WORST_CASE(?,O(n^1)) * Step 1: InnermostRuleRemoval 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: InnermostRuleRemoval + Details: Arguments of following rules are not normal-forms. first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z))) All above mentioned rules can be savely removed. * Step 2: 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() 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_ = WIDP} + Details: We add the following weak innermost dependency pairs: Strict DPs activate#(X) -> c_1() activate#(n__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2))) activate#(n__from(X)) -> c_3(from#(activate(X))) activate#(n__s(X)) -> c_4(s#(activate(X))) first#(X1,X2) -> c_5() first#(0(),X) -> c_6() from#(X) -> c_7() from#(X) -> c_8() s#(X) -> c_9() Weak DPs and mark the set of starting terms. * Step 3: WeightGap 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#(n__from(X)) -> c_3(from#(activate(X))) activate#(n__s(X)) -> c_4(s#(activate(X))) first#(X1,X2) -> c_5() first#(0(),X) -> c_6() from#(X) -> c_7() from#(X) -> c_8() s#(X) -> c_9() - 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() 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_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0} - 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 = WgOnTrs} + 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(first) = {1,2}, uargs(from) = {1}, uargs(s) = {1}, uargs(first#) = {1,2}, uargs(from#) = {1}, uargs(s#) = {1}, uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [13] p(activate) = [3] x1 + [8] p(cons) = [0] p(first) = [1] x1 + [1] x2 + [6] p(from) = [1] x1 + [8] p(n__first) = [1] x1 + [1] x2 + [5] p(n__from) = [1] x1 + [3] p(n__s) = [1] x1 + [1] p(nil) = [0] p(s) = [1] x1 + [2] p(activate#) = [3] x1 + [9] p(first#) = [1] x1 + [1] x2 + [0] p(from#) = [1] x1 + [0] p(s#) = [1] x1 + [0] p(c_1) = [0] p(c_2) = [1] x1 + [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] Following rules are strictly oriented: activate#(X) = [3] X + [9] > [0] = c_1() activate#(n__first(X1,X2)) = [3] X1 + [3] X2 + [24] > [3] X1 + [3] X2 + [16] = c_2(first#(activate(X1),activate(X2))) activate#(n__from(X)) = [3] X + [18] > [3] X + [8] = c_3(from#(activate(X))) activate#(n__s(X)) = [3] X + [12] > [3] X + [8] = c_4(s#(activate(X))) first#(0(),X) = [1] X + [13] > [0] = c_6() activate(X) = [3] X + [8] > [1] X + [0] = X activate(n__first(X1,X2)) = [3] X1 + [3] X2 + [23] > [3] X1 + [3] X2 + [22] = first(activate(X1),activate(X2)) activate(n__from(X)) = [3] X + [17] > [3] X + [16] = from(activate(X)) activate(n__s(X)) = [3] X + [11] > [3] X + [10] = s(activate(X)) first(X1,X2) = [1] X1 + [1] X2 + [6] > [1] X1 + [1] X2 + [5] = n__first(X1,X2) first(0(),X) = [1] X + [19] > [0] = nil() from(X) = [1] X + [8] > [0] = cons(X,n__from(n__s(X))) from(X) = [1] X + [8] > [1] X + [3] = n__from(X) s(X) = [1] X + [2] > [1] X + [1] = n__s(X) Following rules are (at-least) weakly oriented: first#(X1,X2) = [1] X1 + [1] X2 + [0] >= [0] = c_5() from#(X) = [1] X + [0] >= [0] = c_7() from#(X) = [1] X + [0] >= [0] = c_8() s#(X) = [1] X + [0] >= [0] = c_9() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: first#(X1,X2) -> c_5() from#(X) -> c_7() from#(X) -> c_8() s#(X) -> c_9() - Weak DPs: activate#(X) -> c_1() activate#(n__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2))) activate#(n__from(X)) -> c_3(from#(activate(X))) activate#(n__s(X)) -> c_4(s#(activate(X))) first#(0(),X) -> c_6() - 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() 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_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0} - 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:first#(X1,X2) -> c_5() 2:S:from#(X) -> c_7() 3:S:from#(X) -> c_8() 4:S:s#(X) -> c_9() 5:W:activate#(X) -> c_1() 6:W:activate#(n__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2))) -->_1 first#(0(),X) -> c_6():9 -->_1 first#(X1,X2) -> c_5():1 7:W:activate#(n__from(X)) -> c_3(from#(activate(X))) -->_1 from#(X) -> c_8():3 -->_1 from#(X) -> c_7():2 8:W:activate#(n__s(X)) -> c_4(s#(activate(X))) -->_1 s#(X) -> c_9():4 9:W:first#(0(),X) -> c_6() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 9: first#(0(),X) -> c_6() 5: activate#(X) -> c_1() * Step 5: Decompose WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: first#(X1,X2) -> c_5() from#(X) -> c_7() from#(X) -> c_8() s#(X) -> c_9() - Weak DPs: activate#(n__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2))) activate#(n__from(X)) -> c_3(from#(activate(X))) activate#(n__s(X)) -> c_4(s#(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() 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_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,first#,from#,s#} and constructors {0,cons ,n__first,n__from,n__s,nil} + Applied Processor: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: first#(X1,X2) -> c_5() - Weak DPs: activate#(n__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2))) activate#(n__from(X)) -> c_3(from#(activate(X))) activate#(n__s(X)) -> c_4(s#(activate(X))) from#(X) -> c_7() from#(X) -> c_8() s#(X) -> c_9() - 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() 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_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,first#,from#,s#} and constructors {0,cons ,n__first,n__from,n__s,nil} Problem (S) - Strict DPs: from#(X) -> c_7() from#(X) -> c_8() s#(X) -> c_9() - Weak DPs: activate#(n__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2))) activate#(n__from(X)) -> c_3(from#(activate(X))) activate#(n__s(X)) -> c_4(s#(activate(X))) first#(X1,X2) -> c_5() - 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() 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_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,first#,from#,s#} and constructors {0,cons ,n__first,n__from,n__s,nil} ** Step 5.a:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: first#(X1,X2) -> c_5() - Weak DPs: activate#(n__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2))) activate#(n__from(X)) -> c_3(from#(activate(X))) activate#(n__s(X)) -> c_4(s#(activate(X))) from#(X) -> c_7() from#(X) -> c_8() s#(X) -> c_9() - 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() 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_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0} - 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:first#(X1,X2) -> c_5() 2:W:from#(X) -> c_7() 3:W:from#(X) -> c_8() 4:W:s#(X) -> c_9() 6:W:activate#(n__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2))) -->_1 first#(X1,X2) -> c_5():1 7:W:activate#(n__from(X)) -> c_3(from#(activate(X))) -->_1 from#(X) -> c_7():2 -->_1 from#(X) -> c_8():3 8:W:activate#(n__s(X)) -> c_4(s#(activate(X))) -->_1 s#(X) -> c_9():4 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 8: activate#(n__s(X)) -> c_4(s#(activate(X))) 7: activate#(n__from(X)) -> c_3(from#(activate(X))) 4: s#(X) -> c_9() 3: from#(X) -> c_8() 2: from#(X) -> c_7() ** Step 5.a:2: Trivial WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: first#(X1,X2) -> c_5() - Weak DPs: activate#(n__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2))) - 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() 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_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,first#,from#,s#} and constructors {0,cons ,n__first,n__from,n__s,nil} + Applied Processor: Trivial + Details: Consider the dependency graph 1:S:first#(X1,X2) -> c_5() 6:W:activate#(n__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2))) -->_1 first#(X1,X2) -> c_5():1 The dependency graph contains no loops, we remove all dependency pairs. ** Step 5.a:3: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - 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() 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_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0} - 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). ** Step 5.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: from#(X) -> c_7() from#(X) -> c_8() s#(X) -> c_9() - Weak DPs: activate#(n__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2))) activate#(n__from(X)) -> c_3(from#(activate(X))) activate#(n__s(X)) -> c_4(s#(activate(X))) first#(X1,X2) -> c_5() - 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() 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_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0} - 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:from#(X) -> c_7() 2:S:from#(X) -> c_8() 3:S:s#(X) -> c_9() 4:W:activate#(n__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2))) -->_1 first#(X1,X2) -> c_5():7 5:W:activate#(n__from(X)) -> c_3(from#(activate(X))) -->_1 from#(X) -> c_8():2 -->_1 from#(X) -> c_7():1 6:W:activate#(n__s(X)) -> c_4(s#(activate(X))) -->_1 s#(X) -> c_9():3 7:W:first#(X1,X2) -> c_5() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 4: activate#(n__first(X1,X2)) -> c_2(first#(activate(X1),activate(X2))) 7: first#(X1,X2) -> c_5() ** Step 5.b:2: Decompose WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: from#(X) -> c_7() from#(X) -> c_8() s#(X) -> c_9() - Weak DPs: activate#(n__from(X)) -> c_3(from#(activate(X))) activate#(n__s(X)) -> c_4(s#(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() 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_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,first#,from#,s#} and constructors {0,cons ,n__first,n__from,n__s,nil} + Applied Processor: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: from#(X) -> c_7() - Weak DPs: activate#(n__from(X)) -> c_3(from#(activate(X))) activate#(n__s(X)) -> c_4(s#(activate(X))) from#(X) -> c_8() s#(X) -> c_9() - 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() 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_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,first#,from#,s#} and constructors {0,cons ,n__first,n__from,n__s,nil} Problem (S) - Strict DPs: from#(X) -> c_8() s#(X) -> c_9() - Weak DPs: activate#(n__from(X)) -> c_3(from#(activate(X))) activate#(n__s(X)) -> c_4(s#(activate(X))) from#(X) -> c_7() - 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() 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_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,first#,from#,s#} and constructors {0,cons ,n__first,n__from,n__s,nil} *** Step 5.b:2.a:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: from#(X) -> c_7() - Weak DPs: activate#(n__from(X)) -> c_3(from#(activate(X))) activate#(n__s(X)) -> c_4(s#(activate(X))) from#(X) -> c_8() s#(X) -> c_9() - 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() 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_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0} - 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:from#(X) -> c_7() 2:W:from#(X) -> c_8() 3:W:s#(X) -> c_9() 5:W:activate#(n__from(X)) -> c_3(from#(activate(X))) -->_1 from#(X) -> c_7():1 -->_1 from#(X) -> c_8():2 6:W:activate#(n__s(X)) -> c_4(s#(activate(X))) -->_1 s#(X) -> c_9():3 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 6: activate#(n__s(X)) -> c_4(s#(activate(X))) 3: s#(X) -> c_9() 2: from#(X) -> c_8() *** Step 5.b:2.a:2: Trivial WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: from#(X) -> c_7() - Weak DPs: activate#(n__from(X)) -> c_3(from#(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() 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_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,first#,from#,s#} and constructors {0,cons ,n__first,n__from,n__s,nil} + Applied Processor: Trivial + Details: Consider the dependency graph 1:S:from#(X) -> c_7() 5:W:activate#(n__from(X)) -> c_3(from#(activate(X))) -->_1 from#(X) -> c_7():1 The dependency graph contains no loops, we remove all dependency pairs. *** Step 5.b:2.a:3: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - 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() 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_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0} - 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). *** Step 5.b:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: from#(X) -> c_8() s#(X) -> c_9() - Weak DPs: activate#(n__from(X)) -> c_3(from#(activate(X))) activate#(n__s(X)) -> c_4(s#(activate(X))) from#(X) -> c_7() - 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() 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_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0} - 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:from#(X) -> c_8() 2:S:s#(X) -> c_9() 3:W:activate#(n__from(X)) -> c_3(from#(activate(X))) -->_1 from#(X) -> c_7():5 -->_1 from#(X) -> c_8():1 4:W:activate#(n__s(X)) -> c_4(s#(activate(X))) -->_1 s#(X) -> c_9():2 5:W:from#(X) -> c_7() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 5: from#(X) -> c_7() *** Step 5.b:2.b:2: Decompose WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: from#(X) -> c_8() s#(X) -> c_9() - Weak DPs: activate#(n__from(X)) -> c_3(from#(activate(X))) activate#(n__s(X)) -> c_4(s#(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() 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_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,first#,from#,s#} and constructors {0,cons ,n__first,n__from,n__s,nil} + Applied Processor: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: from#(X) -> c_8() - Weak DPs: activate#(n__from(X)) -> c_3(from#(activate(X))) activate#(n__s(X)) -> c_4(s#(activate(X))) s#(X) -> c_9() - 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() 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_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,first#,from#,s#} and constructors {0,cons ,n__first,n__from,n__s,nil} Problem (S) - Strict DPs: s#(X) -> c_9() - Weak DPs: activate#(n__from(X)) -> c_3(from#(activate(X))) activate#(n__s(X)) -> c_4(s#(activate(X))) from#(X) -> c_8() - 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() 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_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,first#,from#,s#} and constructors {0,cons ,n__first,n__from,n__s,nil} **** Step 5.b:2.b:2.a:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: from#(X) -> c_8() - Weak DPs: activate#(n__from(X)) -> c_3(from#(activate(X))) activate#(n__s(X)) -> c_4(s#(activate(X))) s#(X) -> c_9() - 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() 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_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0} - 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:from#(X) -> c_8() 2:W:s#(X) -> c_9() 3:W:activate#(n__from(X)) -> c_3(from#(activate(X))) -->_1 from#(X) -> c_8():1 4:W:activate#(n__s(X)) -> c_4(s#(activate(X))) -->_1 s#(X) -> c_9():2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 4: activate#(n__s(X)) -> c_4(s#(activate(X))) 2: s#(X) -> c_9() **** Step 5.b:2.b:2.a:2: Trivial WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: from#(X) -> c_8() - Weak DPs: activate#(n__from(X)) -> c_3(from#(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() 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_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,first#,from#,s#} and constructors {0,cons ,n__first,n__from,n__s,nil} + Applied Processor: Trivial + Details: Consider the dependency graph 1:S:from#(X) -> c_8() 3:W:activate#(n__from(X)) -> c_3(from#(activate(X))) -->_1 from#(X) -> c_8():1 The dependency graph contains no loops, we remove all dependency pairs. **** Step 5.b:2.b:2.a:3: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - 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() 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_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0} - 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). **** Step 5.b:2.b:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: s#(X) -> c_9() - Weak DPs: activate#(n__from(X)) -> c_3(from#(activate(X))) activate#(n__s(X)) -> c_4(s#(activate(X))) from#(X) -> c_8() - 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() 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_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0} - 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:s#(X) -> c_9() 2:W:activate#(n__from(X)) -> c_3(from#(activate(X))) -->_1 from#(X) -> c_8():4 3:W:activate#(n__s(X)) -> c_4(s#(activate(X))) -->_1 s#(X) -> c_9():1 4:W:from#(X) -> c_8() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: activate#(n__from(X)) -> c_3(from#(activate(X))) 4: from#(X) -> c_8() **** Step 5.b:2.b:2.b:2: Trivial WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: s#(X) -> c_9() - Weak DPs: activate#(n__s(X)) -> c_4(s#(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() 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_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,first#,from#,s#} and constructors {0,cons ,n__first,n__from,n__s,nil} + Applied Processor: Trivial + Details: Consider the dependency graph 1:S:s#(X) -> c_9() 3:W:activate#(n__s(X)) -> c_4(s#(activate(X))) -->_1 s#(X) -> c_9():1 The dependency graph contains no loops, we remove all dependency pairs. **** Step 5.b:2.b:2.b:3: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - 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() 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_2/1,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0} - 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))