WORST_CASE(Omega(n^1),O(n^1)) * Step 1: Sum WORST_CASE(Omega(n^1),O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__natsFrom(X)) -> natsFrom(activate(X)) activate(n__s(X)) -> s(activate(X)) afterNth(N,XS) -> snd(splitAt(N,XS)) fst(pair(XS,YS)) -> XS head(cons(N,XS)) -> N natsFrom(N) -> cons(N,n__natsFrom(n__s(N))) natsFrom(X) -> n__natsFrom(X) s(X) -> n__s(X) sel(N,XS) -> head(afterNth(N,XS)) snd(pair(XS,YS)) -> YS splitAt(0(),XS) -> pair(nil(),XS) splitAt(s(N),cons(X,XS)) -> u(splitAt(N,activate(XS)),N,X,activate(XS)) tail(cons(N,XS)) -> activate(XS) take(N,XS) -> fst(splitAt(N,XS)) u(pair(YS,ZS),N,X,XS) -> pair(cons(activate(X),YS),ZS) - Signature: {activate/1,afterNth/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4} / {0/0,cons/2 ,n__natsFrom/1,n__s/1,nil/0,pair/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate,afterNth,fst,head,natsFrom,s,sel,snd,splitAt ,tail,take,u} and constructors {0,cons,n__natsFrom,n__s,nil,pair} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__natsFrom(X)) -> natsFrom(activate(X)) activate(n__s(X)) -> s(activate(X)) afterNth(N,XS) -> snd(splitAt(N,XS)) fst(pair(XS,YS)) -> XS head(cons(N,XS)) -> N natsFrom(N) -> cons(N,n__natsFrom(n__s(N))) natsFrom(X) -> n__natsFrom(X) s(X) -> n__s(X) sel(N,XS) -> head(afterNth(N,XS)) snd(pair(XS,YS)) -> YS splitAt(0(),XS) -> pair(nil(),XS) splitAt(s(N),cons(X,XS)) -> u(splitAt(N,activate(XS)),N,X,activate(XS)) tail(cons(N,XS)) -> activate(XS) take(N,XS) -> fst(splitAt(N,XS)) u(pair(YS,ZS),N,X,XS) -> pair(cons(activate(X),YS),ZS) - Signature: {activate/1,afterNth/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4} / {0/0,cons/2 ,n__natsFrom/1,n__s/1,nil/0,pair/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate,afterNth,fst,head,natsFrom,s,sel,snd,splitAt ,tail,take,u} and constructors {0,cons,n__natsFrom,n__s,nil,pair} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: activate(x){x -> n__natsFrom(x)} = activate(n__natsFrom(x)) ->^+ natsFrom(activate(x)) = C[activate(x) = activate(x){}] ** Step 1.b:1: DependencyPairs WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__natsFrom(X)) -> natsFrom(activate(X)) activate(n__s(X)) -> s(activate(X)) afterNth(N,XS) -> snd(splitAt(N,XS)) fst(pair(XS,YS)) -> XS head(cons(N,XS)) -> N natsFrom(N) -> cons(N,n__natsFrom(n__s(N))) natsFrom(X) -> n__natsFrom(X) s(X) -> n__s(X) sel(N,XS) -> head(afterNth(N,XS)) snd(pair(XS,YS)) -> YS splitAt(0(),XS) -> pair(nil(),XS) splitAt(s(N),cons(X,XS)) -> u(splitAt(N,activate(XS)),N,X,activate(XS)) tail(cons(N,XS)) -> activate(XS) take(N,XS) -> fst(splitAt(N,XS)) u(pair(YS,ZS),N,X,XS) -> pair(cons(activate(X),YS),ZS) - Signature: {activate/1,afterNth/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4} / {0/0,cons/2 ,n__natsFrom/1,n__s/1,nil/0,pair/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate,afterNth,fst,head,natsFrom,s,sel,snd,splitAt ,tail,take,u} and constructors {0,cons,n__natsFrom,n__s,nil,pair} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs activate#(X) -> c_1() activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X)) afterNth#(N,XS) -> c_4(snd#(splitAt(N,XS)),splitAt#(N,XS)) fst#(pair(XS,YS)) -> c_5() head#(cons(N,XS)) -> c_6() natsFrom#(N) -> c_7() natsFrom#(X) -> c_8() s#(X) -> c_9() sel#(N,XS) -> c_10(head#(afterNth(N,XS)),afterNth#(N,XS)) snd#(pair(XS,YS)) -> c_11() splitAt#(0(),XS) -> c_12() splitAt#(s(N),cons(X,XS)) -> c_13(u#(splitAt(N,activate(XS)),N,X,activate(XS)) ,splitAt#(N,activate(XS)) ,activate#(XS) ,activate#(XS)) tail#(cons(N,XS)) -> c_14(activate#(XS)) take#(N,XS) -> c_15(fst#(splitAt(N,XS)),splitAt#(N,XS)) u#(pair(YS,ZS),N,X,XS) -> c_16(activate#(X)) Weak DPs and mark the set of starting terms. ** Step 1.b:2: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(X) -> c_1() activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X)) afterNth#(N,XS) -> c_4(snd#(splitAt(N,XS)),splitAt#(N,XS)) fst#(pair(XS,YS)) -> c_5() head#(cons(N,XS)) -> c_6() natsFrom#(N) -> c_7() natsFrom#(X) -> c_8() s#(X) -> c_9() sel#(N,XS) -> c_10(head#(afterNth(N,XS)),afterNth#(N,XS)) snd#(pair(XS,YS)) -> c_11() splitAt#(0(),XS) -> c_12() splitAt#(s(N),cons(X,XS)) -> c_13(u#(splitAt(N,activate(XS)),N,X,activate(XS)) ,splitAt#(N,activate(XS)) ,activate#(XS) ,activate#(XS)) tail#(cons(N,XS)) -> c_14(activate#(XS)) take#(N,XS) -> c_15(fst#(splitAt(N,XS)),splitAt#(N,XS)) u#(pair(YS,ZS),N,X,XS) -> c_16(activate#(X)) - Weak TRS: activate(X) -> X activate(n__natsFrom(X)) -> natsFrom(activate(X)) activate(n__s(X)) -> s(activate(X)) afterNth(N,XS) -> snd(splitAt(N,XS)) fst(pair(XS,YS)) -> XS head(cons(N,XS)) -> N natsFrom(N) -> cons(N,n__natsFrom(n__s(N))) natsFrom(X) -> n__natsFrom(X) s(X) -> n__s(X) sel(N,XS) -> head(afterNth(N,XS)) snd(pair(XS,YS)) -> YS splitAt(0(),XS) -> pair(nil(),XS) splitAt(s(N),cons(X,XS)) -> u(splitAt(N,activate(XS)),N,X,activate(XS)) tail(cons(N,XS)) -> activate(XS) take(N,XS) -> fst(splitAt(N,XS)) u(pair(YS,ZS),N,X,XS) -> pair(cons(activate(X),YS),ZS) - Signature: {activate/1,afterNth/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1 ,afterNth#/2,fst#/1,head#/1,natsFrom#/1,s#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2 ,n__natsFrom/1,n__s/1,nil/0,pair/2,c_1/0,c_2/2,c_3/2,c_4/2,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/2,c_11/0 ,c_12/0,c_13/4,c_14/1,c_15/2,c_16/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,afterNth#,fst#,head#,natsFrom#,s#,sel#,snd# ,splitAt#,tail#,take#,u#} and constructors {0,cons,n__natsFrom,n__s,nil,pair} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,5,6,7,8,9,11,12,13} by application of Pre({1,5,6,7,8,9,11,12,13}) = {2,3,4,10,14,15,16}. Here rules are labelled as follows: 1: activate#(X) -> c_1() 2: activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X)),activate#(X)) 3: activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X)) 4: afterNth#(N,XS) -> c_4(snd#(splitAt(N,XS)),splitAt#(N,XS)) 5: fst#(pair(XS,YS)) -> c_5() 6: head#(cons(N,XS)) -> c_6() 7: natsFrom#(N) -> c_7() 8: natsFrom#(X) -> c_8() 9: s#(X) -> c_9() 10: sel#(N,XS) -> c_10(head#(afterNth(N,XS)),afterNth#(N,XS)) 11: snd#(pair(XS,YS)) -> c_11() 12: splitAt#(0(),XS) -> c_12() 13: splitAt#(s(N),cons(X,XS)) -> c_13(u#(splitAt(N,activate(XS)),N,X,activate(XS)) ,splitAt#(N,activate(XS)) ,activate#(XS) ,activate#(XS)) 14: tail#(cons(N,XS)) -> c_14(activate#(XS)) 15: take#(N,XS) -> c_15(fst#(splitAt(N,XS)),splitAt#(N,XS)) 16: u#(pair(YS,ZS),N,X,XS) -> c_16(activate#(X)) ** Step 1.b:3: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X)) afterNth#(N,XS) -> c_4(snd#(splitAt(N,XS)),splitAt#(N,XS)) sel#(N,XS) -> c_10(head#(afterNth(N,XS)),afterNth#(N,XS)) tail#(cons(N,XS)) -> c_14(activate#(XS)) take#(N,XS) -> c_15(fst#(splitAt(N,XS)),splitAt#(N,XS)) u#(pair(YS,ZS),N,X,XS) -> c_16(activate#(X)) - Weak DPs: activate#(X) -> c_1() fst#(pair(XS,YS)) -> c_5() head#(cons(N,XS)) -> c_6() natsFrom#(N) -> c_7() natsFrom#(X) -> c_8() s#(X) -> c_9() snd#(pair(XS,YS)) -> c_11() splitAt#(0(),XS) -> c_12() splitAt#(s(N),cons(X,XS)) -> c_13(u#(splitAt(N,activate(XS)),N,X,activate(XS)) ,splitAt#(N,activate(XS)) ,activate#(XS) ,activate#(XS)) - Weak TRS: activate(X) -> X activate(n__natsFrom(X)) -> natsFrom(activate(X)) activate(n__s(X)) -> s(activate(X)) afterNth(N,XS) -> snd(splitAt(N,XS)) fst(pair(XS,YS)) -> XS head(cons(N,XS)) -> N natsFrom(N) -> cons(N,n__natsFrom(n__s(N))) natsFrom(X) -> n__natsFrom(X) s(X) -> n__s(X) sel(N,XS) -> head(afterNth(N,XS)) snd(pair(XS,YS)) -> YS splitAt(0(),XS) -> pair(nil(),XS) splitAt(s(N),cons(X,XS)) -> u(splitAt(N,activate(XS)),N,X,activate(XS)) tail(cons(N,XS)) -> activate(XS) take(N,XS) -> fst(splitAt(N,XS)) u(pair(YS,ZS),N,X,XS) -> pair(cons(activate(X),YS),ZS) - Signature: {activate/1,afterNth/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1 ,afterNth#/2,fst#/1,head#/1,natsFrom#/1,s#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2 ,n__natsFrom/1,n__s/1,nil/0,pair/2,c_1/0,c_2/2,c_3/2,c_4/2,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/2,c_11/0 ,c_12/0,c_13/4,c_14/1,c_15/2,c_16/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,afterNth#,fst#,head#,natsFrom#,s#,sel#,snd# ,splitAt#,tail#,take#,u#} and constructors {0,cons,n__natsFrom,n__s,nil,pair} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {3,6} by application of Pre({3,6}) = {4}. Here rules are labelled as follows: 1: activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X)),activate#(X)) 2: activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X)) 3: afterNth#(N,XS) -> c_4(snd#(splitAt(N,XS)),splitAt#(N,XS)) 4: sel#(N,XS) -> c_10(head#(afterNth(N,XS)),afterNth#(N,XS)) 5: tail#(cons(N,XS)) -> c_14(activate#(XS)) 6: take#(N,XS) -> c_15(fst#(splitAt(N,XS)),splitAt#(N,XS)) 7: u#(pair(YS,ZS),N,X,XS) -> c_16(activate#(X)) 8: activate#(X) -> c_1() 9: fst#(pair(XS,YS)) -> c_5() 10: head#(cons(N,XS)) -> c_6() 11: natsFrom#(N) -> c_7() 12: natsFrom#(X) -> c_8() 13: s#(X) -> c_9() 14: snd#(pair(XS,YS)) -> c_11() 15: splitAt#(0(),XS) -> c_12() 16: splitAt#(s(N),cons(X,XS)) -> c_13(u#(splitAt(N,activate(XS)),N,X,activate(XS)) ,splitAt#(N,activate(XS)) ,activate#(XS) ,activate#(XS)) ** Step 1.b:4: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X)) sel#(N,XS) -> c_10(head#(afterNth(N,XS)),afterNth#(N,XS)) tail#(cons(N,XS)) -> c_14(activate#(XS)) u#(pair(YS,ZS),N,X,XS) -> c_16(activate#(X)) - Weak DPs: activate#(X) -> c_1() afterNth#(N,XS) -> c_4(snd#(splitAt(N,XS)),splitAt#(N,XS)) fst#(pair(XS,YS)) -> c_5() head#(cons(N,XS)) -> c_6() natsFrom#(N) -> c_7() natsFrom#(X) -> c_8() s#(X) -> c_9() snd#(pair(XS,YS)) -> c_11() splitAt#(0(),XS) -> c_12() splitAt#(s(N),cons(X,XS)) -> c_13(u#(splitAt(N,activate(XS)),N,X,activate(XS)) ,splitAt#(N,activate(XS)) ,activate#(XS) ,activate#(XS)) take#(N,XS) -> c_15(fst#(splitAt(N,XS)),splitAt#(N,XS)) - Weak TRS: activate(X) -> X activate(n__natsFrom(X)) -> natsFrom(activate(X)) activate(n__s(X)) -> s(activate(X)) afterNth(N,XS) -> snd(splitAt(N,XS)) fst(pair(XS,YS)) -> XS head(cons(N,XS)) -> N natsFrom(N) -> cons(N,n__natsFrom(n__s(N))) natsFrom(X) -> n__natsFrom(X) s(X) -> n__s(X) sel(N,XS) -> head(afterNth(N,XS)) snd(pair(XS,YS)) -> YS splitAt(0(),XS) -> pair(nil(),XS) splitAt(s(N),cons(X,XS)) -> u(splitAt(N,activate(XS)),N,X,activate(XS)) tail(cons(N,XS)) -> activate(XS) take(N,XS) -> fst(splitAt(N,XS)) u(pair(YS,ZS),N,X,XS) -> pair(cons(activate(X),YS),ZS) - Signature: {activate/1,afterNth/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1 ,afterNth#/2,fst#/1,head#/1,natsFrom#/1,s#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2 ,n__natsFrom/1,n__s/1,nil/0,pair/2,c_1/0,c_2/2,c_3/2,c_4/2,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/2,c_11/0 ,c_12/0,c_13/4,c_14/1,c_15/2,c_16/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,afterNth#,fst#,head#,natsFrom#,s#,sel#,snd# ,splitAt#,tail#,take#,u#} and constructors {0,cons,n__natsFrom,n__s,nil,pair} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {3} by application of Pre({3}) = {}. Here rules are labelled as follows: 1: activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X)),activate#(X)) 2: activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X)) 3: sel#(N,XS) -> c_10(head#(afterNth(N,XS)),afterNth#(N,XS)) 4: tail#(cons(N,XS)) -> c_14(activate#(XS)) 5: u#(pair(YS,ZS),N,X,XS) -> c_16(activate#(X)) 6: activate#(X) -> c_1() 7: afterNth#(N,XS) -> c_4(snd#(splitAt(N,XS)),splitAt#(N,XS)) 8: fst#(pair(XS,YS)) -> c_5() 9: head#(cons(N,XS)) -> c_6() 10: natsFrom#(N) -> c_7() 11: natsFrom#(X) -> c_8() 12: s#(X) -> c_9() 13: snd#(pair(XS,YS)) -> c_11() 14: splitAt#(0(),XS) -> c_12() 15: splitAt#(s(N),cons(X,XS)) -> c_13(u#(splitAt(N,activate(XS)),N,X,activate(XS)) ,splitAt#(N,activate(XS)) ,activate#(XS) ,activate#(XS)) 16: take#(N,XS) -> c_15(fst#(splitAt(N,XS)),splitAt#(N,XS)) ** Step 1.b:5: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X)) tail#(cons(N,XS)) -> c_14(activate#(XS)) u#(pair(YS,ZS),N,X,XS) -> c_16(activate#(X)) - Weak DPs: activate#(X) -> c_1() afterNth#(N,XS) -> c_4(snd#(splitAt(N,XS)),splitAt#(N,XS)) fst#(pair(XS,YS)) -> c_5() head#(cons(N,XS)) -> c_6() natsFrom#(N) -> c_7() natsFrom#(X) -> c_8() s#(X) -> c_9() sel#(N,XS) -> c_10(head#(afterNth(N,XS)),afterNth#(N,XS)) snd#(pair(XS,YS)) -> c_11() splitAt#(0(),XS) -> c_12() splitAt#(s(N),cons(X,XS)) -> c_13(u#(splitAt(N,activate(XS)),N,X,activate(XS)) ,splitAt#(N,activate(XS)) ,activate#(XS) ,activate#(XS)) take#(N,XS) -> c_15(fst#(splitAt(N,XS)),splitAt#(N,XS)) - Weak TRS: activate(X) -> X activate(n__natsFrom(X)) -> natsFrom(activate(X)) activate(n__s(X)) -> s(activate(X)) afterNth(N,XS) -> snd(splitAt(N,XS)) fst(pair(XS,YS)) -> XS head(cons(N,XS)) -> N natsFrom(N) -> cons(N,n__natsFrom(n__s(N))) natsFrom(X) -> n__natsFrom(X) s(X) -> n__s(X) sel(N,XS) -> head(afterNth(N,XS)) snd(pair(XS,YS)) -> YS splitAt(0(),XS) -> pair(nil(),XS) splitAt(s(N),cons(X,XS)) -> u(splitAt(N,activate(XS)),N,X,activate(XS)) tail(cons(N,XS)) -> activate(XS) take(N,XS) -> fst(splitAt(N,XS)) u(pair(YS,ZS),N,X,XS) -> pair(cons(activate(X),YS),ZS) - Signature: {activate/1,afterNth/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1 ,afterNth#/2,fst#/1,head#/1,natsFrom#/1,s#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2 ,n__natsFrom/1,n__s/1,nil/0,pair/2,c_1/0,c_2/2,c_3/2,c_4/2,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/2,c_11/0 ,c_12/0,c_13/4,c_14/1,c_15/2,c_16/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,afterNth#,fst#,head#,natsFrom#,s#,sel#,snd# ,splitAt#,tail#,take#,u#} and constructors {0,cons,n__natsFrom,n__s,nil,pair} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X)),activate#(X)) -->_2 activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X)):2 -->_1 natsFrom#(X) -> c_8():10 -->_1 natsFrom#(N) -> c_7():9 -->_2 activate#(X) -> c_1():5 -->_2 activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X)),activate#(X)):1 2:S:activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X)) -->_1 s#(X) -> c_9():11 -->_2 activate#(X) -> c_1():5 -->_2 activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X)):2 -->_2 activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X)),activate#(X)):1 3:S:tail#(cons(N,XS)) -> c_14(activate#(XS)) -->_1 activate#(X) -> c_1():5 -->_1 activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X)):2 -->_1 activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X)),activate#(X)):1 4:S:u#(pair(YS,ZS),N,X,XS) -> c_16(activate#(X)) -->_1 activate#(X) -> c_1():5 -->_1 activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X)):2 -->_1 activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X)),activate#(X)):1 5:W:activate#(X) -> c_1() 6:W:afterNth#(N,XS) -> c_4(snd#(splitAt(N,XS)),splitAt#(N,XS)) -->_2 splitAt#(0(),XS) -> c_12():14 -->_1 snd#(pair(XS,YS)) -> c_11():13 7:W:fst#(pair(XS,YS)) -> c_5() 8:W:head#(cons(N,XS)) -> c_6() 9:W:natsFrom#(N) -> c_7() 10:W:natsFrom#(X) -> c_8() 11:W:s#(X) -> c_9() 12:W:sel#(N,XS) -> c_10(head#(afterNth(N,XS)),afterNth#(N,XS)) -->_1 head#(cons(N,XS)) -> c_6():8 -->_2 afterNth#(N,XS) -> c_4(snd#(splitAt(N,XS)),splitAt#(N,XS)):6 13:W:snd#(pair(XS,YS)) -> c_11() 14:W:splitAt#(0(),XS) -> c_12() 15:W:splitAt#(s(N),cons(X,XS)) -> c_13(u#(splitAt(N,activate(XS)),N,X,activate(XS)) ,splitAt#(N,activate(XS)) ,activate#(XS) ,activate#(XS)) 16:W:take#(N,XS) -> c_15(fst#(splitAt(N,XS)),splitAt#(N,XS)) -->_2 splitAt#(0(),XS) -> c_12():14 -->_1 fst#(pair(XS,YS)) -> c_5():7 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 16: take#(N,XS) -> c_15(fst#(splitAt(N,XS)),splitAt#(N,XS)) 15: splitAt#(s(N),cons(X,XS)) -> c_13(u#(splitAt(N,activate(XS)),N,X,activate(XS)) ,splitAt#(N,activate(XS)) ,activate#(XS) ,activate#(XS)) 12: sel#(N,XS) -> c_10(head#(afterNth(N,XS)),afterNth#(N,XS)) 8: head#(cons(N,XS)) -> c_6() 7: fst#(pair(XS,YS)) -> c_5() 6: afterNth#(N,XS) -> c_4(snd#(splitAt(N,XS)),splitAt#(N,XS)) 13: snd#(pair(XS,YS)) -> c_11() 14: splitAt#(0(),XS) -> c_12() 9: natsFrom#(N) -> c_7() 10: natsFrom#(X) -> c_8() 5: activate#(X) -> c_1() 11: s#(X) -> c_9() ** Step 1.b:6: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X)) tail#(cons(N,XS)) -> c_14(activate#(XS)) u#(pair(YS,ZS),N,X,XS) -> c_16(activate#(X)) - Weak TRS: activate(X) -> X activate(n__natsFrom(X)) -> natsFrom(activate(X)) activate(n__s(X)) -> s(activate(X)) afterNth(N,XS) -> snd(splitAt(N,XS)) fst(pair(XS,YS)) -> XS head(cons(N,XS)) -> N natsFrom(N) -> cons(N,n__natsFrom(n__s(N))) natsFrom(X) -> n__natsFrom(X) s(X) -> n__s(X) sel(N,XS) -> head(afterNth(N,XS)) snd(pair(XS,YS)) -> YS splitAt(0(),XS) -> pair(nil(),XS) splitAt(s(N),cons(X,XS)) -> u(splitAt(N,activate(XS)),N,X,activate(XS)) tail(cons(N,XS)) -> activate(XS) take(N,XS) -> fst(splitAt(N,XS)) u(pair(YS,ZS),N,X,XS) -> pair(cons(activate(X),YS),ZS) - Signature: {activate/1,afterNth/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1 ,afterNth#/2,fst#/1,head#/1,natsFrom#/1,s#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2 ,n__natsFrom/1,n__s/1,nil/0,pair/2,c_1/0,c_2/2,c_3/2,c_4/2,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/2,c_11/0 ,c_12/0,c_13/4,c_14/1,c_15/2,c_16/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,afterNth#,fst#,head#,natsFrom#,s#,sel#,snd# ,splitAt#,tail#,take#,u#} and constructors {0,cons,n__natsFrom,n__s,nil,pair} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X)),activate#(X)) -->_2 activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X)):2 -->_2 activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X)),activate#(X)):1 2:S:activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X)) -->_2 activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X)):2 -->_2 activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X)),activate#(X)):1 3:S:tail#(cons(N,XS)) -> c_14(activate#(XS)) -->_1 activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X)):2 -->_1 activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X)),activate#(X)):1 4:S:u#(pair(YS,ZS),N,X,XS) -> c_16(activate#(X)) -->_1 activate#(n__s(X)) -> c_3(s#(activate(X)),activate#(X)):2 -->_1 activate#(n__natsFrom(X)) -> c_2(natsFrom#(activate(X)),activate#(X)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: activate#(n__natsFrom(X)) -> c_2(activate#(X)) activate#(n__s(X)) -> c_3(activate#(X)) ** Step 1.b:7: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__natsFrom(X)) -> c_2(activate#(X)) activate#(n__s(X)) -> c_3(activate#(X)) tail#(cons(N,XS)) -> c_14(activate#(XS)) u#(pair(YS,ZS),N,X,XS) -> c_16(activate#(X)) - Weak TRS: activate(X) -> X activate(n__natsFrom(X)) -> natsFrom(activate(X)) activate(n__s(X)) -> s(activate(X)) afterNth(N,XS) -> snd(splitAt(N,XS)) fst(pair(XS,YS)) -> XS head(cons(N,XS)) -> N natsFrom(N) -> cons(N,n__natsFrom(n__s(N))) natsFrom(X) -> n__natsFrom(X) s(X) -> n__s(X) sel(N,XS) -> head(afterNth(N,XS)) snd(pair(XS,YS)) -> YS splitAt(0(),XS) -> pair(nil(),XS) splitAt(s(N),cons(X,XS)) -> u(splitAt(N,activate(XS)),N,X,activate(XS)) tail(cons(N,XS)) -> activate(XS) take(N,XS) -> fst(splitAt(N,XS)) u(pair(YS,ZS),N,X,XS) -> pair(cons(activate(X),YS),ZS) - Signature: {activate/1,afterNth/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1 ,afterNth#/2,fst#/1,head#/1,natsFrom#/1,s#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2 ,n__natsFrom/1,n__s/1,nil/0,pair/2,c_1/0,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/2,c_11/0 ,c_12/0,c_13/4,c_14/1,c_15/2,c_16/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,afterNth#,fst#,head#,natsFrom#,s#,sel#,snd# ,splitAt#,tail#,take#,u#} and constructors {0,cons,n__natsFrom,n__s,nil,pair} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: activate#(n__natsFrom(X)) -> c_2(activate#(X)) activate#(n__s(X)) -> c_3(activate#(X)) tail#(cons(N,XS)) -> c_14(activate#(XS)) u#(pair(YS,ZS),N,X,XS) -> c_16(activate#(X)) ** Step 1.b:8: RemoveHeads WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__natsFrom(X)) -> c_2(activate#(X)) activate#(n__s(X)) -> c_3(activate#(X)) tail#(cons(N,XS)) -> c_14(activate#(XS)) u#(pair(YS,ZS),N,X,XS) -> c_16(activate#(X)) - Signature: {activate/1,afterNth/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1 ,afterNth#/2,fst#/1,head#/1,natsFrom#/1,s#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2 ,n__natsFrom/1,n__s/1,nil/0,pair/2,c_1/0,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/2,c_11/0 ,c_12/0,c_13/4,c_14/1,c_15/2,c_16/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,afterNth#,fst#,head#,natsFrom#,s#,sel#,snd# ,splitAt#,tail#,take#,u#} and constructors {0,cons,n__natsFrom,n__s,nil,pair} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:activate#(n__natsFrom(X)) -> c_2(activate#(X)) -->_1 activate#(n__s(X)) -> c_3(activate#(X)):2 -->_1 activate#(n__natsFrom(X)) -> c_2(activate#(X)):1 2:S:activate#(n__s(X)) -> c_3(activate#(X)) -->_1 activate#(n__s(X)) -> c_3(activate#(X)):2 -->_1 activate#(n__natsFrom(X)) -> c_2(activate#(X)):1 3:S:tail#(cons(N,XS)) -> c_14(activate#(XS)) -->_1 activate#(n__s(X)) -> c_3(activate#(X)):2 -->_1 activate#(n__natsFrom(X)) -> c_2(activate#(X)):1 4:S:u#(pair(YS,ZS),N,X,XS) -> c_16(activate#(X)) -->_1 activate#(n__s(X)) -> c_3(activate#(X)):2 -->_1 activate#(n__natsFrom(X)) -> c_2(activate#(X)):1 Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts). [(3,tail#(cons(N,XS)) -> c_14(activate#(XS))),(4,u#(pair(YS,ZS),N,X,XS) -> c_16(activate#(X)))] ** Step 1.b:9: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__natsFrom(X)) -> c_2(activate#(X)) activate#(n__s(X)) -> c_3(activate#(X)) - Signature: {activate/1,afterNth/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1 ,afterNth#/2,fst#/1,head#/1,natsFrom#/1,s#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2 ,n__natsFrom/1,n__s/1,nil/0,pair/2,c_1/0,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/2,c_11/0 ,c_12/0,c_13/4,c_14/1,c_15/2,c_16/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,afterNth#,fst#,head#,natsFrom#,s#,sel#,snd# ,splitAt#,tail#,take#,u#} and constructors {0,cons,n__natsFrom,n__s,nil,pair} + 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_2) = {1}, uargs(c_3) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(activate) = [0] p(afterNth) = [0] p(cons) = [1] x1 + [1] x2 + [1] p(fst) = [0] p(head) = [0] p(n__natsFrom) = [1] x1 + [0] p(n__s) = [1] x1 + [1] p(natsFrom) = [0] p(nil) = [0] p(pair) = [1] x2 + [0] p(s) = [0] p(sel) = [0] p(snd) = [0] p(splitAt) = [0] p(tail) = [0] p(take) = [0] p(u) = [0] p(activate#) = [9] x1 + [0] p(afterNth#) = [0] p(fst#) = [0] p(head#) = [0] p(natsFrom#) = [0] p(s#) = [0] p(sel#) = [0] p(snd#) = [0] p(splitAt#) = [0] p(tail#) = [0] p(take#) = [0] p(u#) = [0] p(c_1) = [0] p(c_2) = [1] x1 + [0] p(c_3) = [1] x1 + [0] p(c_4) = [0] p(c_5) = [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] p(c_15) = [0] p(c_16) = [0] Following rules are strictly oriented: activate#(n__s(X)) = [9] X + [9] > [9] X + [0] = c_3(activate#(X)) Following rules are (at-least) weakly oriented: activate#(n__natsFrom(X)) = [9] X + [0] >= [9] X + [0] = c_2(activate#(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 1.b:10: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(n__natsFrom(X)) -> c_2(activate#(X)) - Weak DPs: activate#(n__s(X)) -> c_3(activate#(X)) - Signature: {activate/1,afterNth/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1 ,afterNth#/2,fst#/1,head#/1,natsFrom#/1,s#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2 ,n__natsFrom/1,n__s/1,nil/0,pair/2,c_1/0,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/2,c_11/0 ,c_12/0,c_13/4,c_14/1,c_15/2,c_16/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,afterNth#,fst#,head#,natsFrom#,s#,sel#,snd# ,splitAt#,tail#,take#,u#} and constructors {0,cons,n__natsFrom,n__s,nil,pair} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_3) = {1} Following symbols are considered usable: {activate#,afterNth#,fst#,head#,natsFrom#,s#,sel#,snd#,splitAt#,tail#,take#,u#} TcT has computed the following interpretation: p(0) = [1] p(activate) = [1] x1 + [1] p(afterNth) = [1] x1 + [2] x2 + [1] p(cons) = [1] x1 + [0] p(fst) = [1] p(head) = [0] p(n__natsFrom) = [1] x1 + [8] p(n__s) = [1] x1 + [0] p(natsFrom) = [0] p(nil) = [0] p(pair) = [1] x1 + [1] x2 + [0] p(s) = [0] p(sel) = [0] p(snd) = [0] p(splitAt) = [0] p(tail) = [0] p(take) = [0] p(u) = [0] p(activate#) = [2] x1 + [8] p(afterNth#) = [0] p(fst#) = [0] p(head#) = [0] p(natsFrom#) = [0] p(s#) = [4] x1 + [0] p(sel#) = [1] x1 + [4] x2 + [0] p(snd#) = [1] x1 + [0] p(splitAt#) = [1] x1 + [4] x2 + [2] p(tail#) = [1] p(take#) = [1] x2 + [2] p(u#) = [1] x1 + [4] x2 + [1] x3 + [1] p(c_1) = [0] p(c_2) = [1] x1 + [12] p(c_3) = [1] x1 + [0] p(c_4) = [2] x2 + [8] p(c_5) = [0] p(c_6) = [2] p(c_7) = [0] p(c_8) = [0] p(c_9) = [1] p(c_10) = [1] p(c_11) = [2] p(c_12) = [0] p(c_13) = [1] x1 + [1] x2 + [1] x3 + [2] p(c_14) = [1] x1 + [2] p(c_15) = [2] p(c_16) = [1] x1 + [1] Following rules are strictly oriented: activate#(n__natsFrom(X)) = [2] X + [24] > [2] X + [20] = c_2(activate#(X)) Following rules are (at-least) weakly oriented: activate#(n__s(X)) = [2] X + [8] >= [2] X + [8] = c_3(activate#(X)) ** Step 1.b:11: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: activate#(n__natsFrom(X)) -> c_2(activate#(X)) activate#(n__s(X)) -> c_3(activate#(X)) - Signature: {activate/1,afterNth/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2,u/4,activate#/1 ,afterNth#/2,fst#/1,head#/1,natsFrom#/1,s#/1,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2,u#/4} / {0/0,cons/2 ,n__natsFrom/1,n__s/1,nil/0,pair/2,c_1/0,c_2/1,c_3/1,c_4/2,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/2,c_11/0 ,c_12/0,c_13/4,c_14/1,c_15/2,c_16/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,afterNth#,fst#,head#,natsFrom#,s#,sel#,snd# ,splitAt#,tail#,take#,u#} and constructors {0,cons,n__natsFrom,n__s,nil,pair} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^1))