MAYBE * Step 1: DependencyPairs MAYBE + Considered Problem: - Strict TRS: activate(X) -> X activate(n__take(X1,X2)) -> take(X1,X2) activate(n__zeros()) -> zeros() and(tt(),X) -> activate(X) length(cons(N,L)) -> s(length(activate(L))) length(nil()) -> 0() take(X1,X2) -> n__take(X1,X2) take(0(),IL) -> nil() take(s(M),cons(N,IL)) -> cons(N,n__take(M,activate(IL))) zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() - Signature: {activate/1,and/2,length/1,take/2,zeros/0} / {0/0,cons/2,n__take/2,n__zeros/0,nil/0,s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate,and,length,take,zeros} and constructors {0,cons ,n__take,n__zeros,nil,s,tt} + Applied Processor: DependencyPairs {dpKind_ = WIDP} + Details: We add the following weak innermost dependency pairs: Strict DPs activate#(X) -> c_1() activate#(n__take(X1,X2)) -> c_2(take#(X1,X2)) activate#(n__zeros()) -> c_3(zeros#()) and#(tt(),X) -> c_4(activate#(X)) length#(cons(N,L)) -> c_5(length#(activate(L))) length#(nil()) -> c_6() take#(X1,X2) -> c_7() take#(0(),IL) -> c_8() take#(s(M),cons(N,IL)) -> c_9(activate#(IL)) zeros#() -> c_10() zeros#() -> c_11() Weak DPs and mark the set of starting terms. * Step 2: UsableRules MAYBE + Considered Problem: - Strict DPs: activate#(X) -> c_1() activate#(n__take(X1,X2)) -> c_2(take#(X1,X2)) activate#(n__zeros()) -> c_3(zeros#()) and#(tt(),X) -> c_4(activate#(X)) length#(cons(N,L)) -> c_5(length#(activate(L))) length#(nil()) -> c_6() take#(X1,X2) -> c_7() take#(0(),IL) -> c_8() take#(s(M),cons(N,IL)) -> c_9(activate#(IL)) zeros#() -> c_10() zeros#() -> c_11() - Strict TRS: activate(X) -> X activate(n__take(X1,X2)) -> take(X1,X2) activate(n__zeros()) -> zeros() and(tt(),X) -> activate(X) length(cons(N,L)) -> s(length(activate(L))) length(nil()) -> 0() take(X1,X2) -> n__take(X1,X2) take(0(),IL) -> nil() take(s(M),cons(N,IL)) -> cons(N,n__take(M,activate(IL))) zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() - Signature: {activate/1,and/2,length/1,take/2,zeros/0,activate#/1,and#/2,length#/1,take#/2,zeros#/0} / {0/0,cons/2 ,n__take/2,n__zeros/0,nil/0,s/1,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,and#,length#,take#,zeros#} and constructors {0 ,cons,n__take,n__zeros,nil,s,tt} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: activate(X) -> X activate(n__take(X1,X2)) -> take(X1,X2) activate(n__zeros()) -> zeros() take(X1,X2) -> n__take(X1,X2) take(0(),IL) -> nil() take(s(M),cons(N,IL)) -> cons(N,n__take(M,activate(IL))) zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() activate#(X) -> c_1() activate#(n__take(X1,X2)) -> c_2(take#(X1,X2)) activate#(n__zeros()) -> c_3(zeros#()) and#(tt(),X) -> c_4(activate#(X)) length#(cons(N,L)) -> c_5(length#(activate(L))) length#(nil()) -> c_6() take#(X1,X2) -> c_7() take#(0(),IL) -> c_8() take#(s(M),cons(N,IL)) -> c_9(activate#(IL)) zeros#() -> c_10() zeros#() -> c_11() * Step 3: WeightGap MAYBE + Considered Problem: - Strict DPs: activate#(X) -> c_1() activate#(n__take(X1,X2)) -> c_2(take#(X1,X2)) activate#(n__zeros()) -> c_3(zeros#()) and#(tt(),X) -> c_4(activate#(X)) length#(cons(N,L)) -> c_5(length#(activate(L))) length#(nil()) -> c_6() take#(X1,X2) -> c_7() take#(0(),IL) -> c_8() take#(s(M),cons(N,IL)) -> c_9(activate#(IL)) zeros#() -> c_10() zeros#() -> c_11() - Strict TRS: activate(X) -> X activate(n__take(X1,X2)) -> take(X1,X2) activate(n__zeros()) -> zeros() take(X1,X2) -> n__take(X1,X2) take(0(),IL) -> nil() take(s(M),cons(N,IL)) -> cons(N,n__take(M,activate(IL))) zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() - Signature: {activate/1,and/2,length/1,take/2,zeros/0,activate#/1,and#/2,length#/1,take#/2,zeros#/0} / {0/0,cons/2 ,n__take/2,n__zeros/0,nil/0,s/1,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,and#,length#,take#,zeros#} and constructors {0 ,cons,n__take,n__zeros,nil,s,tt} + 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(cons) = {2}, uargs(n__take) = {2}, uargs(length#) = {1}, uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1}, uargs(c_9) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1] p(activate) = [1] x1 + [3] p(and) = [1] x1 + [1] x2 + [4] p(cons) = [1] x2 + [1] p(length) = [1] x1 + [2] p(n__take) = [1] x1 + [1] x2 + [4] p(n__zeros) = [0] p(nil) = [0] p(s) = [1] x1 + [8] p(take) = [1] x1 + [1] x2 + [6] p(tt) = [0] p(zeros) = [2] p(activate#) = [0] p(and#) = [0] p(length#) = [1] x1 + [0] p(take#) = [0] p(zeros#) = [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) = [1] x1 + [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [1] x1 + [0] p(c_10) = [0] p(c_11) = [0] Following rules are strictly oriented: activate(X) = [1] X + [3] > [1] X + [0] = X activate(n__take(X1,X2)) = [1] X1 + [1] X2 + [7] > [1] X1 + [1] X2 + [6] = take(X1,X2) activate(n__zeros()) = [3] > [2] = zeros() take(X1,X2) = [1] X1 + [1] X2 + [6] > [1] X1 + [1] X2 + [4] = n__take(X1,X2) take(0(),IL) = [1] IL + [7] > [0] = nil() take(s(M),cons(N,IL)) = [1] IL + [1] M + [15] > [1] IL + [1] M + [8] = cons(N,n__take(M,activate(IL))) zeros() = [2] > [1] = cons(0(),n__zeros()) zeros() = [2] > [0] = n__zeros() Following rules are (at-least) weakly oriented: activate#(X) = [0] >= [0] = c_1() activate#(n__take(X1,X2)) = [0] >= [0] = c_2(take#(X1,X2)) activate#(n__zeros()) = [0] >= [0] = c_3(zeros#()) and#(tt(),X) = [0] >= [0] = c_4(activate#(X)) length#(cons(N,L)) = [1] L + [1] >= [1] L + [3] = c_5(length#(activate(L))) length#(nil()) = [0] >= [0] = c_6() take#(X1,X2) = [0] >= [0] = c_7() take#(0(),IL) = [0] >= [0] = c_8() take#(s(M),cons(N,IL)) = [0] >= [0] = c_9(activate#(IL)) zeros#() = [0] >= [0] = c_10() zeros#() = [0] >= [0] = c_11() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: activate#(X) -> c_1() activate#(n__take(X1,X2)) -> c_2(take#(X1,X2)) activate#(n__zeros()) -> c_3(zeros#()) and#(tt(),X) -> c_4(activate#(X)) length#(cons(N,L)) -> c_5(length#(activate(L))) length#(nil()) -> c_6() take#(X1,X2) -> c_7() take#(0(),IL) -> c_8() take#(s(M),cons(N,IL)) -> c_9(activate#(IL)) zeros#() -> c_10() zeros#() -> c_11() - Weak TRS: activate(X) -> X activate(n__take(X1,X2)) -> take(X1,X2) activate(n__zeros()) -> zeros() take(X1,X2) -> n__take(X1,X2) take(0(),IL) -> nil() take(s(M),cons(N,IL)) -> cons(N,n__take(M,activate(IL))) zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() - Signature: {activate/1,and/2,length/1,take/2,zeros/0,activate#/1,and#/2,length#/1,take#/2,zeros#/0} / {0/0,cons/2 ,n__take/2,n__zeros/0,nil/0,s/1,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,and#,length#,take#,zeros#} and constructors {0 ,cons,n__take,n__zeros,nil,s,tt} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,6,7,8,10,11} by application of Pre({1,6,7,8,10,11}) = {2,3,4,5,9}. Here rules are labelled as follows: 1: activate#(X) -> c_1() 2: activate#(n__take(X1,X2)) -> c_2(take#(X1,X2)) 3: activate#(n__zeros()) -> c_3(zeros#()) 4: and#(tt(),X) -> c_4(activate#(X)) 5: length#(cons(N,L)) -> c_5(length#(activate(L))) 6: length#(nil()) -> c_6() 7: take#(X1,X2) -> c_7() 8: take#(0(),IL) -> c_8() 9: take#(s(M),cons(N,IL)) -> c_9(activate#(IL)) 10: zeros#() -> c_10() 11: zeros#() -> c_11() * Step 5: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: activate#(n__take(X1,X2)) -> c_2(take#(X1,X2)) activate#(n__zeros()) -> c_3(zeros#()) and#(tt(),X) -> c_4(activate#(X)) length#(cons(N,L)) -> c_5(length#(activate(L))) take#(s(M),cons(N,IL)) -> c_9(activate#(IL)) - Weak DPs: activate#(X) -> c_1() length#(nil()) -> c_6() take#(X1,X2) -> c_7() take#(0(),IL) -> c_8() zeros#() -> c_10() zeros#() -> c_11() - Weak TRS: activate(X) -> X activate(n__take(X1,X2)) -> take(X1,X2) activate(n__zeros()) -> zeros() take(X1,X2) -> n__take(X1,X2) take(0(),IL) -> nil() take(s(M),cons(N,IL)) -> cons(N,n__take(M,activate(IL))) zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() - Signature: {activate/1,and/2,length/1,take/2,zeros/0,activate#/1,and#/2,length#/1,take#/2,zeros#/0} / {0/0,cons/2 ,n__take/2,n__zeros/0,nil/0,s/1,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,and#,length#,take#,zeros#} and constructors {0 ,cons,n__take,n__zeros,nil,s,tt} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {2} by application of Pre({2}) = {3,5}. Here rules are labelled as follows: 1: activate#(n__take(X1,X2)) -> c_2(take#(X1,X2)) 2: activate#(n__zeros()) -> c_3(zeros#()) 3: and#(tt(),X) -> c_4(activate#(X)) 4: length#(cons(N,L)) -> c_5(length#(activate(L))) 5: take#(s(M),cons(N,IL)) -> c_9(activate#(IL)) 6: activate#(X) -> c_1() 7: length#(nil()) -> c_6() 8: take#(X1,X2) -> c_7() 9: take#(0(),IL) -> c_8() 10: zeros#() -> c_10() 11: zeros#() -> c_11() * Step 6: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: activate#(n__take(X1,X2)) -> c_2(take#(X1,X2)) and#(tt(),X) -> c_4(activate#(X)) length#(cons(N,L)) -> c_5(length#(activate(L))) take#(s(M),cons(N,IL)) -> c_9(activate#(IL)) - Weak DPs: activate#(X) -> c_1() activate#(n__zeros()) -> c_3(zeros#()) length#(nil()) -> c_6() take#(X1,X2) -> c_7() take#(0(),IL) -> c_8() zeros#() -> c_10() zeros#() -> c_11() - Weak TRS: activate(X) -> X activate(n__take(X1,X2)) -> take(X1,X2) activate(n__zeros()) -> zeros() take(X1,X2) -> n__take(X1,X2) take(0(),IL) -> nil() take(s(M),cons(N,IL)) -> cons(N,n__take(M,activate(IL))) zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() - Signature: {activate/1,and/2,length/1,take/2,zeros/0,activate#/1,and#/2,length#/1,take#/2,zeros#/0} / {0/0,cons/2 ,n__take/2,n__zeros/0,nil/0,s/1,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,and#,length#,take#,zeros#} and constructors {0 ,cons,n__take,n__zeros,nil,s,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:activate#(n__take(X1,X2)) -> c_2(take#(X1,X2)) -->_1 take#(s(M),cons(N,IL)) -> c_9(activate#(IL)):4 -->_1 take#(0(),IL) -> c_8():9 -->_1 take#(X1,X2) -> c_7():8 2:S:and#(tt(),X) -> c_4(activate#(X)) -->_1 activate#(n__zeros()) -> c_3(zeros#()):6 -->_1 activate#(X) -> c_1():5 -->_1 activate#(n__take(X1,X2)) -> c_2(take#(X1,X2)):1 3:S:length#(cons(N,L)) -> c_5(length#(activate(L))) -->_1 length#(nil()) -> c_6():7 -->_1 length#(cons(N,L)) -> c_5(length#(activate(L))):3 4:S:take#(s(M),cons(N,IL)) -> c_9(activate#(IL)) -->_1 activate#(n__zeros()) -> c_3(zeros#()):6 -->_1 activate#(X) -> c_1():5 -->_1 activate#(n__take(X1,X2)) -> c_2(take#(X1,X2)):1 5:W:activate#(X) -> c_1() 6:W:activate#(n__zeros()) -> c_3(zeros#()) -->_1 zeros#() -> c_11():11 -->_1 zeros#() -> c_10():10 7:W:length#(nil()) -> c_6() 8:W:take#(X1,X2) -> c_7() 9:W:take#(0(),IL) -> c_8() 10:W:zeros#() -> c_10() 11:W:zeros#() -> c_11() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 7: length#(nil()) -> c_6() 8: take#(X1,X2) -> c_7() 9: take#(0(),IL) -> c_8() 5: activate#(X) -> c_1() 6: activate#(n__zeros()) -> c_3(zeros#()) 10: zeros#() -> c_10() 11: zeros#() -> c_11() * Step 7: RemoveHeads MAYBE + Considered Problem: - Strict DPs: activate#(n__take(X1,X2)) -> c_2(take#(X1,X2)) and#(tt(),X) -> c_4(activate#(X)) length#(cons(N,L)) -> c_5(length#(activate(L))) take#(s(M),cons(N,IL)) -> c_9(activate#(IL)) - Weak TRS: activate(X) -> X activate(n__take(X1,X2)) -> take(X1,X2) activate(n__zeros()) -> zeros() take(X1,X2) -> n__take(X1,X2) take(0(),IL) -> nil() take(s(M),cons(N,IL)) -> cons(N,n__take(M,activate(IL))) zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() - Signature: {activate/1,and/2,length/1,take/2,zeros/0,activate#/1,and#/2,length#/1,take#/2,zeros#/0} / {0/0,cons/2 ,n__take/2,n__zeros/0,nil/0,s/1,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,and#,length#,take#,zeros#} and constructors {0 ,cons,n__take,n__zeros,nil,s,tt} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:activate#(n__take(X1,X2)) -> c_2(take#(X1,X2)) -->_1 take#(s(M),cons(N,IL)) -> c_9(activate#(IL)):4 2:S:and#(tt(),X) -> c_4(activate#(X)) -->_1 activate#(n__take(X1,X2)) -> c_2(take#(X1,X2)):1 3:S:length#(cons(N,L)) -> c_5(length#(activate(L))) -->_1 length#(cons(N,L)) -> c_5(length#(activate(L))):3 4:S:take#(s(M),cons(N,IL)) -> c_9(activate#(IL)) -->_1 activate#(n__take(X1,X2)) -> c_2(take#(X1,X2)):1 Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts). [(2,and#(tt(),X) -> c_4(activate#(X)))] * Step 8: Decompose MAYBE + Considered Problem: - Strict DPs: activate#(n__take(X1,X2)) -> c_2(take#(X1,X2)) length#(cons(N,L)) -> c_5(length#(activate(L))) take#(s(M),cons(N,IL)) -> c_9(activate#(IL)) - Weak TRS: activate(X) -> X activate(n__take(X1,X2)) -> take(X1,X2) activate(n__zeros()) -> zeros() take(X1,X2) -> n__take(X1,X2) take(0(),IL) -> nil() take(s(M),cons(N,IL)) -> cons(N,n__take(M,activate(IL))) zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() - Signature: {activate/1,and/2,length/1,take/2,zeros/0,activate#/1,and#/2,length#/1,take#/2,zeros#/0} / {0/0,cons/2 ,n__take/2,n__zeros/0,nil/0,s/1,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,and#,length#,take#,zeros#} and constructors {0 ,cons,n__take,n__zeros,nil,s,tt} + 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: activate#(n__take(X1,X2)) -> c_2(take#(X1,X2)) take#(s(M),cons(N,IL)) -> c_9(activate#(IL)) - Weak DPs: length#(cons(N,L)) -> c_5(length#(activate(L))) - Weak TRS: activate(X) -> X activate(n__take(X1,X2)) -> take(X1,X2) activate(n__zeros()) -> zeros() take(X1,X2) -> n__take(X1,X2) take(0(),IL) -> nil() take(s(M),cons(N,IL)) -> cons(N,n__take(M,activate(IL))) zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() - Signature: {activate/1,and/2,length/1,take/2,zeros/0,activate#/1,and#/2,length#/1,take#/2,zeros#/0} / {0/0,cons/2 ,n__take/2,n__zeros/0,nil/0,s/1,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,and#,length#,take# ,zeros#} and constructors {0,cons,n__take,n__zeros,nil,s,tt} Problem (S) - Strict DPs: length#(cons(N,L)) -> c_5(length#(activate(L))) - Weak DPs: activate#(n__take(X1,X2)) -> c_2(take#(X1,X2)) take#(s(M),cons(N,IL)) -> c_9(activate#(IL)) - Weak TRS: activate(X) -> X activate(n__take(X1,X2)) -> take(X1,X2) activate(n__zeros()) -> zeros() take(X1,X2) -> n__take(X1,X2) take(0(),IL) -> nil() take(s(M),cons(N,IL)) -> cons(N,n__take(M,activate(IL))) zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() - Signature: {activate/1,and/2,length/1,take/2,zeros/0,activate#/1,and#/2,length#/1,take#/2,zeros#/0} / {0/0,cons/2 ,n__take/2,n__zeros/0,nil/0,s/1,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,and#,length#,take# ,zeros#} and constructors {0,cons,n__take,n__zeros,nil,s,tt} ** Step 8.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__take(X1,X2)) -> c_2(take#(X1,X2)) take#(s(M),cons(N,IL)) -> c_9(activate#(IL)) - Weak DPs: length#(cons(N,L)) -> c_5(length#(activate(L))) - Weak TRS: activate(X) -> X activate(n__take(X1,X2)) -> take(X1,X2) activate(n__zeros()) -> zeros() take(X1,X2) -> n__take(X1,X2) take(0(),IL) -> nil() take(s(M),cons(N,IL)) -> cons(N,n__take(M,activate(IL))) zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() - Signature: {activate/1,and/2,length/1,take/2,zeros/0,activate#/1,and#/2,length#/1,take#/2,zeros#/0} / {0/0,cons/2 ,n__take/2,n__zeros/0,nil/0,s/1,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,and#,length#,take#,zeros#} and constructors {0 ,cons,n__take,n__zeros,nil,s,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:activate#(n__take(X1,X2)) -> c_2(take#(X1,X2)) -->_1 take#(s(M),cons(N,IL)) -> c_9(activate#(IL)):4 3:W:length#(cons(N,L)) -> c_5(length#(activate(L))) -->_1 length#(cons(N,L)) -> c_5(length#(activate(L))):3 4:S:take#(s(M),cons(N,IL)) -> c_9(activate#(IL)) -->_1 activate#(n__take(X1,X2)) -> c_2(take#(X1,X2)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: length#(cons(N,L)) -> c_5(length#(activate(L))) ** Step 8.a:2: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__take(X1,X2)) -> c_2(take#(X1,X2)) take#(s(M),cons(N,IL)) -> c_9(activate#(IL)) - Weak TRS: activate(X) -> X activate(n__take(X1,X2)) -> take(X1,X2) activate(n__zeros()) -> zeros() take(X1,X2) -> n__take(X1,X2) take(0(),IL) -> nil() take(s(M),cons(N,IL)) -> cons(N,n__take(M,activate(IL))) zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() - Signature: {activate/1,and/2,length/1,take/2,zeros/0,activate#/1,and#/2,length#/1,take#/2,zeros#/0} / {0/0,cons/2 ,n__take/2,n__zeros/0,nil/0,s/1,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,and#,length#,take#,zeros#} and constructors {0 ,cons,n__take,n__zeros,nil,s,tt} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: activate#(n__take(X1,X2)) -> c_2(take#(X1,X2)) take#(s(M),cons(N,IL)) -> c_9(activate#(IL)) ** Step 8.a:3: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__take(X1,X2)) -> c_2(take#(X1,X2)) take#(s(M),cons(N,IL)) -> c_9(activate#(IL)) - Signature: {activate/1,and/2,length/1,take/2,zeros/0,activate#/1,and#/2,length#/1,take#/2,zeros#/0} / {0/0,cons/2 ,n__take/2,n__zeros/0,nil/0,s/1,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,and#,length#,take#,zeros#} and constructors {0 ,cons,n__take,n__zeros,nil,s,tt} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: activate#(n__take(X1,X2)) -> c_2(take#(X1,X2)) 2: take#(s(M),cons(N,IL)) -> c_9(activate#(IL)) The strictly oriented rules are moved into the weak component. *** Step 8.a:3.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__take(X1,X2)) -> c_2(take#(X1,X2)) take#(s(M),cons(N,IL)) -> c_9(activate#(IL)) - Signature: {activate/1,and/2,length/1,take/2,zeros/0,activate#/1,and#/2,length#/1,take#/2,zeros#/0} / {0/0,cons/2 ,n__take/2,n__zeros/0,nil/0,s/1,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,and#,length#,take#,zeros#} and constructors {0 ,cons,n__take,n__zeros,nil,s,tt} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_9) = {1} Following symbols are considered usable: {activate#,and#,length#,take#,zeros#} TcT has computed the following interpretation: p(0) = [0] p(activate) = [0] p(and) = [0] p(cons) = [1] x1 + [1] x2 + [0] p(length) = [1] x1 + [0] p(n__take) = [1] x1 + [1] x2 + [0] p(n__zeros) = [2] p(nil) = [1] p(s) = [1] x1 + [3] p(take) = [1] x1 + [0] p(tt) = [0] p(zeros) = [0] p(activate#) = [8] x1 + [9] p(and#) = [0] p(length#) = [2] x1 + [1] p(take#) = [8] x1 + [8] x2 + [0] p(zeros#) = [1] p(c_1) = [1] p(c_2) = [1] x1 + [1] p(c_3) = [1] x1 + [1] p(c_4) = [1] x1 + [0] p(c_5) = [8] p(c_6) = [0] p(c_7) = [0] p(c_8) = [4] p(c_9) = [1] x1 + [2] p(c_10) = [2] p(c_11) = [0] Following rules are strictly oriented: activate#(n__take(X1,X2)) = [8] X1 + [8] X2 + [9] > [8] X1 + [8] X2 + [1] = c_2(take#(X1,X2)) take#(s(M),cons(N,IL)) = [8] IL + [8] M + [8] N + [24] > [8] IL + [11] = c_9(activate#(IL)) Following rules are (at-least) weakly oriented: *** Step 8.a:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: activate#(n__take(X1,X2)) -> c_2(take#(X1,X2)) take#(s(M),cons(N,IL)) -> c_9(activate#(IL)) - Signature: {activate/1,and/2,length/1,take/2,zeros/0,activate#/1,and#/2,length#/1,take#/2,zeros#/0} / {0/0,cons/2 ,n__take/2,n__zeros/0,nil/0,s/1,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,and#,length#,take#,zeros#} and constructors {0 ,cons,n__take,n__zeros,nil,s,tt} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () *** Step 8.a:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: activate#(n__take(X1,X2)) -> c_2(take#(X1,X2)) take#(s(M),cons(N,IL)) -> c_9(activate#(IL)) - Signature: {activate/1,and/2,length/1,take/2,zeros/0,activate#/1,and#/2,length#/1,take#/2,zeros#/0} / {0/0,cons/2 ,n__take/2,n__zeros/0,nil/0,s/1,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,and#,length#,take#,zeros#} and constructors {0 ,cons,n__take,n__zeros,nil,s,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:activate#(n__take(X1,X2)) -> c_2(take#(X1,X2)) -->_1 take#(s(M),cons(N,IL)) -> c_9(activate#(IL)):2 2:W:take#(s(M),cons(N,IL)) -> c_9(activate#(IL)) -->_1 activate#(n__take(X1,X2)) -> c_2(take#(X1,X2)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: activate#(n__take(X1,X2)) -> c_2(take#(X1,X2)) 2: take#(s(M),cons(N,IL)) -> c_9(activate#(IL)) *** Step 8.a:3.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Signature: {activate/1,and/2,length/1,take/2,zeros/0,activate#/1,and#/2,length#/1,take#/2,zeros#/0} / {0/0,cons/2 ,n__take/2,n__zeros/0,nil/0,s/1,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,and#,length#,take#,zeros#} and constructors {0 ,cons,n__take,n__zeros,nil,s,tt} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 8.b:1: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: length#(cons(N,L)) -> c_5(length#(activate(L))) - Weak DPs: activate#(n__take(X1,X2)) -> c_2(take#(X1,X2)) take#(s(M),cons(N,IL)) -> c_9(activate#(IL)) - Weak TRS: activate(X) -> X activate(n__take(X1,X2)) -> take(X1,X2) activate(n__zeros()) -> zeros() take(X1,X2) -> n__take(X1,X2) take(0(),IL) -> nil() take(s(M),cons(N,IL)) -> cons(N,n__take(M,activate(IL))) zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() - Signature: {activate/1,and/2,length/1,take/2,zeros/0,activate#/1,and#/2,length#/1,take#/2,zeros#/0} / {0/0,cons/2 ,n__take/2,n__zeros/0,nil/0,s/1,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,and#,length#,take#,zeros#} and constructors {0 ,cons,n__take,n__zeros,nil,s,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:length#(cons(N,L)) -> c_5(length#(activate(L))) -->_1 length#(cons(N,L)) -> c_5(length#(activate(L))):1 2:W:activate#(n__take(X1,X2)) -> c_2(take#(X1,X2)) -->_1 take#(s(M),cons(N,IL)) -> c_9(activate#(IL)):3 3:W:take#(s(M),cons(N,IL)) -> c_9(activate#(IL)) -->_1 activate#(n__take(X1,X2)) -> c_2(take#(X1,X2)):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: activate#(n__take(X1,X2)) -> c_2(take#(X1,X2)) 3: take#(s(M),cons(N,IL)) -> c_9(activate#(IL)) ** Step 8.b:2: Failure MAYBE + Considered Problem: - Strict DPs: length#(cons(N,L)) -> c_5(length#(activate(L))) - Weak TRS: activate(X) -> X activate(n__take(X1,X2)) -> take(X1,X2) activate(n__zeros()) -> zeros() take(X1,X2) -> n__take(X1,X2) take(0(),IL) -> nil() take(s(M),cons(N,IL)) -> cons(N,n__take(M,activate(IL))) zeros() -> cons(0(),n__zeros()) zeros() -> n__zeros() - Signature: {activate/1,and/2,length/1,take/2,zeros/0,activate#/1,and#/2,length#/1,take#/2,zeros#/0} / {0/0,cons/2 ,n__take/2,n__zeros/0,nil/0,s/1,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,and#,length#,take#,zeros#} and constructors {0 ,cons,n__take,n__zeros,nil,s,tt} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE