WORST_CASE(Omega(n^1),O(n^1)) * Step 1: Sum WORST_CASE(Omega(n^1),O(n^1)) + Considered Problem: - Strict TRS: U11(tt(),N,X,XS) -> U12(splitAt(activate(N),activate(XS)),activate(X)) U12(pair(YS,ZS),X) -> pair(cons(activate(X),YS),ZS) activate(X) -> X activate(n__natsFrom(X)) -> natsFrom(activate(X)) activate(n__s(X)) -> s(activate(X)) afterNth(N,XS) -> snd(splitAt(N,XS)) and(tt(),X) -> activate(X) fst(pair(X,Y)) -> X 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(X,Y)) -> Y splitAt(0(),XS) -> pair(nil(),XS) splitAt(s(N),cons(X,XS)) -> U11(tt(),N,X,activate(XS)) tail(cons(N,XS)) -> activate(XS) take(N,XS) -> fst(splitAt(N,XS)) - Signature: {U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1 ,take/2} / {0/0,cons/2,n__natsFrom/1,n__s/1,nil/0,pair/2,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {U11,U12,activate,afterNth,and,fst,head,natsFrom,s,sel,snd ,splitAt,tail,take} and constructors {0,cons,n__natsFrom,n__s,nil,pair,tt} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: U11(tt(),N,X,XS) -> U12(splitAt(activate(N),activate(XS)),activate(X)) U12(pair(YS,ZS),X) -> pair(cons(activate(X),YS),ZS) activate(X) -> X activate(n__natsFrom(X)) -> natsFrom(activate(X)) activate(n__s(X)) -> s(activate(X)) afterNth(N,XS) -> snd(splitAt(N,XS)) and(tt(),X) -> activate(X) fst(pair(X,Y)) -> X 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(X,Y)) -> Y splitAt(0(),XS) -> pair(nil(),XS) splitAt(s(N),cons(X,XS)) -> U11(tt(),N,X,activate(XS)) tail(cons(N,XS)) -> activate(XS) take(N,XS) -> fst(splitAt(N,XS)) - Signature: {U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1 ,take/2} / {0/0,cons/2,n__natsFrom/1,n__s/1,nil/0,pair/2,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {U11,U12,activate,afterNth,and,fst,head,natsFrom,s,sel,snd ,splitAt,tail,take} and constructors {0,cons,n__natsFrom,n__s,nil,pair,tt} + 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: U11(tt(),N,X,XS) -> U12(splitAt(activate(N),activate(XS)),activate(X)) U12(pair(YS,ZS),X) -> pair(cons(activate(X),YS),ZS) activate(X) -> X activate(n__natsFrom(X)) -> natsFrom(activate(X)) activate(n__s(X)) -> s(activate(X)) afterNth(N,XS) -> snd(splitAt(N,XS)) and(tt(),X) -> activate(X) fst(pair(X,Y)) -> X 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(X,Y)) -> Y splitAt(0(),XS) -> pair(nil(),XS) splitAt(s(N),cons(X,XS)) -> U11(tt(),N,X,activate(XS)) tail(cons(N,XS)) -> activate(XS) take(N,XS) -> fst(splitAt(N,XS)) - Signature: {U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1 ,take/2} / {0/0,cons/2,n__natsFrom/1,n__s/1,nil/0,pair/2,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {U11,U12,activate,afterNth,and,fst,head,natsFrom,s,sel,snd ,splitAt,tail,take} and constructors {0,cons,n__natsFrom,n__s,nil,pair,tt} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs U11#(tt(),N,X,XS) -> c_1(U12#(splitAt(activate(N),activate(XS)),activate(X)) ,splitAt#(activate(N),activate(XS)) ,activate#(N) ,activate#(XS) ,activate#(X)) U12#(pair(YS,ZS),X) -> c_2(activate#(X)) activate#(X) -> c_3() activate#(n__natsFrom(X)) -> c_4(natsFrom#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) afterNth#(N,XS) -> c_6(snd#(splitAt(N,XS)),splitAt#(N,XS)) and#(tt(),X) -> c_7(activate#(X)) fst#(pair(X,Y)) -> c_8() head#(cons(N,XS)) -> c_9() natsFrom#(N) -> c_10() natsFrom#(X) -> c_11() s#(X) -> c_12() sel#(N,XS) -> c_13(head#(afterNth(N,XS)),afterNth#(N,XS)) snd#(pair(X,Y)) -> c_14() splitAt#(0(),XS) -> c_15() splitAt#(s(N),cons(X,XS)) -> c_16(U11#(tt(),N,X,activate(XS)),activate#(XS)) tail#(cons(N,XS)) -> c_17(activate#(XS)) take#(N,XS) -> c_18(fst#(splitAt(N,XS)),splitAt#(N,XS)) Weak DPs and mark the set of starting terms. ** Step 1.b:2: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),N,X,XS) -> c_1(U12#(splitAt(activate(N),activate(XS)),activate(X)) ,splitAt#(activate(N),activate(XS)) ,activate#(N) ,activate#(XS) ,activate#(X)) U12#(pair(YS,ZS),X) -> c_2(activate#(X)) activate#(X) -> c_3() activate#(n__natsFrom(X)) -> c_4(natsFrom#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) afterNth#(N,XS) -> c_6(snd#(splitAt(N,XS)),splitAt#(N,XS)) and#(tt(),X) -> c_7(activate#(X)) fst#(pair(X,Y)) -> c_8() head#(cons(N,XS)) -> c_9() natsFrom#(N) -> c_10() natsFrom#(X) -> c_11() s#(X) -> c_12() sel#(N,XS) -> c_13(head#(afterNth(N,XS)),afterNth#(N,XS)) snd#(pair(X,Y)) -> c_14() splitAt#(0(),XS) -> c_15() splitAt#(s(N),cons(X,XS)) -> c_16(U11#(tt(),N,X,activate(XS)),activate#(XS)) tail#(cons(N,XS)) -> c_17(activate#(XS)) take#(N,XS) -> c_18(fst#(splitAt(N,XS)),splitAt#(N,XS)) - Weak TRS: U11(tt(),N,X,XS) -> U12(splitAt(activate(N),activate(XS)),activate(X)) U12(pair(YS,ZS),X) -> pair(cons(activate(X),YS),ZS) activate(X) -> X activate(n__natsFrom(X)) -> natsFrom(activate(X)) activate(n__s(X)) -> s(activate(X)) afterNth(N,XS) -> snd(splitAt(N,XS)) and(tt(),X) -> activate(X) fst(pair(X,Y)) -> X 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(X,Y)) -> Y splitAt(0(),XS) -> pair(nil(),XS) splitAt(s(N),cons(X,XS)) -> U11(tt(),N,X,activate(XS)) tail(cons(N,XS)) -> activate(XS) take(N,XS) -> fst(splitAt(N,XS)) - Signature: {U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2 ,U11#/4,U12#/2,activate#/1,afterNth#/2,and#/2,fst#/1,head#/1,natsFrom#/1,s#/1,sel#/2,snd#/1,splitAt#/2 ,tail#/1,take#/2} / {0/0,cons/2,n__natsFrom/1,n__s/1,nil/0,pair/2,tt/0,c_1/5,c_2/1,c_3/0,c_4/2,c_5/2,c_6/2 ,c_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/2,c_14/0,c_15/0,c_16/2,c_17/1,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom# ,s#,sel#,snd#,splitAt#,tail#,take#} and constructors {0,cons,n__natsFrom,n__s,nil,pair,tt} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {3,8,9,10,11,12,14,15,16} by application of Pre({3,8,9,10,11,12,14,15,16}) = {1,2,4,5,6,7,13,17,18}. Here rules are labelled as follows: 1: U11#(tt(),N,X,XS) -> c_1(U12#(splitAt(activate(N),activate(XS)),activate(X)) ,splitAt#(activate(N),activate(XS)) ,activate#(N) ,activate#(XS) ,activate#(X)) 2: U12#(pair(YS,ZS),X) -> c_2(activate#(X)) 3: activate#(X) -> c_3() 4: activate#(n__natsFrom(X)) -> c_4(natsFrom#(activate(X)),activate#(X)) 5: activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) 6: afterNth#(N,XS) -> c_6(snd#(splitAt(N,XS)),splitAt#(N,XS)) 7: and#(tt(),X) -> c_7(activate#(X)) 8: fst#(pair(X,Y)) -> c_8() 9: head#(cons(N,XS)) -> c_9() 10: natsFrom#(N) -> c_10() 11: natsFrom#(X) -> c_11() 12: s#(X) -> c_12() 13: sel#(N,XS) -> c_13(head#(afterNth(N,XS)),afterNth#(N,XS)) 14: snd#(pair(X,Y)) -> c_14() 15: splitAt#(0(),XS) -> c_15() 16: splitAt#(s(N),cons(X,XS)) -> c_16(U11#(tt(),N,X,activate(XS)),activate#(XS)) 17: tail#(cons(N,XS)) -> c_17(activate#(XS)) 18: take#(N,XS) -> c_18(fst#(splitAt(N,XS)),splitAt#(N,XS)) ** Step 1.b:3: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),N,X,XS) -> c_1(U12#(splitAt(activate(N),activate(XS)),activate(X)) ,splitAt#(activate(N),activate(XS)) ,activate#(N) ,activate#(XS) ,activate#(X)) U12#(pair(YS,ZS),X) -> c_2(activate#(X)) activate#(n__natsFrom(X)) -> c_4(natsFrom#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) afterNth#(N,XS) -> c_6(snd#(splitAt(N,XS)),splitAt#(N,XS)) and#(tt(),X) -> c_7(activate#(X)) sel#(N,XS) -> c_13(head#(afterNth(N,XS)),afterNth#(N,XS)) tail#(cons(N,XS)) -> c_17(activate#(XS)) take#(N,XS) -> c_18(fst#(splitAt(N,XS)),splitAt#(N,XS)) - Weak DPs: activate#(X) -> c_3() fst#(pair(X,Y)) -> c_8() head#(cons(N,XS)) -> c_9() natsFrom#(N) -> c_10() natsFrom#(X) -> c_11() s#(X) -> c_12() snd#(pair(X,Y)) -> c_14() splitAt#(0(),XS) -> c_15() splitAt#(s(N),cons(X,XS)) -> c_16(U11#(tt(),N,X,activate(XS)),activate#(XS)) - Weak TRS: U11(tt(),N,X,XS) -> U12(splitAt(activate(N),activate(XS)),activate(X)) U12(pair(YS,ZS),X) -> pair(cons(activate(X),YS),ZS) activate(X) -> X activate(n__natsFrom(X)) -> natsFrom(activate(X)) activate(n__s(X)) -> s(activate(X)) afterNth(N,XS) -> snd(splitAt(N,XS)) and(tt(),X) -> activate(X) fst(pair(X,Y)) -> X 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(X,Y)) -> Y splitAt(0(),XS) -> pair(nil(),XS) splitAt(s(N),cons(X,XS)) -> U11(tt(),N,X,activate(XS)) tail(cons(N,XS)) -> activate(XS) take(N,XS) -> fst(splitAt(N,XS)) - Signature: {U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2 ,U11#/4,U12#/2,activate#/1,afterNth#/2,and#/2,fst#/1,head#/1,natsFrom#/1,s#/1,sel#/2,snd#/1,splitAt#/2 ,tail#/1,take#/2} / {0/0,cons/2,n__natsFrom/1,n__s/1,nil/0,pair/2,tt/0,c_1/5,c_2/1,c_3/0,c_4/2,c_5/2,c_6/2 ,c_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/2,c_14/0,c_15/0,c_16/2,c_17/1,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom# ,s#,sel#,snd#,splitAt#,tail#,take#} and constructors {0,cons,n__natsFrom,n__s,nil,pair,tt} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {5,9} by application of Pre({5,9}) = {7}. Here rules are labelled as follows: 1: U11#(tt(),N,X,XS) -> c_1(U12#(splitAt(activate(N),activate(XS)),activate(X)) ,splitAt#(activate(N),activate(XS)) ,activate#(N) ,activate#(XS) ,activate#(X)) 2: U12#(pair(YS,ZS),X) -> c_2(activate#(X)) 3: activate#(n__natsFrom(X)) -> c_4(natsFrom#(activate(X)),activate#(X)) 4: activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) 5: afterNth#(N,XS) -> c_6(snd#(splitAt(N,XS)),splitAt#(N,XS)) 6: and#(tt(),X) -> c_7(activate#(X)) 7: sel#(N,XS) -> c_13(head#(afterNth(N,XS)),afterNth#(N,XS)) 8: tail#(cons(N,XS)) -> c_17(activate#(XS)) 9: take#(N,XS) -> c_18(fst#(splitAt(N,XS)),splitAt#(N,XS)) 10: activate#(X) -> c_3() 11: fst#(pair(X,Y)) -> c_8() 12: head#(cons(N,XS)) -> c_9() 13: natsFrom#(N) -> c_10() 14: natsFrom#(X) -> c_11() 15: s#(X) -> c_12() 16: snd#(pair(X,Y)) -> c_14() 17: splitAt#(0(),XS) -> c_15() 18: splitAt#(s(N),cons(X,XS)) -> c_16(U11#(tt(),N,X,activate(XS)),activate#(XS)) ** Step 1.b:4: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),N,X,XS) -> c_1(U12#(splitAt(activate(N),activate(XS)),activate(X)) ,splitAt#(activate(N),activate(XS)) ,activate#(N) ,activate#(XS) ,activate#(X)) U12#(pair(YS,ZS),X) -> c_2(activate#(X)) activate#(n__natsFrom(X)) -> c_4(natsFrom#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) and#(tt(),X) -> c_7(activate#(X)) sel#(N,XS) -> c_13(head#(afterNth(N,XS)),afterNth#(N,XS)) tail#(cons(N,XS)) -> c_17(activate#(XS)) - Weak DPs: activate#(X) -> c_3() afterNth#(N,XS) -> c_6(snd#(splitAt(N,XS)),splitAt#(N,XS)) fst#(pair(X,Y)) -> c_8() head#(cons(N,XS)) -> c_9() natsFrom#(N) -> c_10() natsFrom#(X) -> c_11() s#(X) -> c_12() snd#(pair(X,Y)) -> c_14() splitAt#(0(),XS) -> c_15() splitAt#(s(N),cons(X,XS)) -> c_16(U11#(tt(),N,X,activate(XS)),activate#(XS)) take#(N,XS) -> c_18(fst#(splitAt(N,XS)),splitAt#(N,XS)) - Weak TRS: U11(tt(),N,X,XS) -> U12(splitAt(activate(N),activate(XS)),activate(X)) U12(pair(YS,ZS),X) -> pair(cons(activate(X),YS),ZS) activate(X) -> X activate(n__natsFrom(X)) -> natsFrom(activate(X)) activate(n__s(X)) -> s(activate(X)) afterNth(N,XS) -> snd(splitAt(N,XS)) and(tt(),X) -> activate(X) fst(pair(X,Y)) -> X 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(X,Y)) -> Y splitAt(0(),XS) -> pair(nil(),XS) splitAt(s(N),cons(X,XS)) -> U11(tt(),N,X,activate(XS)) tail(cons(N,XS)) -> activate(XS) take(N,XS) -> fst(splitAt(N,XS)) - Signature: {U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2 ,U11#/4,U12#/2,activate#/1,afterNth#/2,and#/2,fst#/1,head#/1,natsFrom#/1,s#/1,sel#/2,snd#/1,splitAt#/2 ,tail#/1,take#/2} / {0/0,cons/2,n__natsFrom/1,n__s/1,nil/0,pair/2,tt/0,c_1/5,c_2/1,c_3/0,c_4/2,c_5/2,c_6/2 ,c_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/2,c_14/0,c_15/0,c_16/2,c_17/1,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom# ,s#,sel#,snd#,splitAt#,tail#,take#} and constructors {0,cons,n__natsFrom,n__s,nil,pair,tt} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {6} by application of Pre({6}) = {}. Here rules are labelled as follows: 1: U11#(tt(),N,X,XS) -> c_1(U12#(splitAt(activate(N),activate(XS)),activate(X)) ,splitAt#(activate(N),activate(XS)) ,activate#(N) ,activate#(XS) ,activate#(X)) 2: U12#(pair(YS,ZS),X) -> c_2(activate#(X)) 3: activate#(n__natsFrom(X)) -> c_4(natsFrom#(activate(X)),activate#(X)) 4: activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) 5: and#(tt(),X) -> c_7(activate#(X)) 6: sel#(N,XS) -> c_13(head#(afterNth(N,XS)),afterNth#(N,XS)) 7: tail#(cons(N,XS)) -> c_17(activate#(XS)) 8: activate#(X) -> c_3() 9: afterNth#(N,XS) -> c_6(snd#(splitAt(N,XS)),splitAt#(N,XS)) 10: fst#(pair(X,Y)) -> c_8() 11: head#(cons(N,XS)) -> c_9() 12: natsFrom#(N) -> c_10() 13: natsFrom#(X) -> c_11() 14: s#(X) -> c_12() 15: snd#(pair(X,Y)) -> c_14() 16: splitAt#(0(),XS) -> c_15() 17: splitAt#(s(N),cons(X,XS)) -> c_16(U11#(tt(),N,X,activate(XS)),activate#(XS)) 18: take#(N,XS) -> c_18(fst#(splitAt(N,XS)),splitAt#(N,XS)) ** Step 1.b:5: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),N,X,XS) -> c_1(U12#(splitAt(activate(N),activate(XS)),activate(X)) ,splitAt#(activate(N),activate(XS)) ,activate#(N) ,activate#(XS) ,activate#(X)) U12#(pair(YS,ZS),X) -> c_2(activate#(X)) activate#(n__natsFrom(X)) -> c_4(natsFrom#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) and#(tt(),X) -> c_7(activate#(X)) tail#(cons(N,XS)) -> c_17(activate#(XS)) - Weak DPs: activate#(X) -> c_3() afterNth#(N,XS) -> c_6(snd#(splitAt(N,XS)),splitAt#(N,XS)) fst#(pair(X,Y)) -> c_8() head#(cons(N,XS)) -> c_9() natsFrom#(N) -> c_10() natsFrom#(X) -> c_11() s#(X) -> c_12() sel#(N,XS) -> c_13(head#(afterNth(N,XS)),afterNth#(N,XS)) snd#(pair(X,Y)) -> c_14() splitAt#(0(),XS) -> c_15() splitAt#(s(N),cons(X,XS)) -> c_16(U11#(tt(),N,X,activate(XS)),activate#(XS)) take#(N,XS) -> c_18(fst#(splitAt(N,XS)),splitAt#(N,XS)) - Weak TRS: U11(tt(),N,X,XS) -> U12(splitAt(activate(N),activate(XS)),activate(X)) U12(pair(YS,ZS),X) -> pair(cons(activate(X),YS),ZS) activate(X) -> X activate(n__natsFrom(X)) -> natsFrom(activate(X)) activate(n__s(X)) -> s(activate(X)) afterNth(N,XS) -> snd(splitAt(N,XS)) and(tt(),X) -> activate(X) fst(pair(X,Y)) -> X 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(X,Y)) -> Y splitAt(0(),XS) -> pair(nil(),XS) splitAt(s(N),cons(X,XS)) -> U11(tt(),N,X,activate(XS)) tail(cons(N,XS)) -> activate(XS) take(N,XS) -> fst(splitAt(N,XS)) - Signature: {U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2 ,U11#/4,U12#/2,activate#/1,afterNth#/2,and#/2,fst#/1,head#/1,natsFrom#/1,s#/1,sel#/2,snd#/1,splitAt#/2 ,tail#/1,take#/2} / {0/0,cons/2,n__natsFrom/1,n__s/1,nil/0,pair/2,tt/0,c_1/5,c_2/1,c_3/0,c_4/2,c_5/2,c_6/2 ,c_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/2,c_14/0,c_15/0,c_16/2,c_17/1,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom# ,s#,sel#,snd#,splitAt#,tail#,take#} and constructors {0,cons,n__natsFrom,n__s,nil,pair,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:U11#(tt(),N,X,XS) -> c_1(U12#(splitAt(activate(N),activate(XS)),activate(X)) ,splitAt#(activate(N),activate(XS)) ,activate#(N) ,activate#(XS) ,activate#(X)) -->_5 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):4 -->_4 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):4 -->_3 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):4 -->_5 activate#(n__natsFrom(X)) -> c_4(natsFrom#(activate(X)),activate#(X)):3 -->_4 activate#(n__natsFrom(X)) -> c_4(natsFrom#(activate(X)),activate#(X)):3 -->_3 activate#(n__natsFrom(X)) -> c_4(natsFrom#(activate(X)),activate#(X)):3 -->_1 U12#(pair(YS,ZS),X) -> c_2(activate#(X)):2 -->_2 splitAt#(0(),XS) -> c_15():16 -->_5 activate#(X) -> c_3():7 -->_4 activate#(X) -> c_3():7 -->_3 activate#(X) -> c_3():7 2:S:U12#(pair(YS,ZS),X) -> c_2(activate#(X)) -->_1 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):4 -->_1 activate#(n__natsFrom(X)) -> c_4(natsFrom#(activate(X)),activate#(X)):3 -->_1 activate#(X) -> c_3():7 3:S:activate#(n__natsFrom(X)) -> c_4(natsFrom#(activate(X)),activate#(X)) -->_2 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):4 -->_1 natsFrom#(X) -> c_11():12 -->_1 natsFrom#(N) -> c_10():11 -->_2 activate#(X) -> c_3():7 -->_2 activate#(n__natsFrom(X)) -> c_4(natsFrom#(activate(X)),activate#(X)):3 4:S:activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) -->_1 s#(X) -> c_12():13 -->_2 activate#(X) -> c_3():7 -->_2 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):4 -->_2 activate#(n__natsFrom(X)) -> c_4(natsFrom#(activate(X)),activate#(X)):3 5:S:and#(tt(),X) -> c_7(activate#(X)) -->_1 activate#(X) -> c_3():7 -->_1 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):4 -->_1 activate#(n__natsFrom(X)) -> c_4(natsFrom#(activate(X)),activate#(X)):3 6:S:tail#(cons(N,XS)) -> c_17(activate#(XS)) -->_1 activate#(X) -> c_3():7 -->_1 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):4 -->_1 activate#(n__natsFrom(X)) -> c_4(natsFrom#(activate(X)),activate#(X)):3 7:W:activate#(X) -> c_3() 8:W:afterNth#(N,XS) -> c_6(snd#(splitAt(N,XS)),splitAt#(N,XS)) -->_2 splitAt#(0(),XS) -> c_15():16 -->_1 snd#(pair(X,Y)) -> c_14():15 9:W:fst#(pair(X,Y)) -> c_8() 10:W:head#(cons(N,XS)) -> c_9() 11:W:natsFrom#(N) -> c_10() 12:W:natsFrom#(X) -> c_11() 13:W:s#(X) -> c_12() 14:W:sel#(N,XS) -> c_13(head#(afterNth(N,XS)),afterNth#(N,XS)) -->_1 head#(cons(N,XS)) -> c_9():10 -->_2 afterNth#(N,XS) -> c_6(snd#(splitAt(N,XS)),splitAt#(N,XS)):8 15:W:snd#(pair(X,Y)) -> c_14() 16:W:splitAt#(0(),XS) -> c_15() 17:W:splitAt#(s(N),cons(X,XS)) -> c_16(U11#(tt(),N,X,activate(XS)),activate#(XS)) 18:W:take#(N,XS) -> c_18(fst#(splitAt(N,XS)),splitAt#(N,XS)) -->_2 splitAt#(0(),XS) -> c_15():16 -->_1 fst#(pair(X,Y)) -> c_8():9 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 18: take#(N,XS) -> c_18(fst#(splitAt(N,XS)),splitAt#(N,XS)) 17: splitAt#(s(N),cons(X,XS)) -> c_16(U11#(tt(),N,X,activate(XS)),activate#(XS)) 14: sel#(N,XS) -> c_13(head#(afterNth(N,XS)),afterNth#(N,XS)) 10: head#(cons(N,XS)) -> c_9() 9: fst#(pair(X,Y)) -> c_8() 8: afterNth#(N,XS) -> c_6(snd#(splitAt(N,XS)),splitAt#(N,XS)) 15: snd#(pair(X,Y)) -> c_14() 16: splitAt#(0(),XS) -> c_15() 11: natsFrom#(N) -> c_10() 12: natsFrom#(X) -> c_11() 7: activate#(X) -> c_3() 13: s#(X) -> c_12() ** Step 1.b:6: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),N,X,XS) -> c_1(U12#(splitAt(activate(N),activate(XS)),activate(X)) ,splitAt#(activate(N),activate(XS)) ,activate#(N) ,activate#(XS) ,activate#(X)) U12#(pair(YS,ZS),X) -> c_2(activate#(X)) activate#(n__natsFrom(X)) -> c_4(natsFrom#(activate(X)),activate#(X)) activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) and#(tt(),X) -> c_7(activate#(X)) tail#(cons(N,XS)) -> c_17(activate#(XS)) - Weak TRS: U11(tt(),N,X,XS) -> U12(splitAt(activate(N),activate(XS)),activate(X)) U12(pair(YS,ZS),X) -> pair(cons(activate(X),YS),ZS) activate(X) -> X activate(n__natsFrom(X)) -> natsFrom(activate(X)) activate(n__s(X)) -> s(activate(X)) afterNth(N,XS) -> snd(splitAt(N,XS)) and(tt(),X) -> activate(X) fst(pair(X,Y)) -> X 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(X,Y)) -> Y splitAt(0(),XS) -> pair(nil(),XS) splitAt(s(N),cons(X,XS)) -> U11(tt(),N,X,activate(XS)) tail(cons(N,XS)) -> activate(XS) take(N,XS) -> fst(splitAt(N,XS)) - Signature: {U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2 ,U11#/4,U12#/2,activate#/1,afterNth#/2,and#/2,fst#/1,head#/1,natsFrom#/1,s#/1,sel#/2,snd#/1,splitAt#/2 ,tail#/1,take#/2} / {0/0,cons/2,n__natsFrom/1,n__s/1,nil/0,pair/2,tt/0,c_1/5,c_2/1,c_3/0,c_4/2,c_5/2,c_6/2 ,c_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/2,c_14/0,c_15/0,c_16/2,c_17/1,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom# ,s#,sel#,snd#,splitAt#,tail#,take#} and constructors {0,cons,n__natsFrom,n__s,nil,pair,tt} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:U11#(tt(),N,X,XS) -> c_1(U12#(splitAt(activate(N),activate(XS)),activate(X)) ,splitAt#(activate(N),activate(XS)) ,activate#(N) ,activate#(XS) ,activate#(X)) -->_5 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):4 -->_4 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):4 -->_3 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):4 -->_5 activate#(n__natsFrom(X)) -> c_4(natsFrom#(activate(X)),activate#(X)):3 -->_4 activate#(n__natsFrom(X)) -> c_4(natsFrom#(activate(X)),activate#(X)):3 -->_3 activate#(n__natsFrom(X)) -> c_4(natsFrom#(activate(X)),activate#(X)):3 -->_1 U12#(pair(YS,ZS),X) -> c_2(activate#(X)):2 2:S:U12#(pair(YS,ZS),X) -> c_2(activate#(X)) -->_1 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):4 -->_1 activate#(n__natsFrom(X)) -> c_4(natsFrom#(activate(X)),activate#(X)):3 3:S:activate#(n__natsFrom(X)) -> c_4(natsFrom#(activate(X)),activate#(X)) -->_2 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):4 -->_2 activate#(n__natsFrom(X)) -> c_4(natsFrom#(activate(X)),activate#(X)):3 4:S:activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)) -->_2 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):4 -->_2 activate#(n__natsFrom(X)) -> c_4(natsFrom#(activate(X)),activate#(X)):3 5:S:and#(tt(),X) -> c_7(activate#(X)) -->_1 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):4 -->_1 activate#(n__natsFrom(X)) -> c_4(natsFrom#(activate(X)),activate#(X)):3 6:S:tail#(cons(N,XS)) -> c_17(activate#(XS)) -->_1 activate#(n__s(X)) -> c_5(s#(activate(X)),activate#(X)):4 -->_1 activate#(n__natsFrom(X)) -> c_4(natsFrom#(activate(X)),activate#(X)):3 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: U11#(tt(),N,X,XS) -> c_1(U12#(splitAt(activate(N),activate(XS)),activate(X)) ,activate#(N) ,activate#(XS) ,activate#(X)) activate#(n__natsFrom(X)) -> c_4(activate#(X)) activate#(n__s(X)) -> c_5(activate#(X)) ** Step 1.b:7: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),N,X,XS) -> c_1(U12#(splitAt(activate(N),activate(XS)),activate(X)) ,activate#(N) ,activate#(XS) ,activate#(X)) U12#(pair(YS,ZS),X) -> c_2(activate#(X)) activate#(n__natsFrom(X)) -> c_4(activate#(X)) activate#(n__s(X)) -> c_5(activate#(X)) and#(tt(),X) -> c_7(activate#(X)) tail#(cons(N,XS)) -> c_17(activate#(XS)) - Weak TRS: U11(tt(),N,X,XS) -> U12(splitAt(activate(N),activate(XS)),activate(X)) U12(pair(YS,ZS),X) -> pair(cons(activate(X),YS),ZS) activate(X) -> X activate(n__natsFrom(X)) -> natsFrom(activate(X)) activate(n__s(X)) -> s(activate(X)) afterNth(N,XS) -> snd(splitAt(N,XS)) and(tt(),X) -> activate(X) fst(pair(X,Y)) -> X 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(X,Y)) -> Y splitAt(0(),XS) -> pair(nil(),XS) splitAt(s(N),cons(X,XS)) -> U11(tt(),N,X,activate(XS)) tail(cons(N,XS)) -> activate(XS) take(N,XS) -> fst(splitAt(N,XS)) - Signature: {U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2 ,U11#/4,U12#/2,activate#/1,afterNth#/2,and#/2,fst#/1,head#/1,natsFrom#/1,s#/1,sel#/2,snd#/1,splitAt#/2 ,tail#/1,take#/2} / {0/0,cons/2,n__natsFrom/1,n__s/1,nil/0,pair/2,tt/0,c_1/4,c_2/1,c_3/0,c_4/1,c_5/1,c_6/2 ,c_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/2,c_14/0,c_15/0,c_16/2,c_17/1,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom# ,s#,sel#,snd#,splitAt#,tail#,take#} and constructors {0,cons,n__natsFrom,n__s,nil,pair,tt} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: U11(tt(),N,X,XS) -> U12(splitAt(activate(N),activate(XS)),activate(X)) U12(pair(YS,ZS),X) -> pair(cons(activate(X),YS),ZS) activate(X) -> X activate(n__natsFrom(X)) -> natsFrom(activate(X)) activate(n__s(X)) -> s(activate(X)) natsFrom(N) -> cons(N,n__natsFrom(n__s(N))) natsFrom(X) -> n__natsFrom(X) s(X) -> n__s(X) splitAt(0(),XS) -> pair(nil(),XS) splitAt(s(N),cons(X,XS)) -> U11(tt(),N,X,activate(XS)) U11#(tt(),N,X,XS) -> c_1(U12#(splitAt(activate(N),activate(XS)),activate(X)) ,activate#(N) ,activate#(XS) ,activate#(X)) U12#(pair(YS,ZS),X) -> c_2(activate#(X)) activate#(n__natsFrom(X)) -> c_4(activate#(X)) activate#(n__s(X)) -> c_5(activate#(X)) and#(tt(),X) -> c_7(activate#(X)) tail#(cons(N,XS)) -> c_17(activate#(XS)) ** Step 1.b:8: RemoveHeads WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),N,X,XS) -> c_1(U12#(splitAt(activate(N),activate(XS)),activate(X)) ,activate#(N) ,activate#(XS) ,activate#(X)) U12#(pair(YS,ZS),X) -> c_2(activate#(X)) activate#(n__natsFrom(X)) -> c_4(activate#(X)) activate#(n__s(X)) -> c_5(activate#(X)) and#(tt(),X) -> c_7(activate#(X)) tail#(cons(N,XS)) -> c_17(activate#(XS)) - Weak TRS: U11(tt(),N,X,XS) -> U12(splitAt(activate(N),activate(XS)),activate(X)) U12(pair(YS,ZS),X) -> pair(cons(activate(X),YS),ZS) activate(X) -> X activate(n__natsFrom(X)) -> natsFrom(activate(X)) activate(n__s(X)) -> s(activate(X)) natsFrom(N) -> cons(N,n__natsFrom(n__s(N))) natsFrom(X) -> n__natsFrom(X) s(X) -> n__s(X) splitAt(0(),XS) -> pair(nil(),XS) splitAt(s(N),cons(X,XS)) -> U11(tt(),N,X,activate(XS)) - Signature: {U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2 ,U11#/4,U12#/2,activate#/1,afterNth#/2,and#/2,fst#/1,head#/1,natsFrom#/1,s#/1,sel#/2,snd#/1,splitAt#/2 ,tail#/1,take#/2} / {0/0,cons/2,n__natsFrom/1,n__s/1,nil/0,pair/2,tt/0,c_1/4,c_2/1,c_3/0,c_4/1,c_5/1,c_6/2 ,c_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/2,c_14/0,c_15/0,c_16/2,c_17/1,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom# ,s#,sel#,snd#,splitAt#,tail#,take#} and constructors {0,cons,n__natsFrom,n__s,nil,pair,tt} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:U11#(tt(),N,X,XS) -> c_1(U12#(splitAt(activate(N),activate(XS)),activate(X)) ,activate#(N) ,activate#(XS) ,activate#(X)) -->_4 activate#(n__s(X)) -> c_5(activate#(X)):4 -->_3 activate#(n__s(X)) -> c_5(activate#(X)):4 -->_2 activate#(n__s(X)) -> c_5(activate#(X)):4 -->_4 activate#(n__natsFrom(X)) -> c_4(activate#(X)):3 -->_3 activate#(n__natsFrom(X)) -> c_4(activate#(X)):3 -->_2 activate#(n__natsFrom(X)) -> c_4(activate#(X)):3 -->_1 U12#(pair(YS,ZS),X) -> c_2(activate#(X)):2 2:S:U12#(pair(YS,ZS),X) -> c_2(activate#(X)) -->_1 activate#(n__s(X)) -> c_5(activate#(X)):4 -->_1 activate#(n__natsFrom(X)) -> c_4(activate#(X)):3 3:S:activate#(n__natsFrom(X)) -> c_4(activate#(X)) -->_1 activate#(n__s(X)) -> c_5(activate#(X)):4 -->_1 activate#(n__natsFrom(X)) -> c_4(activate#(X)):3 4:S:activate#(n__s(X)) -> c_5(activate#(X)) -->_1 activate#(n__s(X)) -> c_5(activate#(X)):4 -->_1 activate#(n__natsFrom(X)) -> c_4(activate#(X)):3 5:S:and#(tt(),X) -> c_7(activate#(X)) -->_1 activate#(n__s(X)) -> c_5(activate#(X)):4 -->_1 activate#(n__natsFrom(X)) -> c_4(activate#(X)):3 6:S:tail#(cons(N,XS)) -> c_17(activate#(XS)) -->_1 activate#(n__s(X)) -> c_5(activate#(X)):4 -->_1 activate#(n__natsFrom(X)) -> c_4(activate#(X)):3 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). [(5,and#(tt(),X) -> c_7(activate#(X))),(6,tail#(cons(N,XS)) -> c_17(activate#(XS)))] ** Step 1.b:9: NaturalPI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),N,X,XS) -> c_1(U12#(splitAt(activate(N),activate(XS)),activate(X)) ,activate#(N) ,activate#(XS) ,activate#(X)) U12#(pair(YS,ZS),X) -> c_2(activate#(X)) activate#(n__natsFrom(X)) -> c_4(activate#(X)) activate#(n__s(X)) -> c_5(activate#(X)) - Weak TRS: U11(tt(),N,X,XS) -> U12(splitAt(activate(N),activate(XS)),activate(X)) U12(pair(YS,ZS),X) -> pair(cons(activate(X),YS),ZS) activate(X) -> X activate(n__natsFrom(X)) -> natsFrom(activate(X)) activate(n__s(X)) -> s(activate(X)) natsFrom(N) -> cons(N,n__natsFrom(n__s(N))) natsFrom(X) -> n__natsFrom(X) s(X) -> n__s(X) splitAt(0(),XS) -> pair(nil(),XS) splitAt(s(N),cons(X,XS)) -> U11(tt(),N,X,activate(XS)) - Signature: {U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2 ,U11#/4,U12#/2,activate#/1,afterNth#/2,and#/2,fst#/1,head#/1,natsFrom#/1,s#/1,sel#/2,snd#/1,splitAt#/2 ,tail#/1,take#/2} / {0/0,cons/2,n__natsFrom/1,n__s/1,nil/0,pair/2,tt/0,c_1/4,c_2/1,c_3/0,c_4/1,c_5/1,c_6/2 ,c_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/2,c_14/0,c_15/0,c_16/2,c_17/1,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom# ,s#,sel#,snd#,splitAt#,tail#,take#} and constructors {0,cons,n__natsFrom,n__s,nil,pair,tt} + Applied Processor: NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(linear): The following argument positions are considered usable: uargs(c_1) = {1,2,3,4}, uargs(c_2) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1} Following symbols are considered usable: {activate,natsFrom,s,U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom#,s#,sel#,snd#,splitAt#,tail# ,take#} TcT has computed the following interpretation: p(0) = 1 p(U11) = 2 + 2*x3 p(U12) = 4 + 2*x1 p(activate) = 4 + 2*x1 p(afterNth) = 4*x2 p(and) = 4 p(cons) = 0 p(fst) = x1 p(head) = 1 + 2*x1 p(n__natsFrom) = 2 + x1 p(n__s) = 2 + x1 p(natsFrom) = 3 + x1 p(nil) = 0 p(pair) = x1 p(s) = 2 + x1 p(sel) = 1 + x2 p(snd) = 2 p(splitAt) = 4 p(tail) = 2 + x1 p(take) = 1 + x1 p(tt) = 2 p(U11#) = 7 + 4*x1 + 7*x2 + 6*x3 + 4*x4 p(U12#) = 2*x2 p(activate#) = x1 p(afterNth#) = 2 + x2 p(and#) = 1 + 4*x1 + 2*x2 p(fst#) = x1 p(head#) = 0 p(natsFrom#) = 1 p(s#) = 4*x1 p(sel#) = 4 + x2 p(snd#) = 0 p(splitAt#) = 4 p(tail#) = 1 p(take#) = 1 p(c_1) = 2 + x1 + x2 + x3 + x4 p(c_2) = x1 p(c_3) = 4 p(c_4) = x1 p(c_5) = x1 p(c_6) = 0 p(c_7) = 2 + x1 p(c_8) = 0 p(c_9) = 0 p(c_10) = 0 p(c_11) = 0 p(c_12) = 0 p(c_13) = 4 + x1 + x2 p(c_14) = 2 p(c_15) = 1 p(c_16) = 1 + x1 p(c_17) = 0 p(c_18) = x2 Following rules are strictly oriented: U11#(tt(),N,X,XS) = 15 + 7*N + 6*X + 4*XS > 10 + N + 5*X + XS = c_1(U12#(splitAt(activate(N),activate(XS)),activate(X)),activate#(N),activate#(XS),activate#(X)) activate#(n__natsFrom(X)) = 2 + X > X = c_4(activate#(X)) activate#(n__s(X)) = 2 + X > X = c_5(activate#(X)) Following rules are (at-least) weakly oriented: U12#(pair(YS,ZS),X) = 2*X >= X = c_2(activate#(X)) activate(X) = 4 + 2*X >= X = X activate(n__natsFrom(X)) = 8 + 2*X >= 7 + 2*X = natsFrom(activate(X)) activate(n__s(X)) = 8 + 2*X >= 6 + 2*X = s(activate(X)) natsFrom(N) = 3 + N >= 0 = cons(N,n__natsFrom(n__s(N))) natsFrom(X) = 3 + X >= 2 + X = n__natsFrom(X) s(X) = 2 + X >= 2 + X = n__s(X) ** Step 1.b:10: NaturalPI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U12#(pair(YS,ZS),X) -> c_2(activate#(X)) - Weak DPs: U11#(tt(),N,X,XS) -> c_1(U12#(splitAt(activate(N),activate(XS)),activate(X)) ,activate#(N) ,activate#(XS) ,activate#(X)) activate#(n__natsFrom(X)) -> c_4(activate#(X)) activate#(n__s(X)) -> c_5(activate#(X)) - Weak TRS: U11(tt(),N,X,XS) -> U12(splitAt(activate(N),activate(XS)),activate(X)) U12(pair(YS,ZS),X) -> pair(cons(activate(X),YS),ZS) activate(X) -> X activate(n__natsFrom(X)) -> natsFrom(activate(X)) activate(n__s(X)) -> s(activate(X)) natsFrom(N) -> cons(N,n__natsFrom(n__s(N))) natsFrom(X) -> n__natsFrom(X) s(X) -> n__s(X) splitAt(0(),XS) -> pair(nil(),XS) splitAt(s(N),cons(X,XS)) -> U11(tt(),N,X,activate(XS)) - Signature: {U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2 ,U11#/4,U12#/2,activate#/1,afterNth#/2,and#/2,fst#/1,head#/1,natsFrom#/1,s#/1,sel#/2,snd#/1,splitAt#/2 ,tail#/1,take#/2} / {0/0,cons/2,n__natsFrom/1,n__s/1,nil/0,pair/2,tt/0,c_1/4,c_2/1,c_3/0,c_4/1,c_5/1,c_6/2 ,c_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/2,c_14/0,c_15/0,c_16/2,c_17/1,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom# ,s#,sel#,snd#,splitAt#,tail#,take#} and constructors {0,cons,n__natsFrom,n__s,nil,pair,tt} + Applied Processor: NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(linear): The following argument positions are considered usable: uargs(c_1) = {1,2,3,4}, uargs(c_2) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1} Following symbols are considered usable: {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom#,s#,sel#,snd#,splitAt#,tail#,take#} TcT has computed the following interpretation: p(0) = 1 p(U11) = 1 + 3*x1 + x2 + x3 + x4 p(U12) = 2 p(activate) = 2 p(afterNth) = 0 p(and) = 0 p(cons) = x2 p(fst) = x1 p(head) = 0 p(n__natsFrom) = x1 p(n__s) = 1 p(natsFrom) = 4*x1 p(nil) = 1 p(pair) = 0 p(s) = 0 p(sel) = 0 p(snd) = 0 p(splitAt) = 2*x1 + 4*x2 p(tail) = 0 p(take) = 0 p(tt) = 2 p(U11#) = 4 + x2 p(U12#) = 3 p(activate#) = 0 p(afterNth#) = 0 p(and#) = 0 p(fst#) = 0 p(head#) = 0 p(natsFrom#) = 0 p(s#) = 0 p(sel#) = 4*x2 p(snd#) = 1 + x1 p(splitAt#) = x1 p(tail#) = 1 p(take#) = 2*x2 p(c_1) = x1 + x2 + x3 + x4 p(c_2) = 2 + x1 p(c_3) = 0 p(c_4) = x1 p(c_5) = x1 p(c_6) = x2 p(c_7) = 4 + x1 p(c_8) = 1 p(c_9) = 0 p(c_10) = 1 p(c_11) = 2 p(c_12) = 4 p(c_13) = 2 p(c_14) = 0 p(c_15) = 0 p(c_16) = 4 + x2 p(c_17) = 2 + x1 p(c_18) = 2 + x2 Following rules are strictly oriented: U12#(pair(YS,ZS),X) = 3 > 2 = c_2(activate#(X)) Following rules are (at-least) weakly oriented: U11#(tt(),N,X,XS) = 4 + N >= 3 = c_1(U12#(splitAt(activate(N),activate(XS)),activate(X)),activate#(N),activate#(XS),activate#(X)) activate#(n__natsFrom(X)) = 0 >= 0 = c_4(activate#(X)) activate#(n__s(X)) = 0 >= 0 = c_5(activate#(X)) ** Step 1.b:11: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: U11#(tt(),N,X,XS) -> c_1(U12#(splitAt(activate(N),activate(XS)),activate(X)) ,activate#(N) ,activate#(XS) ,activate#(X)) U12#(pair(YS,ZS),X) -> c_2(activate#(X)) activate#(n__natsFrom(X)) -> c_4(activate#(X)) activate#(n__s(X)) -> c_5(activate#(X)) - Weak TRS: U11(tt(),N,X,XS) -> U12(splitAt(activate(N),activate(XS)),activate(X)) U12(pair(YS,ZS),X) -> pair(cons(activate(X),YS),ZS) activate(X) -> X activate(n__natsFrom(X)) -> natsFrom(activate(X)) activate(n__s(X)) -> s(activate(X)) natsFrom(N) -> cons(N,n__natsFrom(n__s(N))) natsFrom(X) -> n__natsFrom(X) s(X) -> n__s(X) splitAt(0(),XS) -> pair(nil(),XS) splitAt(s(N),cons(X,XS)) -> U11(tt(),N,X,activate(XS)) - Signature: {U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2 ,U11#/4,U12#/2,activate#/1,afterNth#/2,and#/2,fst#/1,head#/1,natsFrom#/1,s#/1,sel#/2,snd#/1,splitAt#/2 ,tail#/1,take#/2} / {0/0,cons/2,n__natsFrom/1,n__s/1,nil/0,pair/2,tt/0,c_1/4,c_2/1,c_3/0,c_4/1,c_5/1,c_6/2 ,c_7/1,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/2,c_14/0,c_15/0,c_16/2,c_17/1,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom# ,s#,sel#,snd#,splitAt#,tail#,take#} and constructors {0,cons,n__natsFrom,n__s,nil,pair,tt} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^1))