WORST_CASE(Omega(n^1),O(n^1)) * Step 1: Sum WORST_CASE(Omega(n^1),O(n^1)) + Considered Problem: - Strict TRS: 2nd(cons(X,XS)) -> head(activate(XS)) activate(X) -> X activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__take(X1,X2)) -> take(activate(X1),activate(X2)) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) head(cons(X,XS)) -> X s(X) -> n__s(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,s/1,sel/2,take/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {2nd,activate,from,head,s,sel,take} and constructors {0 ,cons,n__from,n__s,n__take,nil} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: 2nd(cons(X,XS)) -> head(activate(XS)) activate(X) -> X activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__take(X1,X2)) -> take(activate(X1),activate(X2)) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) head(cons(X,XS)) -> X s(X) -> n__s(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,s/1,sel/2,take/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {2nd,activate,from,head,s,sel,take} and constructors {0 ,cons,n__from,n__s,n__take,nil} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: activate(x){x -> n__from(x)} = activate(n__from(x)) ->^+ from(activate(x)) = C[activate(x) = activate(x){}] ** Step 1.b: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(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__take(X1,X2)) -> take(activate(X1),activate(X2)) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) head(cons(X,XS)) -> X s(X) -> n__s(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,s/1,sel/2,take/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {2nd,activate,from,head,s,sel,take} and constructors {0 ,cons,n__from,n__s,n__take,nil} + 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#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)) activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) from#(X) -> c_6() from#(X) -> c_7() head#(cons(X,XS)) -> c_8() s#(X) -> c_9() sel#(0(),cons(X,XS)) -> c_10() sel#(s(N),cons(X,XS)) -> c_11(sel#(N,activate(XS)),activate#(XS)) take#(X1,X2) -> c_12() take#(0(),XS) -> c_13() take#(s(N),cons(X,XS)) -> c_14(activate#(XS)) Weak DPs and mark the set of starting terms. ** Step 1.b:2: PredecessorEstimation WORST_CASE(?,O(n^1)) + 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#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)) activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) from#(X) -> c_6() from#(X) -> c_7() head#(cons(X,XS)) -> c_8() s#(X) -> c_9() sel#(0(),cons(X,XS)) -> c_10() sel#(s(N),cons(X,XS)) -> c_11(sel#(N,activate(XS)),activate#(XS)) take#(X1,X2) -> c_12() take#(0(),XS) -> c_13() take#(s(N),cons(X,XS)) -> c_14(activate#(XS)) - Weak TRS: 2nd(cons(X,XS)) -> head(activate(XS)) activate(X) -> X activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__take(X1,X2)) -> take(activate(X1),activate(X2)) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) head(cons(X,XS)) -> X s(X) -> n__s(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,s/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,s#/1,sel#/2 ,take#/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0,c_1/2,c_2/0,c_3/2,c_4/2,c_5/3,c_6/0,c_7/0,c_8/0 ,c_9/0,c_10/0,c_11/2,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,s#,sel# ,take#} and constructors {0,cons,n__from,n__s,n__take,nil} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {2,6,7,8,9,10,11,12,13,14} by application of Pre({2,6,7,8,9,10,11,12,13,14}) = {1,3,4,5}. 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#(activate(X)),activate#(X)) 4: activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)) 5: activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) 6: from#(X) -> c_6() 7: from#(X) -> c_7() 8: head#(cons(X,XS)) -> c_8() 9: s#(X) -> c_9() 10: sel#(0(),cons(X,XS)) -> c_10() 11: sel#(s(N),cons(X,XS)) -> c_11(sel#(N,activate(XS)),activate#(XS)) 12: take#(X1,X2) -> c_12() 13: take#(0(),XS) -> c_13() 14: take#(s(N),cons(X,XS)) -> c_14(activate#(XS)) ** Step 1.b:3: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: 2nd#(cons(X,XS)) -> c_1(head#(activate(XS)),activate#(XS)) activate#(n__from(X)) -> c_3(from#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)) activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) - Weak DPs: activate#(X) -> c_2() from#(X) -> c_6() from#(X) -> c_7() head#(cons(X,XS)) -> c_8() s#(X) -> c_9() sel#(0(),cons(X,XS)) -> c_10() sel#(s(N),cons(X,XS)) -> c_11(sel#(N,activate(XS)),activate#(XS)) take#(X1,X2) -> c_12() take#(0(),XS) -> c_13() take#(s(N),cons(X,XS)) -> c_14(activate#(XS)) - Weak TRS: 2nd(cons(X,XS)) -> head(activate(XS)) activate(X) -> X activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__take(X1,X2)) -> take(activate(X1),activate(X2)) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) head(cons(X,XS)) -> X s(X) -> n__s(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,s/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,s#/1,sel#/2 ,take#/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0,c_1/2,c_2/0,c_3/2,c_4/2,c_5/3,c_6/0,c_7/0,c_8/0 ,c_9/0,c_10/0,c_11/2,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,s#,sel# ,take#} and constructors {0,cons,n__from,n__s,n__take,nil} + Applied Processor: RemoveWeakSuffixes + 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_5(take#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):4 -->_2 activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)):3 -->_2 activate#(n__from(X)) -> c_3(from#(activate(X)),activate#(X)):2 -->_1 head#(cons(X,XS)) -> c_8():8 -->_2 activate#(X) -> c_2():5 2:S:activate#(n__from(X)) -> c_3(from#(activate(X)),activate#(X)) -->_2 activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):4 -->_2 activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)):3 -->_1 from#(X) -> c_7():7 -->_1 from#(X) -> c_6():6 -->_2 activate#(X) -> c_2():5 -->_2 activate#(n__from(X)) -> c_3(from#(activate(X)),activate#(X)):2 3:S:activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)) -->_2 activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):4 -->_1 s#(X) -> c_9():9 -->_2 activate#(X) -> c_2():5 -->_2 activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)):3 -->_2 activate#(n__from(X)) -> c_3(from#(activate(X)),activate#(X)):2 4:S:activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) -->_1 take#(0(),XS) -> c_13():13 -->_1 take#(X1,X2) -> c_12():12 -->_3 activate#(X) -> c_2():5 -->_2 activate#(X) -> c_2():5 -->_3 activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):4 -->_2 activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):4 -->_3 activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)):3 -->_2 activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)):3 -->_3 activate#(n__from(X)) -> c_3(from#(activate(X)),activate#(X)):2 -->_2 activate#(n__from(X)) -> c_3(from#(activate(X)),activate#(X)):2 5:W:activate#(X) -> c_2() 6:W:from#(X) -> c_6() 7:W:from#(X) -> c_7() 8:W:head#(cons(X,XS)) -> c_8() 9:W:s#(X) -> c_9() 10:W:sel#(0(),cons(X,XS)) -> c_10() 11:W:sel#(s(N),cons(X,XS)) -> c_11(sel#(N,activate(XS)),activate#(XS)) 12:W:take#(X1,X2) -> c_12() 13:W:take#(0(),XS) -> c_13() 14:W:take#(s(N),cons(X,XS)) -> c_14(activate#(XS)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 14: take#(s(N),cons(X,XS)) -> c_14(activate#(XS)) 11: sel#(s(N),cons(X,XS)) -> c_11(sel#(N,activate(XS)),activate#(XS)) 10: sel#(0(),cons(X,XS)) -> c_10() 8: head#(cons(X,XS)) -> c_8() 6: from#(X) -> c_6() 7: from#(X) -> c_7() 9: s#(X) -> c_9() 5: activate#(X) -> c_2() 12: take#(X1,X2) -> c_12() 13: take#(0(),XS) -> c_13() ** Step 1.b:4: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: 2nd#(cons(X,XS)) -> c_1(head#(activate(XS)),activate#(XS)) activate#(n__from(X)) -> c_3(from#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)) activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) - Weak TRS: 2nd(cons(X,XS)) -> head(activate(XS)) activate(X) -> X activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__take(X1,X2)) -> take(activate(X1),activate(X2)) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) head(cons(X,XS)) -> X s(X) -> n__s(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,s/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,s#/1,sel#/2 ,take#/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0,c_1/2,c_2/0,c_3/2,c_4/2,c_5/3,c_6/0,c_7/0,c_8/0 ,c_9/0,c_10/0,c_11/2,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,s#,sel# ,take#} and constructors {0,cons,n__from,n__s,n__take,nil} + 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_5(take#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):4 -->_2 activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)):3 -->_2 activate#(n__from(X)) -> c_3(from#(activate(X)),activate#(X)):2 2:S:activate#(n__from(X)) -> c_3(from#(activate(X)),activate#(X)) -->_2 activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):4 -->_2 activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)):3 -->_2 activate#(n__from(X)) -> c_3(from#(activate(X)),activate#(X)):2 3:S:activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)) -->_2 activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):4 -->_2 activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)):3 -->_2 activate#(n__from(X)) -> c_3(from#(activate(X)),activate#(X)):2 4:S:activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)),activate#(X1),activate#(X2)) -->_3 activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):4 -->_2 activate#(n__take(X1,X2)) -> c_5(take#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):4 -->_3 activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)):3 -->_2 activate#(n__s(X)) -> c_4(s#(activate(X)),activate#(X)):3 -->_3 activate#(n__from(X)) -> c_3(from#(activate(X)),activate#(X)):2 -->_2 activate#(n__from(X)) -> c_3(from#(activate(X)),activate#(X)):2 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)) activate#(n__from(X)) -> c_3(activate#(X)) activate#(n__s(X)) -> c_4(activate#(X)) activate#(n__take(X1,X2)) -> c_5(activate#(X1),activate#(X2)) ** Step 1.b:5: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: 2nd#(cons(X,XS)) -> c_1(activate#(XS)) activate#(n__from(X)) -> c_3(activate#(X)) activate#(n__s(X)) -> c_4(activate#(X)) activate#(n__take(X1,X2)) -> c_5(activate#(X1),activate#(X2)) - Weak TRS: 2nd(cons(X,XS)) -> head(activate(XS)) activate(X) -> X activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__take(X1,X2)) -> take(activate(X1),activate(X2)) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) head(cons(X,XS)) -> X s(X) -> n__s(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,s/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,s#/1,sel#/2 ,take#/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/2,c_6/0,c_7/0,c_8/0 ,c_9/0,c_10/0,c_11/2,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,s#,sel# ,take#} and constructors {0,cons,n__from,n__s,n__take,nil} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: 2nd#(cons(X,XS)) -> c_1(activate#(XS)) activate#(n__from(X)) -> c_3(activate#(X)) activate#(n__s(X)) -> c_4(activate#(X)) activate#(n__take(X1,X2)) -> c_5(activate#(X1),activate#(X2)) ** Step 1.b:6: RemoveHeads WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: 2nd#(cons(X,XS)) -> c_1(activate#(XS)) activate#(n__from(X)) -> c_3(activate#(X)) activate#(n__s(X)) -> c_4(activate#(X)) activate#(n__take(X1,X2)) -> c_5(activate#(X1),activate#(X2)) - Signature: {2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,s#/1,sel#/2 ,take#/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/2,c_6/0,c_7/0,c_8/0 ,c_9/0,c_10/0,c_11/2,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,s#,sel# ,take#} and constructors {0,cons,n__from,n__s,n__take,nil} + 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_5(activate#(X1),activate#(X2)):4 -->_1 activate#(n__s(X)) -> c_4(activate#(X)):3 -->_1 activate#(n__from(X)) -> c_3(activate#(X)):2 2:S:activate#(n__from(X)) -> c_3(activate#(X)) -->_1 activate#(n__take(X1,X2)) -> c_5(activate#(X1),activate#(X2)):4 -->_1 activate#(n__s(X)) -> c_4(activate#(X)):3 -->_1 activate#(n__from(X)) -> c_3(activate#(X)):2 3:S:activate#(n__s(X)) -> c_4(activate#(X)) -->_1 activate#(n__take(X1,X2)) -> c_5(activate#(X1),activate#(X2)):4 -->_1 activate#(n__s(X)) -> c_4(activate#(X)):3 -->_1 activate#(n__from(X)) -> c_3(activate#(X)):2 4:S:activate#(n__take(X1,X2)) -> c_5(activate#(X1),activate#(X2)) -->_2 activate#(n__take(X1,X2)) -> c_5(activate#(X1),activate#(X2)):4 -->_1 activate#(n__take(X1,X2)) -> c_5(activate#(X1),activate#(X2)):4 -->_2 activate#(n__s(X)) -> c_4(activate#(X)):3 -->_1 activate#(n__s(X)) -> c_4(activate#(X)):3 -->_2 activate#(n__from(X)) -> c_3(activate#(X)):2 -->_1 activate#(n__from(X)) -> c_3(activate#(X)):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 1.b:7: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__from(X)) -> c_3(activate#(X)) activate#(n__s(X)) -> c_4(activate#(X)) activate#(n__take(X1,X2)) -> c_5(activate#(X1),activate#(X2)) - Signature: {2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,s#/1,sel#/2 ,take#/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/2,c_6/0,c_7/0,c_8/0 ,c_9/0,c_10/0,c_11/2,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,s#,sel# ,take#} and constructors {0,cons,n__from,n__s,n__take,nil} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1,2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(2nd) = [0] p(activate) = [0] p(cons) = [1] x1 + [1] x2 + [0] p(from) = [0] p(head) = [0] p(n__from) = [1] x1 + [0] p(n__s) = [1] x1 + [0] p(n__take) = [1] x1 + [1] x2 + [1] p(nil) = [0] p(s) = [0] p(sel) = [0] p(take) = [0] p(2nd#) = [0] p(activate#) = [5] x1 + [0] p(from#) = [0] p(head#) = [0] p(s#) = [0] p(sel#) = [0] p(take#) = [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [0] p(c_5) = [1] x1 + [1] x2 + [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [0] p(c_12) = [0] p(c_13) = [0] p(c_14) = [0] Following rules are strictly oriented: activate#(n__take(X1,X2)) = [5] X1 + [5] X2 + [5] > [5] X1 + [5] X2 + [0] = c_5(activate#(X1),activate#(X2)) Following rules are (at-least) weakly oriented: activate#(n__from(X)) = [5] X + [0] >= [5] X + [0] = c_3(activate#(X)) activate#(n__s(X)) = [5] X + [0] >= [5] X + [0] = c_4(activate#(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 1.b:8: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__from(X)) -> c_3(activate#(X)) activate#(n__s(X)) -> c_4(activate#(X)) - Weak DPs: activate#(n__take(X1,X2)) -> c_5(activate#(X1),activate#(X2)) - Signature: {2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,s#/1,sel#/2 ,take#/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/2,c_6/0,c_7/0,c_8/0 ,c_9/0,c_10/0,c_11/2,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,s#,sel# ,take#} and constructors {0,cons,n__from,n__s,n__take,nil} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1,2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(2nd) = [0] p(activate) = [0] p(cons) = [1] x1 + [1] x2 + [0] p(from) = [0] p(head) = [0] p(n__from) = [1] x1 + [0] p(n__s) = [1] x1 + [1] p(n__take) = [1] x1 + [1] x2 + [0] p(nil) = [0] p(s) = [0] p(sel) = [0] p(take) = [0] p(2nd#) = [0] p(activate#) = [5] x1 + [0] p(from#) = [0] p(head#) = [0] p(s#) = [0] p(sel#) = [0] p(take#) = [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [0] p(c_5) = [1] x1 + [1] x2 + [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [0] p(c_12) = [0] p(c_13) = [0] p(c_14) = [0] Following rules are strictly oriented: activate#(n__s(X)) = [5] X + [5] > [5] X + [0] = c_4(activate#(X)) Following rules are (at-least) weakly oriented: activate#(n__from(X)) = [5] X + [0] >= [5] X + [0] = c_3(activate#(X)) activate#(n__take(X1,X2)) = [5] X1 + [5] X2 + [0] >= [5] X1 + [5] X2 + [0] = c_5(activate#(X1),activate#(X2)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 1.b:9: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__from(X)) -> c_3(activate#(X)) - Weak DPs: activate#(n__s(X)) -> c_4(activate#(X)) activate#(n__take(X1,X2)) -> c_5(activate#(X1),activate#(X2)) - Signature: {2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,s#/1,sel#/2 ,take#/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/2,c_6/0,c_7/0,c_8/0 ,c_9/0,c_10/0,c_11/2,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,s#,sel# ,take#} and constructors {0,cons,n__from,n__s,n__take,nil} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1,2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(2nd) = [0] p(activate) = [0] p(cons) = [1] x1 + [1] x2 + [1] p(from) = [2] p(head) = [1] p(n__from) = [1] x1 + [12] p(n__s) = [1] x1 + [6] p(n__take) = [1] x1 + [1] x2 + [8] p(nil) = [0] p(s) = [0] p(sel) = [1] x1 + [0] p(take) = [1] x1 + [1] p(2nd#) = [2] x1 + [2] p(activate#) = [2] x1 + [6] p(from#) = [0] p(head#) = [0] p(s#) = [0] p(sel#) = [0] p(take#) = [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [1] x1 + [5] p(c_4) = [1] x1 + [0] p(c_5) = [1] x1 + [1] x2 + [7] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [0] p(c_12) = [2] p(c_13) = [2] p(c_14) = [1] x1 + [1] Following rules are strictly oriented: activate#(n__from(X)) = [2] X + [30] > [2] X + [11] = c_3(activate#(X)) Following rules are (at-least) weakly oriented: activate#(n__s(X)) = [2] X + [18] >= [2] X + [6] = c_4(activate#(X)) activate#(n__take(X1,X2)) = [2] X1 + [2] X2 + [22] >= [2] X1 + [2] X2 + [19] = c_5(activate#(X1),activate#(X2)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 1.b:10: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: activate#(n__from(X)) -> c_3(activate#(X)) activate#(n__s(X)) -> c_4(activate#(X)) activate#(n__take(X1,X2)) -> c_5(activate#(X1),activate#(X2)) - Signature: {2nd/1,activate/1,from/1,head/1,s/1,sel/2,take/2,2nd#/1,activate#/1,from#/1,head#/1,s#/1,sel#/2 ,take#/2} / {0/0,cons/2,n__from/1,n__s/1,n__take/2,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/2,c_6/0,c_7/0,c_8/0 ,c_9/0,c_10/0,c_11/2,c_12/0,c_13/0,c_14/1} - Obligation: innermost runtime complexity wrt. defined symbols {2nd#,activate#,from#,head#,s#,sel# ,take#} and constructors {0,cons,n__from,n__s,n__take,nil} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^1))