*** 1 Progress [(?,O(n^1))] *** Considered Problem: Strict DP Rules: Strict TRS 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(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(s(N))) natsFrom(X) -> n__natsFrom(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)) Weak DP Rules: Weak TRS Rules: Signature: {U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/1,sel/2,snd/1,splitAt/2,tail/1,take/2} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,tt/0} Obligation: Innermost basic terms: {U11,U12,activate,afterNth,and,fst,head,natsFrom,sel,snd,splitAt,tail,take}/{0,cons,n__natsFrom,nil,pair,s,tt} Applied Processor: DependencyPairs {dpKind_ = DT} Proof: 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#(X)) afterNth#(N,XS) -> c_5(snd#(splitAt(N,XS)),splitAt#(N,XS)) and#(tt(),X) -> c_6(activate#(X)) fst#(pair(X,Y)) -> c_7() head#(cons(N,XS)) -> c_8() natsFrom#(N) -> c_9() natsFrom#(X) -> c_10() sel#(N,XS) -> c_11(head#(afterNth(N,XS)),afterNth#(N,XS)) snd#(pair(X,Y)) -> c_12() splitAt#(0(),XS) -> c_13() splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)),activate#(XS)) tail#(cons(N,XS)) -> c_15(activate#(XS)) take#(N,XS) -> c_16(fst#(splitAt(N,XS)),splitAt#(N,XS)) Weak DPs and mark the set of starting terms. *** 1.1 Progress [(?,O(n^1))] *** Considered Problem: Strict DP Rules: 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#(X)) afterNth#(N,XS) -> c_5(snd#(splitAt(N,XS)),splitAt#(N,XS)) and#(tt(),X) -> c_6(activate#(X)) fst#(pair(X,Y)) -> c_7() head#(cons(N,XS)) -> c_8() natsFrom#(N) -> c_9() natsFrom#(X) -> c_10() sel#(N,XS) -> c_11(head#(afterNth(N,XS)),afterNth#(N,XS)) snd#(pair(X,Y)) -> c_12() splitAt#(0(),XS) -> c_13() splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)),activate#(XS)) tail#(cons(N,XS)) -> c_15(activate#(XS)) take#(N,XS) -> c_16(fst#(splitAt(N,XS)),splitAt#(N,XS)) Strict TRS Rules: Weak DP Rules: Weak TRS 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(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(s(N))) natsFrom(X) -> n__natsFrom(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,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,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,tt/0,c_1/5,c_2/1,c_3/0,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/2,c_12/0,c_13/0,c_14/2,c_15/1,c_16/2} Obligation: Innermost basic terms: {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom#,sel#,snd#,splitAt#,tail#,take#}/{0,cons,n__natsFrom,nil,pair,s,tt} Applied Processor: UsableRules Proof: 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(X) afterNth(N,XS) -> snd(splitAt(N,XS)) natsFrom(N) -> cons(N,n__natsFrom(s(N))) natsFrom(X) -> n__natsFrom(X) snd(pair(X,Y)) -> Y 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)),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#(X)) afterNth#(N,XS) -> c_5(snd#(splitAt(N,XS)),splitAt#(N,XS)) and#(tt(),X) -> c_6(activate#(X)) fst#(pair(X,Y)) -> c_7() head#(cons(N,XS)) -> c_8() natsFrom#(N) -> c_9() natsFrom#(X) -> c_10() sel#(N,XS) -> c_11(head#(afterNth(N,XS)),afterNth#(N,XS)) snd#(pair(X,Y)) -> c_12() splitAt#(0(),XS) -> c_13() splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)),activate#(XS)) tail#(cons(N,XS)) -> c_15(activate#(XS)) take#(N,XS) -> c_16(fst#(splitAt(N,XS)),splitAt#(N,XS)) *** 1.1.1 Progress [(?,O(n^1))] *** Considered Problem: Strict DP Rules: 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#(X)) afterNth#(N,XS) -> c_5(snd#(splitAt(N,XS)),splitAt#(N,XS)) and#(tt(),X) -> c_6(activate#(X)) fst#(pair(X,Y)) -> c_7() head#(cons(N,XS)) -> c_8() natsFrom#(N) -> c_9() natsFrom#(X) -> c_10() sel#(N,XS) -> c_11(head#(afterNth(N,XS)),afterNth#(N,XS)) snd#(pair(X,Y)) -> c_12() splitAt#(0(),XS) -> c_13() splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)),activate#(XS)) tail#(cons(N,XS)) -> c_15(activate#(XS)) take#(N,XS) -> c_16(fst#(splitAt(N,XS)),splitAt#(N,XS)) Strict TRS Rules: Weak DP Rules: Weak TRS 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(X) afterNth(N,XS) -> snd(splitAt(N,XS)) natsFrom(N) -> cons(N,n__natsFrom(s(N))) natsFrom(X) -> n__natsFrom(X) snd(pair(X,Y)) -> Y 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,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,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,tt/0,c_1/5,c_2/1,c_3/0,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/2,c_12/0,c_13/0,c_14/2,c_15/1,c_16/2} Obligation: Innermost basic terms: {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom#,sel#,snd#,splitAt#,tail#,take#}/{0,cons,n__natsFrom,nil,pair,s,tt} Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} Proof: We estimate the number of application of {3,7,8,9,10,12,13} by application of Pre({3,7,8,9,10,12,13}) = {1,2,4,5,6,11,14,15,16}. 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#(X)) 5: afterNth#(N,XS) -> c_5(snd#(splitAt(N,XS)) ,splitAt#(N,XS)) 6: and#(tt(),X) -> c_6(activate#(X)) 7: fst#(pair(X,Y)) -> c_7() 8: head#(cons(N,XS)) -> c_8() 9: natsFrom#(N) -> c_9() 10: natsFrom#(X) -> c_10() 11: sel#(N,XS) -> c_11(head#(afterNth(N,XS)) ,afterNth#(N,XS)) 12: snd#(pair(X,Y)) -> c_12() 13: splitAt#(0(),XS) -> c_13() 14: splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)) ,activate#(XS)) 15: tail#(cons(N,XS)) -> c_15(activate#(XS)) 16: take#(N,XS) -> c_16(fst#(splitAt(N,XS)) ,splitAt#(N,XS)) *** 1.1.1.1 Progress [(?,O(n^1))] *** Considered Problem: Strict DP Rules: 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#(X)) afterNth#(N,XS) -> c_5(snd#(splitAt(N,XS)),splitAt#(N,XS)) and#(tt(),X) -> c_6(activate#(X)) sel#(N,XS) -> c_11(head#(afterNth(N,XS)),afterNth#(N,XS)) splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)),activate#(XS)) tail#(cons(N,XS)) -> c_15(activate#(XS)) take#(N,XS) -> c_16(fst#(splitAt(N,XS)),splitAt#(N,XS)) Strict TRS Rules: Weak DP Rules: activate#(X) -> c_3() fst#(pair(X,Y)) -> c_7() head#(cons(N,XS)) -> c_8() natsFrom#(N) -> c_9() natsFrom#(X) -> c_10() snd#(pair(X,Y)) -> c_12() splitAt#(0(),XS) -> c_13() Weak TRS 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(X) afterNth(N,XS) -> snd(splitAt(N,XS)) natsFrom(N) -> cons(N,n__natsFrom(s(N))) natsFrom(X) -> n__natsFrom(X) snd(pair(X,Y)) -> Y 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,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,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,tt/0,c_1/5,c_2/1,c_3/0,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/2,c_12/0,c_13/0,c_14/2,c_15/1,c_16/2} Obligation: Innermost basic terms: {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom#,sel#,snd#,splitAt#,tail#,take#}/{0,cons,n__natsFrom,nil,pair,s,tt} Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} Proof: We estimate the number of application of {3} by application of Pre({3}) = {1,2,5,7,8}. 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#(X)) 4: afterNth#(N,XS) -> c_5(snd#(splitAt(N,XS)) ,splitAt#(N,XS)) 5: and#(tt(),X) -> c_6(activate#(X)) 6: sel#(N,XS) -> c_11(head#(afterNth(N,XS)) ,afterNth#(N,XS)) 7: splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)) ,activate#(XS)) 8: tail#(cons(N,XS)) -> c_15(activate#(XS)) 9: take#(N,XS) -> c_16(fst#(splitAt(N,XS)) ,splitAt#(N,XS)) 10: activate#(X) -> c_3() 11: fst#(pair(X,Y)) -> c_7() 12: head#(cons(N,XS)) -> c_8() 13: natsFrom#(N) -> c_9() 14: natsFrom#(X) -> c_10() 15: snd#(pair(X,Y)) -> c_12() 16: splitAt#(0(),XS) -> c_13() *** 1.1.1.1.1 Progress [(?,O(n^1))] *** Considered Problem: Strict DP Rules: 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)) afterNth#(N,XS) -> c_5(snd#(splitAt(N,XS)),splitAt#(N,XS)) and#(tt(),X) -> c_6(activate#(X)) sel#(N,XS) -> c_11(head#(afterNth(N,XS)),afterNth#(N,XS)) splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)),activate#(XS)) tail#(cons(N,XS)) -> c_15(activate#(XS)) take#(N,XS) -> c_16(fst#(splitAt(N,XS)),splitAt#(N,XS)) Strict TRS Rules: Weak DP Rules: activate#(X) -> c_3() activate#(n__natsFrom(X)) -> c_4(natsFrom#(X)) fst#(pair(X,Y)) -> c_7() head#(cons(N,XS)) -> c_8() natsFrom#(N) -> c_9() natsFrom#(X) -> c_10() snd#(pair(X,Y)) -> c_12() splitAt#(0(),XS) -> c_13() Weak TRS 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(X) afterNth(N,XS) -> snd(splitAt(N,XS)) natsFrom(N) -> cons(N,n__natsFrom(s(N))) natsFrom(X) -> n__natsFrom(X) snd(pair(X,Y)) -> Y 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,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,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,tt/0,c_1/5,c_2/1,c_3/0,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/2,c_12/0,c_13/0,c_14/2,c_15/1,c_16/2} Obligation: Innermost basic terms: {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom#,sel#,snd#,splitAt#,tail#,take#}/{0,cons,n__natsFrom,nil,pair,s,tt} Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} Proof: We estimate the number of application of {2,4,7} by application of Pre({2,4,7}) = {1}. 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: afterNth#(N,XS) -> c_5(snd#(splitAt(N,XS)) ,splitAt#(N,XS)) 4: and#(tt(),X) -> c_6(activate#(X)) 5: sel#(N,XS) -> c_11(head#(afterNth(N,XS)) ,afterNth#(N,XS)) 6: splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)) ,activate#(XS)) 7: tail#(cons(N,XS)) -> c_15(activate#(XS)) 8: take#(N,XS) -> c_16(fst#(splitAt(N,XS)) ,splitAt#(N,XS)) 9: activate#(X) -> c_3() 10: activate#(n__natsFrom(X)) -> c_4(natsFrom#(X)) 11: fst#(pair(X,Y)) -> c_7() 12: head#(cons(N,XS)) -> c_8() 13: natsFrom#(N) -> c_9() 14: natsFrom#(X) -> c_10() 15: snd#(pair(X,Y)) -> c_12() 16: splitAt#(0(),XS) -> c_13() *** 1.1.1.1.1.1 Progress [(?,O(n^1))] *** Considered Problem: Strict DP Rules: 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)) afterNth#(N,XS) -> c_5(snd#(splitAt(N,XS)),splitAt#(N,XS)) sel#(N,XS) -> c_11(head#(afterNth(N,XS)),afterNth#(N,XS)) splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)),activate#(XS)) take#(N,XS) -> c_16(fst#(splitAt(N,XS)),splitAt#(N,XS)) Strict TRS Rules: Weak DP Rules: U12#(pair(YS,ZS),X) -> c_2(activate#(X)) activate#(X) -> c_3() activate#(n__natsFrom(X)) -> c_4(natsFrom#(X)) and#(tt(),X) -> c_6(activate#(X)) fst#(pair(X,Y)) -> c_7() head#(cons(N,XS)) -> c_8() natsFrom#(N) -> c_9() natsFrom#(X) -> c_10() snd#(pair(X,Y)) -> c_12() splitAt#(0(),XS) -> c_13() tail#(cons(N,XS)) -> c_15(activate#(XS)) Weak TRS 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(X) afterNth(N,XS) -> snd(splitAt(N,XS)) natsFrom(N) -> cons(N,n__natsFrom(s(N))) natsFrom(X) -> n__natsFrom(X) snd(pair(X,Y)) -> Y 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,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,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,tt/0,c_1/5,c_2/1,c_3/0,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/2,c_12/0,c_13/0,c_14/2,c_15/1,c_16/2} Obligation: Innermost basic terms: {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom#,sel#,snd#,splitAt#,tail#,take#}/{0,cons,n__natsFrom,nil,pair,s,tt} Applied Processor: RemoveWeakSuffixes Proof: 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__natsFrom(X)) -> c_4(natsFrom#(X)):8 -->_4 activate#(n__natsFrom(X)) -> c_4(natsFrom#(X)):8 -->_3 activate#(n__natsFrom(X)) -> c_4(natsFrom#(X)):8 -->_1 U12#(pair(YS,ZS),X) -> c_2(activate#(X)):6 -->_2 splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)),activate#(XS)):4 -->_2 splitAt#(0(),XS) -> c_13():15 -->_5 activate#(X) -> c_3():7 -->_4 activate#(X) -> c_3():7 -->_3 activate#(X) -> c_3():7 2:S:afterNth#(N,XS) -> c_5(snd#(splitAt(N,XS)),splitAt#(N,XS)) -->_2 splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)),activate#(XS)):4 -->_2 splitAt#(0(),XS) -> c_13():15 -->_1 snd#(pair(X,Y)) -> c_12():14 3:S:sel#(N,XS) -> c_11(head#(afterNth(N,XS)),afterNth#(N,XS)) -->_1 head#(cons(N,XS)) -> c_8():11 -->_2 afterNth#(N,XS) -> c_5(snd#(splitAt(N,XS)),splitAt#(N,XS)):2 4:S:splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)),activate#(XS)) -->_2 activate#(n__natsFrom(X)) -> c_4(natsFrom#(X)):8 -->_2 activate#(X) -> c_3():7 -->_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)):1 5:S:take#(N,XS) -> c_16(fst#(splitAt(N,XS)),splitAt#(N,XS)) -->_2 splitAt#(0(),XS) -> c_13():15 -->_1 fst#(pair(X,Y)) -> c_7():10 -->_2 splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)),activate#(XS)):4 6:W:U12#(pair(YS,ZS),X) -> c_2(activate#(X)) -->_1 activate#(n__natsFrom(X)) -> c_4(natsFrom#(X)):8 -->_1 activate#(X) -> c_3():7 7:W:activate#(X) -> c_3() 8:W:activate#(n__natsFrom(X)) -> c_4(natsFrom#(X)) -->_1 natsFrom#(X) -> c_10():13 -->_1 natsFrom#(N) -> c_9():12 9:W:and#(tt(),X) -> c_6(activate#(X)) -->_1 activate#(n__natsFrom(X)) -> c_4(natsFrom#(X)):8 -->_1 activate#(X) -> c_3():7 10:W:fst#(pair(X,Y)) -> c_7() 11:W:head#(cons(N,XS)) -> c_8() 12:W:natsFrom#(N) -> c_9() 13:W:natsFrom#(X) -> c_10() 14:W:snd#(pair(X,Y)) -> c_12() 15:W:splitAt#(0(),XS) -> c_13() 16:W:tail#(cons(N,XS)) -> c_15(activate#(XS)) -->_1 activate#(n__natsFrom(X)) -> c_4(natsFrom#(X)):8 -->_1 activate#(X) -> c_3():7 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 16: tail#(cons(N,XS)) -> c_15(activate#(XS)) 9: and#(tt(),X) -> c_6(activate#(X)) 10: fst#(pair(X,Y)) -> c_7() 11: head#(cons(N,XS)) -> c_8() 14: snd#(pair(X,Y)) -> c_12() 15: splitAt#(0(),XS) -> c_13() 6: U12#(pair(YS,ZS),X) -> c_2(activate#(X)) 7: activate#(X) -> c_3() 8: activate#(n__natsFrom(X)) -> c_4(natsFrom#(X)) 12: natsFrom#(N) -> c_9() 13: natsFrom#(X) -> c_10() *** 1.1.1.1.1.1.1 Progress [(?,O(n^1))] *** Considered Problem: Strict DP Rules: 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)) afterNth#(N,XS) -> c_5(snd#(splitAt(N,XS)),splitAt#(N,XS)) sel#(N,XS) -> c_11(head#(afterNth(N,XS)),afterNth#(N,XS)) splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)),activate#(XS)) take#(N,XS) -> c_16(fst#(splitAt(N,XS)),splitAt#(N,XS)) Strict TRS Rules: Weak DP Rules: Weak TRS 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(X) afterNth(N,XS) -> snd(splitAt(N,XS)) natsFrom(N) -> cons(N,n__natsFrom(s(N))) natsFrom(X) -> n__natsFrom(X) snd(pair(X,Y)) -> Y 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,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,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,tt/0,c_1/5,c_2/1,c_3/0,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/2,c_12/0,c_13/0,c_14/2,c_15/1,c_16/2} Obligation: Innermost basic terms: {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom#,sel#,snd#,splitAt#,tail#,take#}/{0,cons,n__natsFrom,nil,pair,s,tt} Applied Processor: SimplifyRHS Proof: 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)) -->_2 splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)),activate#(XS)):4 2:S:afterNth#(N,XS) -> c_5(snd#(splitAt(N,XS)),splitAt#(N,XS)) -->_2 splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)),activate#(XS)):4 3:S:sel#(N,XS) -> c_11(head#(afterNth(N,XS)),afterNth#(N,XS)) -->_2 afterNth#(N,XS) -> c_5(snd#(splitAt(N,XS)),splitAt#(N,XS)):2 4:S:splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)),activate#(XS)) -->_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)):1 5:S:take#(N,XS) -> c_16(fst#(splitAt(N,XS)),splitAt#(N,XS)) -->_2 splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS)),activate#(XS)):4 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(splitAt#(activate(N),activate(XS))) afterNth#(N,XS) -> c_5(splitAt#(N,XS)) sel#(N,XS) -> c_11(afterNth#(N,XS)) splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS))) take#(N,XS) -> c_16(splitAt#(N,XS)) *** 1.1.1.1.1.1.1.1 Progress [(?,O(n^1))] *** Considered Problem: Strict DP Rules: U11#(tt(),N,X,XS) -> c_1(splitAt#(activate(N),activate(XS))) afterNth#(N,XS) -> c_5(splitAt#(N,XS)) sel#(N,XS) -> c_11(afterNth#(N,XS)) splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS))) take#(N,XS) -> c_16(splitAt#(N,XS)) Strict TRS Rules: Weak DP Rules: Weak TRS 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(X) afterNth(N,XS) -> snd(splitAt(N,XS)) natsFrom(N) -> cons(N,n__natsFrom(s(N))) natsFrom(X) -> n__natsFrom(X) snd(pair(X,Y)) -> Y 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,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,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/0,c_13/0,c_14/1,c_15/1,c_16/1} Obligation: Innermost basic terms: {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom#,sel#,snd#,splitAt#,tail#,take#}/{0,cons,n__natsFrom,nil,pair,s,tt} Applied Processor: UsableRules Proof: We replace rewrite rules by usable rules: activate(X) -> X activate(n__natsFrom(X)) -> natsFrom(X) natsFrom(N) -> cons(N,n__natsFrom(s(N))) natsFrom(X) -> n__natsFrom(X) U11#(tt(),N,X,XS) -> c_1(splitAt#(activate(N),activate(XS))) afterNth#(N,XS) -> c_5(splitAt#(N,XS)) sel#(N,XS) -> c_11(afterNth#(N,XS)) splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS))) take#(N,XS) -> c_16(splitAt#(N,XS)) *** 1.1.1.1.1.1.1.1.1 Progress [(?,O(n^1))] *** Considered Problem: Strict DP Rules: U11#(tt(),N,X,XS) -> c_1(splitAt#(activate(N),activate(XS))) afterNth#(N,XS) -> c_5(splitAt#(N,XS)) sel#(N,XS) -> c_11(afterNth#(N,XS)) splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS))) take#(N,XS) -> c_16(splitAt#(N,XS)) Strict TRS Rules: Weak DP Rules: Weak TRS Rules: activate(X) -> X activate(n__natsFrom(X)) -> natsFrom(X) natsFrom(N) -> cons(N,n__natsFrom(s(N))) natsFrom(X) -> n__natsFrom(X) Signature: {U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/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,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/0,c_13/0,c_14/1,c_15/1,c_16/1} Obligation: Innermost basic terms: {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom#,sel#,snd#,splitAt#,tail#,take#}/{0,cons,n__natsFrom,nil,pair,s,tt} Applied Processor: RemoveHeads Proof: Consider the dependency graph 1:S:U11#(tt(),N,X,XS) -> c_1(splitAt#(activate(N),activate(XS))) -->_1 splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS))):4 2:S:afterNth#(N,XS) -> c_5(splitAt#(N,XS)) -->_1 splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS))):4 3:S:sel#(N,XS) -> c_11(afterNth#(N,XS)) -->_1 afterNth#(N,XS) -> c_5(splitAt#(N,XS)):2 4:S:splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS))) -->_1 U11#(tt(),N,X,XS) -> c_1(splitAt#(activate(N),activate(XS))):1 5:S:take#(N,XS) -> c_16(splitAt#(N,XS)) -->_1 splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS))):4 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,sel#(N,XS) -> c_11(afterNth#(N,XS))),(5,take#(N,XS) -> c_16(splitAt#(N,XS)))] *** 1.1.1.1.1.1.1.1.1.1 Progress [(?,O(n^1))] *** Considered Problem: Strict DP Rules: U11#(tt(),N,X,XS) -> c_1(splitAt#(activate(N),activate(XS))) afterNth#(N,XS) -> c_5(splitAt#(N,XS)) splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS))) Strict TRS Rules: Weak DP Rules: Weak TRS Rules: activate(X) -> X activate(n__natsFrom(X)) -> natsFrom(X) natsFrom(N) -> cons(N,n__natsFrom(s(N))) natsFrom(X) -> n__natsFrom(X) Signature: {U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/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,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/0,c_13/0,c_14/1,c_15/1,c_16/1} Obligation: Innermost basic terms: {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom#,sel#,snd#,splitAt#,tail#,take#}/{0,cons,n__natsFrom,nil,pair,s,tt} Applied Processor: RemoveHeads Proof: Consider the dependency graph 1:S:U11#(tt(),N,X,XS) -> c_1(splitAt#(activate(N),activate(XS))) -->_1 splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS))):4 2:S:afterNth#(N,XS) -> c_5(splitAt#(N,XS)) -->_1 splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS))):4 4:S:splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS))) -->_1 U11#(tt(),N,X,XS) -> c_1(splitAt#(activate(N),activate(XS))):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). [(2,afterNth#(N,XS) -> c_5(splitAt#(N,XS)))] *** 1.1.1.1.1.1.1.1.1.1.1 Progress [(?,O(n^1))] *** Considered Problem: Strict DP Rules: U11#(tt(),N,X,XS) -> c_1(splitAt#(activate(N),activate(XS))) splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS))) Strict TRS Rules: Weak DP Rules: Weak TRS Rules: activate(X) -> X activate(n__natsFrom(X)) -> natsFrom(X) natsFrom(N) -> cons(N,n__natsFrom(s(N))) natsFrom(X) -> n__natsFrom(X) Signature: {U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/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,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/0,c_13/0,c_14/1,c_15/1,c_16/1} Obligation: Innermost basic terms: {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom#,sel#,snd#,splitAt#,tail#,take#}/{0,cons,n__natsFrom,nil,pair,s,tt} Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}} Proof: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly: 1: U11#(tt(),N,X,XS) -> c_1(splitAt#(activate(N) ,activate(XS))) Consider the set of all dependency pairs 1: U11#(tt(),N,X,XS) -> c_1(splitAt#(activate(N) ,activate(XS))) 4: splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt() ,N ,X ,activate(XS))) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {1} These cover all (indirect) predecessors of dependency pairs {1,4} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. *** 1.1.1.1.1.1.1.1.1.1.1.1 Progress [(?,O(n^1))] *** Considered Problem: Strict DP Rules: U11#(tt(),N,X,XS) -> c_1(splitAt#(activate(N),activate(XS))) splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS))) Strict TRS Rules: Weak DP Rules: Weak TRS Rules: activate(X) -> X activate(n__natsFrom(X)) -> natsFrom(X) natsFrom(N) -> cons(N,n__natsFrom(s(N))) natsFrom(X) -> n__natsFrom(X) Signature: {U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/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,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/0,c_13/0,c_14/1,c_15/1,c_16/1} Obligation: Innermost basic terms: {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom#,sel#,snd#,splitAt#,tail#,take#}/{0,cons,n__natsFrom,nil,pair,s,tt} Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy} Proof: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_1) = {1}, uargs(c_14) = {1} Following symbols are considered usable: {activate,natsFrom,U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom#,sel#,snd#,splitAt#,tail#,take#} TcT has computed the following interpretation: p(0) = [2] p(U11) = [8] x2 + [1] x3 + [8] x4 + [1] p(U12) = [2] x1 + [1] x2 + [0] p(activate) = [1] x1 + [0] p(afterNth) = [1] x1 + [1] x2 + [1] p(and) = [2] x2 + [2] p(cons) = [0] p(fst) = [2] x1 + [1] p(head) = [1] x1 + [1] p(n__natsFrom) = [0] p(natsFrom) = [0] p(nil) = [1] p(pair) = [1] x1 + [2] p(s) = [1] x1 + [4] p(sel) = [4] x1 + [1] x2 + [1] p(snd) = [0] p(splitAt) = [1] x1 + [1] x2 + [1] p(tail) = [4] x1 + [0] p(take) = [2] x2 + [1] p(tt) = [2] p(U11#) = [6] x1 + [2] x2 + [4] p(U12#) = [8] x1 + [2] p(activate#) = [1] x1 + [2] p(afterNth#) = [2] p(and#) = [1] x1 + [0] p(fst#) = [4] p(head#) = [1] x1 + [0] p(natsFrom#) = [1] x1 + [2] p(sel#) = [1] p(snd#) = [0] p(splitAt#) = [2] x1 + [9] p(tail#) = [1] x1 + [1] p(take#) = [1] p(c_1) = [1] x1 + [4] p(c_2) = [2] p(c_3) = [1] p(c_4) = [1] x1 + [0] p(c_5) = [8] p(c_6) = [1] x1 + [1] p(c_7) = [0] p(c_8) = [0] p(c_9) = [2] p(c_10) = [1] p(c_11) = [4] x1 + [4] p(c_12) = [1] p(c_13) = [4] p(c_14) = [1] x1 + [1] p(c_15) = [1] x1 + [2] p(c_16) = [1] Following rules are strictly oriented: U11#(tt(),N,X,XS) = [2] N + [16] > [2] N + [13] = c_1(splitAt#(activate(N) ,activate(XS))) Following rules are (at-least) weakly oriented: splitAt#(s(N),cons(X,XS)) = [2] N + [17] >= [2] N + [17] = c_14(U11#(tt() ,N ,X ,activate(XS))) activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__natsFrom(X)) = [0] >= [0] = natsFrom(X) natsFrom(N) = [0] >= [0] = cons(N,n__natsFrom(s(N))) natsFrom(X) = [0] >= [0] = n__natsFrom(X) *** 1.1.1.1.1.1.1.1.1.1.1.1.1 Progress [(?,O(1))] *** Considered Problem: Strict DP Rules: splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS))) Strict TRS Rules: Weak DP Rules: U11#(tt(),N,X,XS) -> c_1(splitAt#(activate(N),activate(XS))) Weak TRS Rules: activate(X) -> X activate(n__natsFrom(X)) -> natsFrom(X) natsFrom(N) -> cons(N,n__natsFrom(s(N))) natsFrom(X) -> n__natsFrom(X) Signature: {U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/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,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/0,c_13/0,c_14/1,c_15/1,c_16/1} Obligation: Innermost basic terms: {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom#,sel#,snd#,splitAt#,tail#,take#}/{0,cons,n__natsFrom,nil,pair,s,tt} Applied Processor: Assumption Proof: () *** 1.1.1.1.1.1.1.1.1.1.1.2 Progress [(O(1),O(1))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: Weak DP Rules: U11#(tt(),N,X,XS) -> c_1(splitAt#(activate(N),activate(XS))) splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS))) Weak TRS Rules: activate(X) -> X activate(n__natsFrom(X)) -> natsFrom(X) natsFrom(N) -> cons(N,n__natsFrom(s(N))) natsFrom(X) -> n__natsFrom(X) Signature: {U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/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,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/0,c_13/0,c_14/1,c_15/1,c_16/1} Obligation: Innermost basic terms: {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom#,sel#,snd#,splitAt#,tail#,take#}/{0,cons,n__natsFrom,nil,pair,s,tt} Applied Processor: RemoveWeakSuffixes Proof: Consider the dependency graph 1:W:U11#(tt(),N,X,XS) -> c_1(splitAt#(activate(N),activate(XS))) -->_1 splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS))):2 2:W:splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt(),N,X,activate(XS))) -->_1 U11#(tt(),N,X,XS) -> c_1(splitAt#(activate(N),activate(XS))):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: U11#(tt(),N,X,XS) -> c_1(splitAt#(activate(N) ,activate(XS))) 2: splitAt#(s(N),cons(X,XS)) -> c_14(U11#(tt() ,N ,X ,activate(XS))) *** 1.1.1.1.1.1.1.1.1.1.1.2.1 Progress [(O(1),O(1))] *** Considered Problem: Strict DP Rules: Strict TRS Rules: Weak DP Rules: Weak TRS Rules: activate(X) -> X activate(n__natsFrom(X)) -> natsFrom(X) natsFrom(N) -> cons(N,n__natsFrom(s(N))) natsFrom(X) -> n__natsFrom(X) Signature: {U11/4,U12/2,activate/1,afterNth/2,and/2,fst/1,head/1,natsFrom/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,sel#/2,snd#/1,splitAt#/2,tail#/1,take#/2} / {0/0,cons/2,n__natsFrom/1,nil/0,pair/2,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/0,c_13/0,c_14/1,c_15/1,c_16/1} Obligation: Innermost basic terms: {U11#,U12#,activate#,afterNth#,and#,fst#,head#,natsFrom#,sel#,snd#,splitAt#,tail#,take#}/{0,cons,n__natsFrom,nil,pair,s,tt} Applied Processor: EmptyProcessor Proof: The problem is already closed. The intended complexity is O(1).