WORST_CASE(?,O(n^2)) * Step 1: DependencyPairs WORST_CASE(?,O(n^2)) + 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_ = DT} + Details: We add the following dependency tuples: Strict DPs 2nd#(cons(X,XS)) -> c_1(head#(activate(XS)),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)),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: PredecessorEstimation WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 2nd#(cons(X,XS)) -> c_1(head#(activate(XS)),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)),activate#(XS)) take#(X1,X2) -> c_10() take#(0(),XS) -> c_11() take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) - Weak 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_1/2,c_2/0,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/2,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 {2,5,6,7,8,10,11} by application of Pre({2,5,6,7,8,10,11}) = {1,3,4,9,12}. Here rules are labelled as follows: 1: 2nd#(cons(X,XS)) -> c_1(head#(activate(XS)),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#(0(),cons(X,XS)) -> c_8() 9: sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS)) 10: take#(X1,X2) -> c_10() 11: take#(0(),XS) -> c_11() 12: take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) * Step 3: PredecessorEstimation WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 2nd#(cons(X,XS)) -> c_1(head#(activate(XS)),activate#(XS)) 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)),activate#(XS)) take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) - Weak DPs: activate#(X) -> c_2() 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() - Weak 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_1/2,c_2/0,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/2,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 {2} by application of Pre({2}) = {1,4,5}. Here rules are labelled as follows: 1: 2nd#(cons(X,XS)) -> c_1(head#(activate(XS)),activate#(XS)) 2: activate#(n__from(X)) -> c_3(from#(X)) 3: activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) 4: sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS)) 5: take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) 6: activate#(X) -> c_2() 7: from#(X) -> c_5() 8: from#(X) -> c_6() 9: head#(cons(X,XS)) -> c_7() 10: sel#(0(),cons(X,XS)) -> c_8() 11: take#(X1,X2) -> c_10() 12: take#(0(),XS) -> c_11() * Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 2nd#(cons(X,XS)) -> c_1(head#(activate(XS)),activate#(XS)) activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS)) take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) - Weak DPs: activate#(X) -> c_2() activate#(n__from(X)) -> c_3(from#(X)) 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() - Weak 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_1/2,c_2/0,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/2,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:2nd#(cons(X,XS)) -> c_1(head#(activate(XS)),activate#(XS)) -->_2 activate#(n__from(X)) -> c_3(from#(X)):6 -->_2 activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)):2 -->_1 head#(cons(X,XS)) -> c_7():9 -->_2 activate#(X) -> c_2():5 2:S:activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) -->_1 take#(s(N),cons(X,XS)) -> c_12(activate#(XS)):4 -->_1 take#(0(),XS) -> c_11():12 -->_1 take#(X1,X2) -> c_10():11 3:S:sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS)) -->_2 activate#(n__from(X)) -> c_3(from#(X)):6 -->_1 sel#(0(),cons(X,XS)) -> c_8():10 -->_2 activate#(X) -> c_2():5 -->_1 sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS)):3 -->_2 activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)):2 4:S:take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) -->_1 activate#(n__from(X)) -> c_3(from#(X)):6 -->_1 activate#(X) -> c_2():5 -->_1 activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)):2 5:W:activate#(X) -> c_2() 6:W:activate#(n__from(X)) -> c_3(from#(X)) -->_1 from#(X) -> c_6():8 -->_1 from#(X) -> c_5():7 7:W:from#(X) -> c_5() 8:W:from#(X) -> c_6() 9:W:head#(cons(X,XS)) -> c_7() 10:W:sel#(0(),cons(X,XS)) -> c_8() 11:W:take#(X1,X2) -> c_10() 12:W:take#(0(),XS) -> c_11() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 10: sel#(0(),cons(X,XS)) -> c_8() 9: head#(cons(X,XS)) -> c_7() 11: take#(X1,X2) -> c_10() 12: take#(0(),XS) -> c_11() 5: activate#(X) -> c_2() 6: activate#(n__from(X)) -> c_3(from#(X)) 7: from#(X) -> c_5() 8: from#(X) -> c_6() * Step 5: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 2nd#(cons(X,XS)) -> c_1(head#(activate(XS)),activate#(XS)) activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS)) take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) - Weak 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_1/2,c_2/0,c_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/2,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:2nd#(cons(X,XS)) -> c_1(head#(activate(XS)),activate#(XS)) -->_2 activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)):2 2:S:activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) -->_1 take#(s(N),cons(X,XS)) -> c_12(activate#(XS)):4 3:S:sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS)) -->_1 sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS)):3 -->_2 activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)):2 4:S:take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) -->_1 activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)):2 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: 2nd#(cons(X,XS)) -> c_1(activate#(XS)) * Step 6: UsableRules WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 2nd#(cons(X,XS)) -> c_1(activate#(XS)) activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS)) take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) - Weak 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_9/2,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(activate#(XS)) activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS)) take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) * Step 7: RemoveHeads WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 2nd#(cons(X,XS)) -> c_1(activate#(XS)) activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),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_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/2,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: RemoveHeads + Details: Consider the dependency graph 1:S:2nd#(cons(X,XS)) -> c_1(activate#(XS)) -->_1 activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)):2 2:S:activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) -->_1 take#(s(N),cons(X,XS)) -> c_12(activate#(XS)):4 3:S:sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS)) -->_1 sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS)):3 -->_2 activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)):2 4:S:take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) -->_1 activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)):2 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). [(1,2nd#(cons(X,XS)) -> c_1(activate#(XS)))] * Step 8: DecomposeDG WORST_CASE(?,O(n^2)) + 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)),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_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/2,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: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS)) and a lower component activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) Further, following extension rules are added to the lower component. sel#(s(N),cons(X,XS)) -> activate#(XS) sel#(s(N),cons(X,XS)) -> sel#(N,activate(XS)) ** Step 8.a:1: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),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_9/2,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:sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS)) -->_1 sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) ** Step 8.a:2: WeightGap 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_3/1,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 = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(cons) = {2}, uargs(n__take) = {2}, uargs(sel#) = {2}, uargs(c_9) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(2nd) = [1] x1 + [1] p(activate) = [1] x1 + [0] p(cons) = [1] x2 + [0] p(from) = [2] p(head) = [0] p(n__from) = [2] p(n__take) = [1] x2 + [2] p(nil) = [2] p(s) = [1] x1 + [2] p(sel) = [4] x1 + [1] x2 + [1] p(take) = [1] x2 + [2] p(2nd#) = [1] x1 + [2] p(activate#) = [1] p(from#) = [1] x1 + [1] p(head#) = [0] p(sel#) = [10] x1 + [1] x2 + [8] p(take#) = [2] x2 + [1] p(c_1) = [2] x1 + [0] p(c_2) = [1] p(c_3) = [1] x1 + [2] p(c_4) = [4] p(c_5) = [0] p(c_6) = [2] p(c_7) = [1] p(c_8) = [1] p(c_9) = [1] x1 + [4] p(c_10) = [1] p(c_11) = [0] p(c_12) = [1] x1 + [1] Following rules are strictly oriented: sel#(s(N),cons(X,XS)) = [10] N + [1] XS + [28] > [10] N + [1] XS + [12] = c_9(sel#(N,activate(XS))) Following rules are (at-least) weakly oriented: activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__from(X)) = [2] >= [2] = from(X) activate(n__take(X1,X2)) = [1] X2 + [2] >= [1] X2 + [2] = take(X1,X2) from(X) = [2] >= [2] = cons(X,n__from(s(X))) from(X) = [2] >= [2] = n__from(X) take(X1,X2) = [1] X2 + [2] >= [1] X2 + [2] = n__take(X1,X2) take(0(),XS) = [1] XS + [2] >= [2] = nil() take(s(N),cons(X,XS)) = [1] XS + [2] >= [1] XS + [2] = cons(X,n__take(N,activate(XS))) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 8.a:3: EmptyProcessor 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_3/1,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: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 8.b:1: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) - Weak DPs: sel#(s(N),cons(X,XS)) -> activate#(XS) sel#(s(N),cons(X,XS)) -> 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_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/2,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 = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(cons) = {2}, uargs(n__take) = {2}, uargs(sel#) = {2}, uargs(c_4) = {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 + [0] p(cons) = [1] x2 + [0] p(from) = [0] p(head) = [0] p(n__from) = [0] p(n__take) = [1] x2 + [0] p(nil) = [0] p(s) = [1] x1 + [0] p(sel) = [0] p(take) = [1] x2 + [0] p(2nd#) = [0] p(activate#) = [1] p(from#) = [0] p(head#) = [0] p(sel#) = [1] x1 + [1] x2 + [3] p(take#) = [0] p(c_1) = [0] p(c_2) = [2] p(c_3) = [1] x1 + [1] p(c_4) = [1] x1 + [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [1] p(c_8) = [1] p(c_9) = [0] p(c_10) = [4] p(c_11) = [1] p(c_12) = [1] x1 + [4] Following rules are strictly oriented: activate#(n__take(X1,X2)) = [1] > [0] = c_4(take#(X1,X2)) Following rules are (at-least) weakly oriented: sel#(s(N),cons(X,XS)) = [1] N + [1] XS + [3] >= [1] = activate#(XS) sel#(s(N),cons(X,XS)) = [1] N + [1] XS + [3] >= [1] N + [1] XS + [3] = sel#(N,activate(XS)) take#(s(N),cons(X,XS)) = [0] >= [5] = c_12(activate#(XS)) activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__from(X)) = [0] >= [0] = from(X) activate(n__take(X1,X2)) = [1] X2 + [0] >= [1] X2 + [0] = take(X1,X2) from(X) = [0] >= [0] = cons(X,n__from(s(X))) from(X) = [0] >= [0] = n__from(X) take(X1,X2) = [1] X2 + [0] >= [1] X2 + [0] = n__take(X1,X2) take(0(),XS) = [1] XS + [0] >= [0] = nil() take(s(N),cons(X,XS)) = [1] XS + [0] >= [1] XS + [0] = cons(X,n__take(N,activate(XS))) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 8.b:2: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: take#(s(N),cons(X,XS)) -> c_12(activate#(XS)) - Weak DPs: activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) sel#(s(N),cons(X,XS)) -> activate#(XS) sel#(s(N),cons(X,XS)) -> 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_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/2,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 = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(cons) = {2}, uargs(n__take) = {2}, uargs(sel#) = {2}, uargs(c_4) = {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 + [0] p(cons) = [1] x2 + [0] p(from) = [0] p(head) = [0] p(n__from) = [0] p(n__take) = [1] x2 + [4] p(nil) = [1] p(s) = [1] x1 + [0] p(sel) = [2] x2 + [0] p(take) = [1] x2 + [4] p(2nd#) = [1] x1 + [1] p(activate#) = [1] x1 + [0] p(from#) = [2] x1 + [1] p(head#) = [1] p(sel#) = [1] x2 + [5] p(take#) = [1] x2 + [1] p(c_1) = [4] p(c_2) = [1] p(c_3) = [1] x1 + [1] p(c_4) = [1] x1 + [3] p(c_5) = [0] p(c_6) = [1] p(c_7) = [0] p(c_8) = [1] p(c_9) = [1] x1 + [0] p(c_10) = [1] p(c_11) = [1] p(c_12) = [1] x1 + [0] Following rules are strictly oriented: take#(s(N),cons(X,XS)) = [1] XS + [1] > [1] XS + [0] = c_12(activate#(XS)) Following rules are (at-least) weakly oriented: activate#(n__take(X1,X2)) = [1] X2 + [4] >= [1] X2 + [4] = c_4(take#(X1,X2)) sel#(s(N),cons(X,XS)) = [1] XS + [5] >= [1] XS + [0] = activate#(XS) sel#(s(N),cons(X,XS)) = [1] XS + [5] >= [1] XS + [5] = sel#(N,activate(XS)) activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__from(X)) = [0] >= [0] = from(X) activate(n__take(X1,X2)) = [1] X2 + [4] >= [1] X2 + [4] = take(X1,X2) from(X) = [0] >= [0] = cons(X,n__from(s(X))) from(X) = [0] >= [0] = n__from(X) take(X1,X2) = [1] X2 + [4] >= [1] X2 + [4] = n__take(X1,X2) take(0(),XS) = [1] XS + [4] >= [1] = nil() take(s(N),cons(X,XS)) = [1] XS + [4] >= [1] XS + [4] = cons(X,n__take(N,activate(XS))) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 8.b:3: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: activate#(n__take(X1,X2)) -> c_4(take#(X1,X2)) sel#(s(N),cons(X,XS)) -> activate#(XS) sel#(s(N),cons(X,XS)) -> 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_3/1,c_4/1,c_5/0,c_6/0,c_7/0,c_8/0,c_9/2,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^2))