__from,n__s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: activate#(n__cons(X1,X2)) -> c_3(activate#(X1)) activate#(n__from(X)) -> c_4(activate#(X)) activate#(n__s(X)) -> c_5(activate#(X)) * Step 7: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__cons(X1,X2)) -> c_3(activate#(X1)) activate#(n__from(X)) -> c_4(activate#(X)) activate#(n__s(X)) -> c_5(activate#(X)) - Signature: {2nd/1,activate/1,cons/2,from/1,s/1,2nd#/1,activate#/1,cons#/2,from#/1,s#/1} / {n__cons/2,n__from/1,n__s/1 ,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,cons#,from#,s#} and constructors {n__cons ,n__from,n__s} + 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(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(2nd) = [0] p(activate) = [4] p(cons) = [1] x1 + [1] x2 + [1] p(from) = [1] x1 + [4] p(n__cons) = [1] x1 + [7] p(n__from) = [1] x1 + [4] p(n__s) = [1] x1 + [6] p(s) = [1] x1 + [0] p(2nd#) = [2] p(activate#) = [3] x1 + [9] p(cons#) = [0] p(from#) = [0] p(s#) = [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [0] p(c_5) = [1] x1 + [1] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] Following rules are strictly oriented: activate#(n__cons(X1,X2)) = [3] X1 + [30] > [3] X1 + [9] = c_3(activate#(X1)) activate#(n__from(X)) = [3] X + [21] > [3] X + [9] = c_4(activate#(X)) activate#(n__s(X)) = [3] X + [27] > [3] X + [10] = c_5(activate#(X)) Following rules are (at-least) weakly oriented: 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: activate#(n__cons(X1,X2)) -> c_3(activate#(X1)) activate#(n__from(X)) -> c_4(activate#(X)) activate#(n__s(X)) -> c_5(activate#(X)) - Signature: {2nd/1,activate/1,cons/2,from/1,s/1,2nd#/1,activate#/1,cons#/2,from#/1,s#/1} / {n__cons/2,n__from/1,n__s/1 ,c_1/1,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,cons#,from#,s#} and constructors {n__cons ,n__from,n__s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))