WORST_CASE(?,O(n^1)) * Step 1: DependencyPairs WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) first(X1,X2) -> n__first(X1,X2) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) - Signature: {activate/1,first/2,from/1,s/1} / {0/0,cons/2,n__first/2,n__from/1,n__s/1,nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate,first,from,s} and constructors {0,cons,n__first ,n__from,n__s,nil} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs activate#(X) -> c_1() activate#(n__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) activate#(n__from(X)) -> c_3(from#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)) first#(X1,X2) -> c_5() first#(0(),X) -> c_6() first#(s(X),cons(Y,Z)) -> c_7(activate#(Z)) from#(X) -> c_8() from#(X) -> c_9() s#(X) -> c_10() Weak DPs and mark the set of starting terms. * Step 2: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(X) -> c_1() activate#(n__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) activate#(n__from(X)) -> c_3(from#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)) first#(X1,X2) -> c_5() first#(0(),X) -> c_6() first#(s(X),cons(Y,Z)) -> c_7(activate#(Z)) from#(X) -> c_8() from#(X) -> c_9() s#(X) -> c_10() - Weak TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) first(X1,X2) -> n__first(X1,X2) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) - Signature: {activate/1,first/2,from/1,s/1,activate#/1,first#/2,from#/1,s#/1} / {0/0,cons/2,n__first/2,n__from/1,n__s/1 ,nil/0,c_1/0,c_2/3,c_3/2,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,first#,from#,s#} and constructors {0,cons ,n__first,n__from,n__s,nil} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,5,6,7,8,9,10} by application of Pre({1,5,6,7,8,9,10}) = {2,3,4}. Here rules are labelled as follows: 1: activate#(X) -> c_1() 2: activate#(n__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) 3: activate#(n__from(X)) -> c_3(from#(activate(X)),activate#(X)) 4: activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)) 5: first#(X1,X2) -> c_5() 6: first#(0(),X) -> c_6() 7: first#(s(X),cons(Y,Z)) -> c_7(activate#(Z)) 8: from#(X) -> c_8() 9: from#(X) -> c_9() 10: s#(X) -> c_10() * Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) activate#(n__from(X)) -> c_3(from#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)) - Weak DPs: activate#(X) -> c_1() first#(X1,X2) -> c_5() first#(0(),X) -> c_6() first#(s(X),cons(Y,Z)) -> c_7(activate#(Z)) from#(X) -> c_8() from#(X) -> c_9() s#(X) -> c_10() - Weak TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) first(X1,X2) -> n__first(X1,X2) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) - Signature: {activate/1,first/2,from/1,s/1,activate#/1,first#/2,from#/1,s#/1} / {0/0,cons/2,n__first/2,n__from/1,n__s/1 ,nil/0,c_1/0,c_2/3,c_3/2,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,first#,from#,s#} and constructors {0,cons ,n__first,n__from,n__s,nil} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:activate#(n__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) -->_3 activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)):3 -->_2 activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)):3 -->_3 activate#(n__from(X)) -> c_3(from#(activate(X)),activate#(X)):2 -->_2 activate#(n__from(X)) -> c_3(from#(activate(X)),activate#(X)):2 -->_1 first#(0(),X) -> c_6():6 -->_1 first#(X1,X2) -> c_5():5 -->_3 activate#(X) -> c_1():4 -->_2 activate#(X) -> c_1():4 -->_3 activate#(n__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):1 -->_2 activate#(n__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):1 2:S:activate#(n__from(X)) -> c_3(from#(activate(X)),activate#(X)) -->_2 activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)):3 -->_1 from#(X) -> c_9():9 -->_1 from#(X) -> c_8():8 -->_2 activate#(X) -> c_1():4 -->_2 activate#(n__from(X)) -> c_3(from#(activate(X)),activate#(X)):2 -->_2 activate#(n__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):1 3:S:activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)) -->_1 s#(X) -> c_10():10 -->_2 activate#(X) -> c_1():4 -->_2 activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)):3 -->_2 activate#(n__from(X)) -> c_3(from#(activate(X)),activate#(X)):2 -->_2 activate#(n__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):1 4:W:activate#(X) -> c_1() 5:W:first#(X1,X2) -> c_5() 6:W:first#(0(),X) -> c_6() 7:W:first#(s(X),cons(Y,Z)) -> c_7(activate#(Z)) 8:W:from#(X) -> c_8() 9:W:from#(X) -> c_9() 10:W:s#(X) -> c_10() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 7: first#(s(X),cons(Y,Z)) -> c_7(activate#(Z)) 5: first#(X1,X2) -> c_5() 6: first#(0(),X) -> c_6() 8: from#(X) -> c_8() 9: from#(X) -> c_9() 4: activate#(X) -> c_1() 10: s#(X) -> c_10() * Step 4: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) activate#(n__from(X)) -> c_3(from#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)) - Weak TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) first(X1,X2) -> n__first(X1,X2) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) - Signature: {activate/1,first/2,from/1,s/1,activate#/1,first#/2,from#/1,s#/1} / {0/0,cons/2,n__first/2,n__from/1,n__s/1 ,nil/0,c_1/0,c_2/3,c_3/2,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,first#,from#,s#} and constructors {0,cons ,n__first,n__from,n__s,nil} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:activate#(n__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) -->_3 activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)):3 -->_2 activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)):3 -->_3 activate#(n__from(X)) -> c_3(from#(activate(X)),activate#(X)):2 -->_2 activate#(n__from(X)) -> c_3(from#(activate(X)),activate#(X)):2 -->_3 activate#(n__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):1 -->_2 activate#(n__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):1 2:S:activate#(n__from(X)) -> c_3(from#(activate(X)),activate#(X)) -->_2 activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)):3 -->_2 activate#(n__from(X)) -> c_3(from#(activate(X)),activate#(X)):2 -->_2 activate#(n__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):1 3:S:activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)) -->_2 activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)):3 -->_2 activate#(n__from(X)) -> c_3(from#(activate(X)),activate#(X)):2 -->_2 activate#(n__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: activate#(n__first(X1,X2)) -> c_2(activate#(X1),activate#(X2)) activate#(n__from(X)) -> c_3(activate#(X)) activate#(n__s(X)) -> c_4(activate#(X)) * Step 5: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__first(X1,X2)) -> c_2(activate#(X1),activate#(X2)) activate#(n__from(X)) -> c_3(activate#(X)) activate#(n__s(X)) -> c_4(activate#(X)) - Weak TRS: activate(X) -> X activate(n__first(X1,X2)) -> first(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) first(X1,X2) -> n__first(X1,X2) first(0(),X) -> nil() first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) s(X) -> n__s(X) - Signature: {activate/1,first/2,from/1,s/1,activate#/1,first#/2,from#/1,s#/1} / {0/0,cons/2,n__first/2,n__from/1,n__s/1 ,nil/0,c_1/0,c_2/2,c_3/1,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,first#,from#,s#} and constructors {0,cons ,n__first,n__from,n__s,nil} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: activate#(n__first(X1,X2)) -> c_2(activate#(X1),activate#(X2)) activate#(n__from(X)) -> c_3(activate#(X)) activate#(n__s(X)) -> c_4(activate#(X)) * Step 6: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__first(X1,X2)) -> c_2(activate#(X1),activate#(X2)) activate#(n__from(X)) -> c_3(activate#(X)) activate#(n__s(X)) -> c_4(activate#(X)) - Signature: {activate/1,first/2,from/1,s/1,activate#/1,first#/2,from#/1,s#/1} / {0/0,cons/2,n__first/2,n__from/1,n__s/1 ,nil/0,c_1/0,c_2/2,c_3/1,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,first#,from#,s#} and constructors {0,cons ,n__first,n__from,n__s,nil} + 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_2) = {1,2}, uargs(c_3) = {1}, uargs(c_4) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(activate) = [0] p(cons) = [1] x1 + [1] x2 + [0] p(first) = [0] p(from) = [0] p(n__first) = [1] x1 + [1] x2 + [2] p(n__from) = [1] x1 + [2] p(n__s) = [1] x1 + [1] p(nil) = [2] p(s) = [0] p(activate#) = [5] x1 + [0] p(first#) = [0] p(from#) = [0] p(s#) = [0] p(c_1) = [0] p(c_2) = [1] x1 + [1] x2 + [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) = [0] p(c_10) = [0] Following rules are strictly oriented: activate#(n__first(X1,X2)) = [5] X1 + [5] X2 + [10] > [5] X1 + [5] X2 + [0] = c_2(activate#(X1),activate#(X2)) activate#(n__from(X)) = [5] X + [10] > [5] X + [0] = c_3(activate#(X)) activate#(n__s(X)) = [5] X + [5] > [5] X + [0] = c_4(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 7: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: activate#(n__first(X1,X2)) -> c_2(activate#(X1),activate#(X2)) activate#(n__from(X)) -> c_3(activate#(X)) activate#(n__s(X)) -> c_4(activate#(X)) - Signature: {activate/1,first/2,from/1,s/1,activate#/1,first#/2,from#/1,s#/1} / {0/0,cons/2,n__first/2,n__from/1,n__s/1 ,nil/0,c_1/0,c_2/2,c_3/1,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,first#,from#,s#} and constructors {0,cons ,n__first,n__from,n__s,nil} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))