WORST_CASE(?,O(n^1)) * Step 1: DependencyPairs WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: 2nd(cons(X,XS)) -> head(activate(XS)) activate(X) -> X activate(n__from(X)) -> from(X) activate(n__take(X1,X2)) -> take(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) head(cons(X,XS)) -> X sel(0(),cons(X,XS)) -> X sel(s(N),cons(X,XS)) -> sel(N,activate(XS)) take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2} / {0/0,cons/2,n__from/1,n__take/2,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd,activate,from,head,sel,take} and constructors {0,cons ,n__from,n__take,nil,s} + Applied Processor: DependencyPairs {dpKind_ = WIDP} + Details: We add the following weak innermost dependency pairs: Strict DPs 2nd#(cons(X,XS)) -> c_1(head#(activate(XS))) activate#(X) -> c_2() activate#(n__from(X)) -> c_3(from#(X)) activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) from#(X) -> c_5() from#(X) -> c_6() head#(cons(X,XS)) -> c_7() sel#(0(),cons(X,XS)) -> c_8() sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) take#(X1,X2) -> c_10() take#(0(),XS) -> c_11() take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) Weak DPs and mark the set of starting terms. * Step 2: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: 2nd#(cons(X,XS)) -> c_1(head#(activate(XS))) activate#(X) -> c_2() activate#(n__from(X)) -> c_3(from#(X)) activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) from#(X) -> c_5() from#(X) -> c_6() head#(cons(X,XS)) -> c_7() sel#(0(),cons(X,XS)) -> c_8() sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) take#(X1,X2) -> c_10() take#(0(),XS) -> c_11() take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) - Strict TRS: 2nd(cons(X,XS)) -> head(activate(XS)) activate(X) -> X activate(n__from(X)) -> from(X) activate(n__take(X1,X2)) -> take(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) head(cons(X,XS)) -> X sel(0(),cons(X,XS)) -> X sel(s(N),cons(X,XS)) -> sel(N,activate(XS)) take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0 ,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel# ,take#} and constructors {0,cons,n__from,n__take,nil,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: activate(X) -> X activate(n__from(X)) -> from(X) activate(n__take(X1,X2)) -> take(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) 2nd#(cons(X,XS)) -> c_1(head#(activate(XS))) activate#(X) -> c_2() activate#(n__from(X)) -> c_3(from#(X)) activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) from#(X) -> c_5() from#(X) -> c_6() head#(cons(X,XS)) -> c_7() sel#(0(),cons(X,XS)) -> c_8() sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) take#(X1,X2) -> c_10() take#(0(),XS) -> c_11() take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) * Step 3: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: 2nd#(cons(X,XS)) -> c_1(head#(activate(XS))) activate#(X) -> c_2() activate#(n__from(X)) -> c_3(from#(X)) activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) from#(X) -> c_5() from#(X) -> c_6() head#(cons(X,XS)) -> c_7() sel#(0(),cons(X,XS)) -> c_8() sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) take#(X1,X2) -> c_10() take#(0(),XS) -> c_11() take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) - Strict TRS: activate(X) -> X activate(n__from(X)) -> from(X) activate(n__take(X1,X2)) -> take(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0 ,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel# ,take#} and constructors {0,cons,n__from,n__take,nil,s} + 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(head#) = {1}, uargs(sel#) = {2}, uargs(c_1) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_9) = {1}, uargs(c_12) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(2nd) = [0] p(activate) = [1] x1 + [5] p(cons) = [1] x2 + [0] p(from) = [3] p(head) = [0] p(n__from) = [2] p(n__take) = [1] x1 + [1] x2 + [0] p(nil) = [0] p(s) = [1] x1 + [5] p(sel) = [1] x1 + [0] p(take) = [1] x1 + [1] x2 + [3] p(2nd#) = [1] x1 + [0] p(activate#) = [0] p(from#) = [0] p(head#) = [1] x1 + [0] p(sel#) = [1] x1 + [1] x2 + [5] p(take#) = [1] p(c_1) = [1] x1 + [0] p(c_2) = [0] p(c_3) = [1] x1 + [3] 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] p(c_10) = [0] p(c_11) = [0] p(c_12) = [1] x1 + [0] Following rules are strictly oriented: sel#(0(),cons(X,XS)) = [1] XS + [5] > [0] = c_8() take#(X1,X2) = [1] > [0] = c_10() take#(0(),XS) = [1] > [0] = c_11() take#(s(N),cons(X,XS)) = [1] > [0] = c_12(activate#(XS)) activate(X) = [1] X + [5] > [1] X + [0] = X activate(n__from(X)) = [7] > [3] = from(X) activate(n__take(X1,X2)) = [1] X1 + [1] X2 + [5] > [1] X1 + [1] X2 + [3] = take(X1,X2) from(X) = [3] > [2] = cons(X,n__from(s(X))) from(X) = [3] > [2] = n__from(X) take(X1,X2) = [1] X1 + [1] X2 + [3] > [1] X1 + [1] X2 + [0] = n__take(X1,X2) take(0(),XS) = [1] XS + [3] > [0] = nil() take(s(N),cons(X,XS)) = [1] N + [1] XS + [8] > [1] N + [1] XS + [5] = cons(X,n__take(N,activate(XS))) Following rules are (at-least) weakly oriented: 2nd#(cons(X,XS)) = [1] XS + [0] >= [1] XS + [5] = c_1(head#(activate(XS))) activate#(X) = [0] >= [0] = c_2() activate#(n__from(X)) = [0] >= [3] = c_3(from#(X)) activate#(n__take(X1,X2)) = [0] >= [1] = c_4(take#(X1,X2)) from#(X) = [0] >= [0] = c_5() from#(X) = [0] >= [0] = c_6() head#(cons(X,XS)) = [1] XS + [0] >= [0] = c_7() sel#(s(N),cons(X,XS)) = [1] N + [1] XS + [10] >= [1] N + [1] XS + [10] = c_9(sel#(N,activate(XS))) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: 2nd#(cons(X,XS)) -> c_1(head#(activate(XS))) activate#(X) -> c_2() activate#(n__from(X)) -> c_3(from#(X)) activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) from#(X) -> c_5() from#(X) -> c_6() head#(cons(X,XS)) -> c_7() sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) - Weak DPs: sel#(0(),cons(X,XS)) -> c_8() take#(X1,X2) -> c_10() take#(0(),XS) -> c_11() take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) activate(n__take(X1,X2)) -> take(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0 ,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel# ,take#} and constructors {0,cons,n__from,n__take,nil,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {5,6,7} by application of Pre({5,6,7}) = {1,3}. Here rules are labelled as follows: 1: 2nd#(cons(X,XS)) -> c_1(head#(activate(XS))) 2: activate#(X) -> c_2() 3: activate#(n__from(X)) -> c_3(from#(X)) 4: activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) 5: from#(X) -> c_5() 6: from#(X) -> c_6() 7: head#(cons(X,XS)) -> c_7() 8: sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) 9: sel#(0(),cons(X,XS)) -> c_8() 10: take#(X1,X2) -> c_10() 11: take#(0(),XS) -> c_11() 12: take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) * Step 5: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: 2nd#(cons(X,XS)) -> c_1(head#(activate(XS))) activate#(X) -> c_2() activate#(n__from(X)) -> c_3(from#(X)) activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) - Weak DPs: from#(X) -> c_5() from#(X) -> c_6() head#(cons(X,XS)) -> c_7() sel#(0(),cons(X,XS)) -> c_8() take#(X1,X2) -> c_10() take#(0(),XS) -> c_11() take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) activate(n__take(X1,X2)) -> take(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0 ,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel# ,take#} and constructors {0,cons,n__from,n__take,nil,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1} by application of Pre({1}) = {}. Here rules are labelled as follows: 1: 2nd#(cons(X,XS)) -> c_1(head#(activate(XS))) 2: activate#(X) -> c_2() 3: activate#(n__from(X)) -> c_3(from#(X)) 4: activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) 5: sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) 6: from#(X) -> c_5() 7: from#(X) -> c_6() 8: head#(cons(X,XS)) -> c_7() 9: sel#(0(),cons(X,XS)) -> c_8() 10: take#(X1,X2) -> c_10() 11: take#(0(),XS) -> c_11() 12: take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) * Step 6: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(X) -> c_2() activate#(n__from(X)) -> c_3(from#(X)) activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) - Weak DPs: 2nd#(cons(X,XS)) -> c_1(head#(activate(XS))) from#(X) -> c_5() from#(X) -> c_6() head#(cons(X,XS)) -> c_7() sel#(0(),cons(X,XS)) -> c_8() take#(X1,X2) -> c_10() take#(0(),XS) -> c_11() take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) activate(n__take(X1,X2)) -> take(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0 ,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel# ,take#} and constructors {0,cons,n__from,n__take,nil,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:activate#(X) -> c_2() 2:S:activate#(n__from(X)) -> c_3(from#(X)) -->_1 from#(X) -> c_6():7 -->_1 from#(X) -> c_5():6 3:S:activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) -->_1 take#(s(N),cons(X,XS)) -> c_12(activate#(XS)):12 -->_1 take#(0(),XS) -> c_11():11 -->_1 take#(X1,X2) -> c_10():10 4:S:sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) -->_1 sel#(0(),cons(X,XS)) -> c_8():9 -->_1 sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))):4 5:W:2nd#(cons(X,XS)) -> c_1(head#(activate(XS))) -->_1 head#(cons(X,XS)) -> c_7():8 6:W:from#(X) -> c_5() 7:W:from#(X) -> c_6() 8:W:head#(cons(X,XS)) -> c_7() 9:W:sel#(0(),cons(X,XS)) -> c_8() 10:W:take#(X1,X2) -> c_10() 11:W:take#(0(),XS) -> c_11() 12:W:take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) -->_1 activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)):3 -->_1 activate#(n__from(X)) -> c_3(from#(X)):2 -->_1 activate#(X) -> c_2():1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 5: 2nd#(cons(X,XS)) -> c_1(head#(activate(XS))) 8: head#(cons(X,XS)) -> c_7() 9: sel#(0(),cons(X,XS)) -> c_8() 10: take#(X1,X2) -> c_10() 11: take#(0(),XS) -> c_11() 6: from#(X) -> c_5() 7: from#(X) -> c_6() * Step 7: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(X) -> c_2() activate#(n__from(X)) -> c_3(from#(X)) activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) - Weak DPs: take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) activate(n__take(X1,X2)) -> take(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0 ,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel# ,take#} and constructors {0,cons,n__from,n__take,nil,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:activate#(X) -> c_2() 2:S:activate#(n__from(X)) -> c_3(from#(X)) 3:S:activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) -->_1 take#(s(N),cons(X,XS)) -> c_12(activate#(XS)):12 4:S:sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) -->_1 sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))):4 12:W:take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) -->_1 activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)):3 -->_1 activate#(n__from(X)) -> c_3(from#(X)):2 -->_1 activate#(X) -> c_2():1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: activate#(n__from(X)) -> c_3() * Step 8: Decompose WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(X) -> c_2() activate#(n__from(X)) -> c_3() activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) - Weak DPs: take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) activate(n__take(X1,X2)) -> take(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0 ,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel# ,take#} and constructors {0,cons,n__from,n__take,nil,s} + 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#(X) -> c_2() - Weak DPs: activate#(n__from(X)) -> c_3() activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) activate(n__take(X1,X2)) -> take(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0 ,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel# ,take#} and constructors {0,cons,n__from,n__take,nil,s} Problem (S) - Strict DPs: activate#(n__from(X)) -> c_3() activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) - Weak DPs: activate#(X) -> c_2() take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) activate(n__take(X1,X2)) -> take(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0 ,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel# ,take#} and constructors {0,cons,n__from,n__take,nil,s} ** Step 8.a:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: activate#(X) -> c_2() - Weak DPs: activate#(n__from(X)) -> c_3() activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) activate(n__take(X1,X2)) -> take(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0 ,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel# ,take#} and constructors {0,cons,n__from,n__take,nil,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:activate#(X) -> c_2() 2:W:activate#(n__from(X)) -> c_3() 3:W:activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) -->_1 take#(s(N),cons(X,XS)) -> c_12(activate#(XS)):5 4:W:sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) -->_1 sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))):4 5:W:take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) -->_1 activate#(X) -> c_2():1 -->_1 activate#(n__from(X)) -> c_3():2 -->_1 activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)):3 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 4: sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) 2: activate#(n__from(X)) -> c_3() ** Step 8.a:2: UsableRules WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: activate#(X) -> c_2() - Weak DPs: activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) activate(n__take(X1,X2)) -> take(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0 ,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel# ,take#} and constructors {0,cons,n__from,n__take,nil,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: activate#(X) -> c_2() activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) ** Step 8.a:3: PredecessorEstimationCP WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: activate#(X) -> c_2() - Weak DPs: activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0 ,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel# ,take#} and constructors {0,cons,n__from,n__take,nil,s} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: activate#(X) -> c_2() The strictly oriented rules are moved into the weak component. *** Step 8.a:3.a:1: NaturalMI WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: activate#(X) -> c_2() - Weak DPs: activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0 ,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel# ,take#} and constructors {0,cons,n__from,n__take,nil,s} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 0, 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 (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_4) = {1}, uargs(c_12) = {1} Following symbols are considered usable: {2nd#,activate#,from#,head#,sel#,take#} TcT has computed the following interpretation: p(0) = [1] p(2nd) = [1] x1 + [8] p(activate) = [1] p(cons) = [0] p(from) = [0] p(head) = [1] x1 + [1] p(n__from) = [8] p(n__take) = [0] p(nil) = [8] p(s) = [0] p(sel) = [0] p(take) = [4] x1 + [2] x2 + [1] p(2nd#) = [1] p(activate#) = [3] p(from#) = [2] p(head#) = [2] p(sel#) = [1] x1 + [0] p(take#) = [3] p(c_1) = [2] p(c_2) = [2] p(c_3) = [2] p(c_4) = [1] x1 + [0] p(c_5) = [1] p(c_6) = [1] p(c_7) = [8] p(c_8) = [8] p(c_9) = [1] x1 + [0] p(c_10) = [2] p(c_11) = [4] p(c_12) = [1] x1 + [0] Following rules are strictly oriented: activate#(X) = [3] > [2] = c_2() Following rules are (at-least) weakly oriented: activate#(n__take(X1,X2)) = [3] >= [3] = c_4(take#(X1,X2)) take#(s(N),cons(X,XS)) = [3] >= [3] = c_12(activate#(XS)) *** Step 8.a:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: activate#(X) -> c_2() activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0 ,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel# ,take#} and constructors {0,cons,n__from,n__take,nil,s} + 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#(X) -> c_2() activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0 ,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel# ,take#} and constructors {0,cons,n__from,n__take,nil,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:activate#(X) -> c_2() 2:W:activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) -->_1 take#(s(N),cons(X,XS)) -> c_12(activate#(XS)):3 3:W:take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) -->_1 activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)):2 -->_1 activate#(X) -> c_2():1 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_4(take#(X1,X2)) 3: take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) 1: activate#(X) -> c_2() *** Step 8.a:3.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0 ,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel# ,take#} and constructors {0,cons,n__from,n__take,nil,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 8.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__from(X)) -> c_3() activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) - Weak DPs: activate#(X) -> c_2() take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) activate(n__take(X1,X2)) -> take(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0 ,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel# ,take#} and constructors {0,cons,n__from,n__take,nil,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:activate#(n__from(X)) -> c_3() 2:S:activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) -->_1 take#(s(N),cons(X,XS)) -> c_12(activate#(XS)):5 3:S:sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) -->_1 sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))):3 4:W:activate#(X) -> c_2() 5:W:take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) -->_1 activate#(X) -> c_2():4 -->_1 activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)):2 -->_1 activate#(n__from(X)) -> c_3():1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 4: activate#(X) -> c_2() ** Step 8.b:2: Decompose WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__from(X)) -> c_3() activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) - Weak DPs: take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) activate(n__take(X1,X2)) -> take(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0 ,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel# ,take#} and constructors {0,cons,n__from,n__take,nil,s} + 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__from(X)) -> c_3() - Weak DPs: activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) activate(n__take(X1,X2)) -> take(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0 ,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel# ,take#} and constructors {0,cons,n__from,n__take,nil,s} Problem (S) - Strict DPs: activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) - Weak DPs: activate#(n__from(X)) -> c_3() take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) activate(n__take(X1,X2)) -> take(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0 ,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel# ,take#} and constructors {0,cons,n__from,n__take,nil,s} *** Step 8.b:2.a:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: activate#(n__from(X)) -> c_3() - Weak DPs: activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) activate(n__take(X1,X2)) -> take(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0 ,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel# ,take#} and constructors {0,cons,n__from,n__take,nil,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:activate#(n__from(X)) -> c_3() 2:W:activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) -->_1 take#(s(N),cons(X,XS)) -> c_12(activate#(XS)):5 3:W:sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) -->_1 sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))):3 5:W:take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) -->_1 activate#(n__from(X)) -> c_3():1 -->_1 activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) *** Step 8.b:2.a:2: UsableRules WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: activate#(n__from(X)) -> c_3() - Weak DPs: activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) activate(n__take(X1,X2)) -> take(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0 ,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel# ,take#} and constructors {0,cons,n__from,n__take,nil,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: activate#(n__from(X)) -> c_3() activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) *** Step 8.b:2.a:3: PredecessorEstimationCP WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: activate#(n__from(X)) -> c_3() - Weak DPs: activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0 ,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel# ,take#} and constructors {0,cons,n__from,n__take,nil,s} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: activate#(n__from(X)) -> c_3() The strictly oriented rules are moved into the weak component. **** Step 8.b:2.a:3.a:1: NaturalMI WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: activate#(n__from(X)) -> c_3() - Weak DPs: activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0 ,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel# ,take#} and constructors {0,cons,n__from,n__take,nil,s} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 0, 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 (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_4) = {1}, uargs(c_12) = {1} Following symbols are considered usable: {2nd#,activate#,from#,head#,sel#,take#} TcT has computed the following interpretation: p(0) = [1] p(2nd) = [2] x1 + [1] p(activate) = [1] x1 + [0] p(cons) = [2] p(from) = [1] p(head) = [8] x1 + [8] p(n__from) = [1] p(n__take) = [0] p(nil) = [0] p(s) = [0] p(sel) = [1] x2 + [1] p(take) = [1] x2 + [0] p(2nd#) = [1] x1 + [1] p(activate#) = [1] p(from#) = [2] x1 + [0] p(head#) = [8] p(sel#) = [0] p(take#) = [1] p(c_1) = [1] x1 + [1] p(c_2) = [1] p(c_3) = [0] p(c_4) = [1] x1 + [0] p(c_5) = [8] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [2] p(c_11) = [0] p(c_12) = [1] x1 + [0] Following rules are strictly oriented: activate#(n__from(X)) = [1] > [0] = c_3() Following rules are (at-least) weakly oriented: activate#(n__take(X1,X2)) = [1] >= [1] = c_4(take#(X1,X2)) take#(s(N),cons(X,XS)) = [1] >= [1] = c_12(activate#(XS)) **** Step 8.b:2.a:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: activate#(n__from(X)) -> c_3() activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0 ,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel# ,take#} and constructors {0,cons,n__from,n__take,nil,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 8.b:2.a:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: activate#(n__from(X)) -> c_3() activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0 ,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel# ,take#} and constructors {0,cons,n__from,n__take,nil,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:activate#(n__from(X)) -> c_3() 2:W:activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) -->_1 take#(s(N),cons(X,XS)) -> c_12(activate#(XS)):3 3:W:take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) -->_1 activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)):2 -->_1 activate#(n__from(X)) -> c_3():1 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_4(take#(X1,X2)) 3: take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) 1: activate#(n__from(X)) -> c_3() **** Step 8.b:2.a:3.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0 ,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel# ,take#} and constructors {0,cons,n__from,n__take,nil,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 8.b:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) - Weak DPs: activate#(n__from(X)) -> c_3() take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) activate(n__take(X1,X2)) -> take(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0 ,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel# ,take#} and constructors {0,cons,n__from,n__take,nil,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) -->_1 take#(s(N),cons(X,XS)) -> c_12(activate#(XS)):4 2:S:sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) -->_1 sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))):2 3:W:activate#(n__from(X)) -> c_3() 4:W:take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) -->_1 activate#(n__from(X)) -> c_3():3 -->_1 activate#(n__take(X1,X2)) -> c_4(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: activate#(n__from(X)) -> c_3() *** Step 8.b:2.b:2: Decompose WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) - Weak DPs: take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) activate(n__take(X1,X2)) -> take(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0 ,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel# ,take#} and constructors {0,cons,n__from,n__take,nil,s} + 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_4(take#(X1,X2)) - Weak DPs: sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) activate(n__take(X1,X2)) -> take(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0 ,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel# ,take#} and constructors {0,cons,n__from,n__take,nil,s} Problem (S) - Strict DPs: sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) - Weak DPs: activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) activate(n__take(X1,X2)) -> take(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0 ,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel# ,take#} and constructors {0,cons,n__from,n__take,nil,s} **** Step 8.b:2.b:2.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) - Weak DPs: sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) activate(n__take(X1,X2)) -> take(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0 ,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel# ,take#} and constructors {0,cons,n__from,n__take,nil,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) -->_1 take#(s(N),cons(X,XS)) -> c_12(activate#(XS)):4 2:W:sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) -->_1 sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))):2 4:W:take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) -->_1 activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) **** Step 8.b:2.b:2.a:2: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) - Weak DPs: take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) activate(n__take(X1,X2)) -> take(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0 ,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel# ,take#} and constructors {0,cons,n__from,n__take,nil,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) **** Step 8.b:2.b:2.a:3: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) - Weak DPs: take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0 ,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel# ,take#} and constructors {0,cons,n__from,n__take,nil,s} + 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_4(take#(X1,X2)) Consider the set of all dependency pairs 1: activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) 2: take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {1} These cover all (indirect) predecessors of dependency pairs {1,2} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ***** Step 8.b:2.b:2.a:3.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) - Weak DPs: take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0 ,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel# ,take#} and constructors {0,cons,n__from,n__take,nil,s} + 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_4) = {1}, uargs(c_12) = {1} Following symbols are considered usable: {2nd#,activate#,from#,head#,sel#,take#} TcT has computed the following interpretation: p(0) = [1] p(2nd) = [1] x1 + [0] p(activate) = [2] p(cons) = [1] x1 + [1] x2 + [12] p(from) = [1] x1 + [0] p(head) = [1] p(n__from) = [1] p(n__take) = [1] x1 + [1] x2 + [9] p(nil) = [0] p(s) = [2] p(sel) = [4] x1 + [1] x2 + [8] p(take) = [2] x1 + [1] x2 + [2] p(2nd#) = [0] p(activate#) = [1] x1 + [8] p(from#) = [1] p(head#) = [1] x1 + [1] p(sel#) = [4] p(take#) = [1] x1 + [1] x2 + [9] p(c_1) = [1] p(c_2) = [1] p(c_3) = [1] p(c_4) = [1] x1 + [5] p(c_5) = [8] p(c_6) = [0] p(c_7) = [0] p(c_8) = [2] p(c_9) = [4] x1 + [0] p(c_10) = [0] p(c_11) = [1] p(c_12) = [1] x1 + [15] Following rules are strictly oriented: activate#(n__take(X1,X2)) = [1] X1 + [1] X2 + [17] > [1] X1 + [1] X2 + [14] = c_4(take#(X1,X2)) Following rules are (at-least) weakly oriented: take#(s(N),cons(X,XS)) = [1] X + [1] XS + [23] >= [1] XS + [23] = c_12(activate#(XS)) ***** Step 8.b:2.b:2.a:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0 ,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel# ,take#} and constructors {0,cons,n__from,n__take,nil,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ***** Step 8.b:2.b:2.a:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0 ,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel# ,take#} and constructors {0,cons,n__from,n__take,nil,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) -->_1 take#(s(N),cons(X,XS)) -> c_12(activate#(XS)):2 2:W:take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) -->_1 activate#(n__take(X1,X2)) -> c_4(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_4(take#(X1,X2)) 2: take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) ***** Step 8.b:2.b:2.a:3.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0 ,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel# ,take#} and constructors {0,cons,n__from,n__take,nil,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). **** Step 8.b:2.b:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) - Weak DPs: activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) activate(n__take(X1,X2)) -> take(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0 ,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel# ,take#} and constructors {0,cons,n__from,n__take,nil,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) -->_1 sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))):1 2:W:activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) -->_1 take#(s(N),cons(X,XS)) -> c_12(activate#(XS)):3 3:W:take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) -->_1 activate#(n__take(X1,X2)) -> c_4(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_4(take#(X1,X2)) 3: take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) **** Step 8.b:2.b:2.b:2: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) activate(n__take(X1,X2)) -> take(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0 ,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel# ,take#} and constructors {0,cons,n__from,n__take,nil,s} + 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: sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) The strictly oriented rules are moved into the weak component. ***** Step 8.b:2.b:2.b:2.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) activate(n__take(X1,X2)) -> take(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0 ,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel# ,take#} and constructors {0,cons,n__from,n__take,nil,s} + 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_9) = {1} Following symbols are considered usable: {2nd#,activate#,from#,head#,sel#,take#} TcT has computed the following interpretation: p(0) = [4] p(2nd) = [1] x1 + [0] p(activate) = [0] p(cons) = [0] p(from) = [4] x1 + [0] p(head) = [4] p(n__from) = [0] p(n__take) = [1] x1 + [9] p(nil) = [0] p(s) = [1] x1 + [2] p(sel) = [1] x1 + [1] p(take) = [0] p(2nd#) = [0] p(activate#) = [1] x1 + [1] p(from#) = [1] x1 + [1] p(head#) = [0] p(sel#) = [8] x1 + [0] p(take#) = [1] p(c_1) = [1] x1 + [2] p(c_2) = [1] p(c_3) = [0] p(c_4) = [2] x1 + [0] p(c_5) = [0] p(c_6) = [1] p(c_7) = [1] p(c_8) = [1] p(c_9) = [1] x1 + [10] p(c_10) = [1] p(c_11) = [4] p(c_12) = [2] x1 + [1] Following rules are strictly oriented: sel#(s(N),cons(X,XS)) = [8] N + [16] > [8] N + [10] = c_9(sel#(N,activate(XS))) Following rules are (at-least) weakly oriented: ***** Step 8.b:2.b:2.b:2.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) activate(n__take(X1,X2)) -> take(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0 ,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel# ,take#} and constructors {0,cons,n__from,n__take,nil,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ***** Step 8.b:2.b:2.b:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) activate(n__take(X1,X2)) -> take(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0 ,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel# ,take#} and constructors {0,cons,n__from,n__take,nil,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) -->_1 sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) ***** Step 8.b:2.b:2.b:2.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) activate(n__take(X1,X2)) -> take(X1,X2) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) take(X1,X2) -> n__take(X1,X2) take(0(),XS) -> nil() take(s(N),cons(X,XS)) -> cons(X,n__take(N,activate(XS))) - Signature: {2nd/1,activate/1,from/1,head/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,sel#/2,take#/2} / {0/0 ,cons/2,n__from/1,n__take/2,nil/0,s/1,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0 ,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,sel# ,take#} and constructors {0,cons,n__from,n__take,nil,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))