105.90/45.88 WORST_CASE(NON_POLY, ?) 105.90/45.89 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 105.90/45.89 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 105.90/45.89 105.90/45.89 105.90/45.89 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 105.90/45.89 105.90/45.89 (0) CpxTRS 105.90/45.89 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 105.90/45.89 (2) TRS for Loop Detection 105.90/45.89 (3) DecreasingLoopProof [LOWER BOUND(ID), 392 ms] 105.90/45.89 (4) BEST 105.90/45.89 (5) proven lower bound 105.90/45.89 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 105.90/45.89 (7) BOUNDS(n^1, INF) 105.90/45.89 (8) TRS for Loop Detection 105.90/45.89 (9) DecreasingLoopProof [FINISHED, 42.6 s] 105.90/45.89 (10) BOUNDS(EXP, INF) 105.90/45.89 105.90/45.89 105.90/45.89 ---------------------------------------- 105.90/45.89 105.90/45.89 (0) 105.90/45.89 Obligation: 105.90/45.89 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 105.90/45.89 105.90/45.89 105.90/45.89 The TRS R consists of the following rules: 105.90/45.89 105.90/45.89 U101(tt, V2) -> U102(isLNat(activate(V2))) 105.90/45.89 U102(tt) -> tt 105.90/45.89 U11(tt, N, XS) -> U12(isLNat(activate(XS)), activate(N), activate(XS)) 105.90/45.89 U111(tt) -> tt 105.90/45.89 U12(tt, N, XS) -> snd(splitAt(activate(N), activate(XS))) 105.90/45.89 U121(tt) -> tt 105.90/45.89 U131(tt, V2) -> U132(isLNat(activate(V2))) 105.90/45.89 U132(tt) -> tt 105.90/45.89 U141(tt, V2) -> U142(isLNat(activate(V2))) 105.90/45.89 U142(tt) -> tt 105.90/45.89 U151(tt, V2) -> U152(isLNat(activate(V2))) 105.90/45.89 U152(tt) -> tt 105.90/45.89 U161(tt, N) -> cons(activate(N), n__natsFrom(s(activate(N)))) 105.90/45.89 U171(tt, N, XS) -> U172(isLNat(activate(XS)), activate(N), activate(XS)) 105.90/45.89 U172(tt, N, XS) -> head(afterNth(activate(N), activate(XS))) 105.90/45.89 U181(tt, Y) -> U182(isLNat(activate(Y)), activate(Y)) 105.90/45.89 U182(tt, Y) -> activate(Y) 105.90/45.89 U191(tt, XS) -> pair(nil, activate(XS)) 105.90/45.89 U201(tt, N, X, XS) -> U202(isNatural(activate(X)), activate(N), activate(X), activate(XS)) 105.90/45.89 U202(tt, N, X, XS) -> U203(isLNat(activate(XS)), activate(N), activate(X), activate(XS)) 105.90/45.89 U203(tt, N, X, XS) -> U204(splitAt(activate(N), activate(XS)), activate(X)) 105.90/45.89 U204(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS) 105.90/45.89 U21(tt, X, Y) -> U22(isLNat(activate(Y)), activate(X)) 105.90/45.89 U211(tt, XS) -> U212(isLNat(activate(XS)), activate(XS)) 105.90/45.89 U212(tt, XS) -> activate(XS) 105.90/45.89 U22(tt, X) -> activate(X) 105.90/45.89 U221(tt, N, XS) -> U222(isLNat(activate(XS)), activate(N), activate(XS)) 105.90/45.89 U222(tt, N, XS) -> fst(splitAt(activate(N), activate(XS))) 105.90/45.89 U31(tt, N, XS) -> U32(isLNat(activate(XS)), activate(N)) 105.90/45.89 U32(tt, N) -> activate(N) 105.90/45.89 U41(tt, V2) -> U42(isLNat(activate(V2))) 105.90/45.89 U42(tt) -> tt 105.90/45.89 U51(tt, V2) -> U52(isLNat(activate(V2))) 105.90/45.89 U52(tt) -> tt 105.90/45.89 U61(tt) -> tt 105.90/45.89 U71(tt) -> tt 105.90/45.89 U81(tt) -> tt 105.90/45.89 U91(tt) -> tt 105.90/45.89 afterNth(N, XS) -> U11(isNatural(N), N, XS) 105.90/45.89 fst(pair(X, Y)) -> U21(isLNat(X), X, Y) 105.90/45.89 head(cons(N, XS)) -> U31(isNatural(N), N, activate(XS)) 105.90/45.89 isLNat(n__nil) -> tt 105.90/45.89 isLNat(n__afterNth(V1, V2)) -> U41(isNatural(activate(V1)), activate(V2)) 105.90/45.89 isLNat(n__cons(V1, V2)) -> U51(isNatural(activate(V1)), activate(V2)) 105.90/45.89 isLNat(n__fst(V1)) -> U61(isPLNat(activate(V1))) 105.90/45.89 isLNat(n__natsFrom(V1)) -> U71(isNatural(activate(V1))) 105.90/45.89 isLNat(n__snd(V1)) -> U81(isPLNat(activate(V1))) 105.90/45.89 isLNat(n__tail(V1)) -> U91(isLNat(activate(V1))) 105.90/45.89 isLNat(n__take(V1, V2)) -> U101(isNatural(activate(V1)), activate(V2)) 105.90/45.89 isNatural(n__0) -> tt 105.90/45.89 isNatural(n__head(V1)) -> U111(isLNat(activate(V1))) 105.90/45.89 isNatural(n__s(V1)) -> U121(isNatural(activate(V1))) 105.90/45.89 isNatural(n__sel(V1, V2)) -> U131(isNatural(activate(V1)), activate(V2)) 105.90/45.89 isPLNat(n__pair(V1, V2)) -> U141(isLNat(activate(V1)), activate(V2)) 105.90/45.89 isPLNat(n__splitAt(V1, V2)) -> U151(isNatural(activate(V1)), activate(V2)) 105.90/45.89 natsFrom(N) -> U161(isNatural(N), N) 105.90/45.89 sel(N, XS) -> U171(isNatural(N), N, XS) 105.90/45.89 snd(pair(X, Y)) -> U181(isLNat(X), Y) 105.90/45.89 splitAt(0, XS) -> U191(isLNat(XS), XS) 105.90/45.89 splitAt(s(N), cons(X, XS)) -> U201(isNatural(N), N, X, activate(XS)) 105.90/45.89 tail(cons(N, XS)) -> U211(isNatural(N), activate(XS)) 105.90/45.89 take(N, XS) -> U221(isNatural(N), N, XS) 105.90/45.89 natsFrom(X) -> n__natsFrom(X) 105.90/45.89 nil -> n__nil 105.90/45.89 afterNth(X1, X2) -> n__afterNth(X1, X2) 105.90/45.89 cons(X1, X2) -> n__cons(X1, X2) 105.90/45.89 fst(X) -> n__fst(X) 105.90/45.89 snd(X) -> n__snd(X) 105.90/45.89 tail(X) -> n__tail(X) 105.90/45.89 take(X1, X2) -> n__take(X1, X2) 105.90/45.89 0 -> n__0 105.90/45.89 head(X) -> n__head(X) 105.90/45.89 s(X) -> n__s(X) 105.90/45.89 sel(X1, X2) -> n__sel(X1, X2) 105.90/45.89 pair(X1, X2) -> n__pair(X1, X2) 105.90/45.89 splitAt(X1, X2) -> n__splitAt(X1, X2) 105.90/45.89 activate(n__natsFrom(X)) -> natsFrom(X) 105.90/45.89 activate(n__nil) -> nil 105.90/45.89 activate(n__afterNth(X1, X2)) -> afterNth(X1, X2) 105.90/45.89 activate(n__cons(X1, X2)) -> cons(X1, X2) 105.90/45.89 activate(n__fst(X)) -> fst(X) 105.90/45.89 activate(n__snd(X)) -> snd(X) 105.90/45.89 activate(n__tail(X)) -> tail(X) 105.90/45.89 activate(n__take(X1, X2)) -> take(X1, X2) 105.90/45.89 activate(n__0) -> 0 105.90/45.89 activate(n__head(X)) -> head(X) 105.90/45.89 activate(n__s(X)) -> s(X) 105.90/45.89 activate(n__sel(X1, X2)) -> sel(X1, X2) 105.90/45.89 activate(n__pair(X1, X2)) -> pair(X1, X2) 105.90/45.89 activate(n__splitAt(X1, X2)) -> splitAt(X1, X2) 105.90/45.89 activate(X) -> X 105.90/45.89 105.90/45.89 S is empty. 105.90/45.89 Rewrite Strategy: FULL 105.90/45.89 ---------------------------------------- 105.90/45.89 105.90/45.89 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 105.90/45.89 Transformed a relative TRS into a decreasing-loop problem. 105.90/45.89 ---------------------------------------- 105.90/45.89 105.90/45.89 (2) 105.90/45.89 Obligation: 105.90/45.89 Analyzing the following TRS for decreasing loops: 105.90/45.89 105.90/45.89 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 105.90/45.89 105.90/45.89 105.90/45.89 The TRS R consists of the following rules: 105.90/45.89 105.90/45.89 U101(tt, V2) -> U102(isLNat(activate(V2))) 105.90/45.89 U102(tt) -> tt 105.90/45.89 U11(tt, N, XS) -> U12(isLNat(activate(XS)), activate(N), activate(XS)) 105.90/45.89 U111(tt) -> tt 105.90/45.89 U12(tt, N, XS) -> snd(splitAt(activate(N), activate(XS))) 105.90/45.89 U121(tt) -> tt 105.90/45.89 U131(tt, V2) -> U132(isLNat(activate(V2))) 105.90/45.89 U132(tt) -> tt 105.90/45.89 U141(tt, V2) -> U142(isLNat(activate(V2))) 105.90/45.89 U142(tt) -> tt 105.90/45.89 U151(tt, V2) -> U152(isLNat(activate(V2))) 105.90/45.89 U152(tt) -> tt 105.90/45.89 U161(tt, N) -> cons(activate(N), n__natsFrom(s(activate(N)))) 105.90/45.89 U171(tt, N, XS) -> U172(isLNat(activate(XS)), activate(N), activate(XS)) 105.90/45.89 U172(tt, N, XS) -> head(afterNth(activate(N), activate(XS))) 105.90/45.89 U181(tt, Y) -> U182(isLNat(activate(Y)), activate(Y)) 105.90/45.89 U182(tt, Y) -> activate(Y) 105.90/45.89 U191(tt, XS) -> pair(nil, activate(XS)) 105.90/45.89 U201(tt, N, X, XS) -> U202(isNatural(activate(X)), activate(N), activate(X), activate(XS)) 105.90/45.89 U202(tt, N, X, XS) -> U203(isLNat(activate(XS)), activate(N), activate(X), activate(XS)) 105.90/45.89 U203(tt, N, X, XS) -> U204(splitAt(activate(N), activate(XS)), activate(X)) 105.90/45.89 U204(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS) 105.90/45.89 U21(tt, X, Y) -> U22(isLNat(activate(Y)), activate(X)) 105.90/45.89 U211(tt, XS) -> U212(isLNat(activate(XS)), activate(XS)) 105.90/45.89 U212(tt, XS) -> activate(XS) 105.90/45.89 U22(tt, X) -> activate(X) 105.90/45.89 U221(tt, N, XS) -> U222(isLNat(activate(XS)), activate(N), activate(XS)) 105.90/45.89 U222(tt, N, XS) -> fst(splitAt(activate(N), activate(XS))) 105.90/45.89 U31(tt, N, XS) -> U32(isLNat(activate(XS)), activate(N)) 105.90/45.89 U32(tt, N) -> activate(N) 105.90/45.89 U41(tt, V2) -> U42(isLNat(activate(V2))) 105.90/45.89 U42(tt) -> tt 105.90/45.89 U51(tt, V2) -> U52(isLNat(activate(V2))) 105.90/45.89 U52(tt) -> tt 105.90/45.89 U61(tt) -> tt 105.90/45.89 U71(tt) -> tt 105.90/45.89 U81(tt) -> tt 105.90/45.89 U91(tt) -> tt 105.90/45.89 afterNth(N, XS) -> U11(isNatural(N), N, XS) 105.90/45.89 fst(pair(X, Y)) -> U21(isLNat(X), X, Y) 105.90/45.89 head(cons(N, XS)) -> U31(isNatural(N), N, activate(XS)) 105.90/45.89 isLNat(n__nil) -> tt 105.90/45.89 isLNat(n__afterNth(V1, V2)) -> U41(isNatural(activate(V1)), activate(V2)) 105.90/45.89 isLNat(n__cons(V1, V2)) -> U51(isNatural(activate(V1)), activate(V2)) 105.90/45.89 isLNat(n__fst(V1)) -> U61(isPLNat(activate(V1))) 105.90/45.89 isLNat(n__natsFrom(V1)) -> U71(isNatural(activate(V1))) 105.90/45.89 isLNat(n__snd(V1)) -> U81(isPLNat(activate(V1))) 105.90/45.89 isLNat(n__tail(V1)) -> U91(isLNat(activate(V1))) 105.90/45.89 isLNat(n__take(V1, V2)) -> U101(isNatural(activate(V1)), activate(V2)) 105.90/45.89 isNatural(n__0) -> tt 105.90/45.89 isNatural(n__head(V1)) -> U111(isLNat(activate(V1))) 105.90/45.89 isNatural(n__s(V1)) -> U121(isNatural(activate(V1))) 105.90/45.89 isNatural(n__sel(V1, V2)) -> U131(isNatural(activate(V1)), activate(V2)) 105.90/45.89 isPLNat(n__pair(V1, V2)) -> U141(isLNat(activate(V1)), activate(V2)) 105.90/45.89 isPLNat(n__splitAt(V1, V2)) -> U151(isNatural(activate(V1)), activate(V2)) 105.90/45.89 natsFrom(N) -> U161(isNatural(N), N) 105.90/45.89 sel(N, XS) -> U171(isNatural(N), N, XS) 105.90/45.89 snd(pair(X, Y)) -> U181(isLNat(X), Y) 105.90/45.89 splitAt(0, XS) -> U191(isLNat(XS), XS) 105.90/45.89 splitAt(s(N), cons(X, XS)) -> U201(isNatural(N), N, X, activate(XS)) 105.90/45.89 tail(cons(N, XS)) -> U211(isNatural(N), activate(XS)) 105.90/45.89 take(N, XS) -> U221(isNatural(N), N, XS) 105.90/45.89 natsFrom(X) -> n__natsFrom(X) 105.90/45.89 nil -> n__nil 105.90/45.89 afterNth(X1, X2) -> n__afterNth(X1, X2) 105.90/45.89 cons(X1, X2) -> n__cons(X1, X2) 105.90/45.89 fst(X) -> n__fst(X) 105.90/45.89 snd(X) -> n__snd(X) 105.90/45.89 tail(X) -> n__tail(X) 105.90/45.89 take(X1, X2) -> n__take(X1, X2) 105.90/45.89 0 -> n__0 105.90/45.89 head(X) -> n__head(X) 105.90/45.89 s(X) -> n__s(X) 105.90/45.89 sel(X1, X2) -> n__sel(X1, X2) 105.90/45.89 pair(X1, X2) -> n__pair(X1, X2) 105.90/45.89 splitAt(X1, X2) -> n__splitAt(X1, X2) 105.90/45.89 activate(n__natsFrom(X)) -> natsFrom(X) 105.90/45.89 activate(n__nil) -> nil 105.90/45.89 activate(n__afterNth(X1, X2)) -> afterNth(X1, X2) 105.90/45.89 activate(n__cons(X1, X2)) -> cons(X1, X2) 105.90/45.89 activate(n__fst(X)) -> fst(X) 105.90/45.89 activate(n__snd(X)) -> snd(X) 105.90/45.89 activate(n__tail(X)) -> tail(X) 105.90/45.89 activate(n__take(X1, X2)) -> take(X1, X2) 105.90/45.89 activate(n__0) -> 0 105.90/45.89 activate(n__head(X)) -> head(X) 105.90/45.89 activate(n__s(X)) -> s(X) 105.90/45.89 activate(n__sel(X1, X2)) -> sel(X1, X2) 105.90/45.89 activate(n__pair(X1, X2)) -> pair(X1, X2) 105.90/45.89 activate(n__splitAt(X1, X2)) -> splitAt(X1, X2) 105.90/45.89 activate(X) -> X 105.90/45.89 105.90/45.89 S is empty. 105.90/45.89 Rewrite Strategy: FULL 105.90/45.89 ---------------------------------------- 105.90/45.89 105.90/45.89 (3) DecreasingLoopProof (LOWER BOUND(ID)) 105.90/45.89 The following loop(s) give(s) rise to the lower bound Omega(n^1): 105.90/45.89 105.90/45.89 The rewrite sequence 105.90/45.89 105.90/45.89 isNatural(n__sel(V1, V2)) ->^+ U131(isNatural(V1), activate(V2)) 105.90/45.89 105.90/45.89 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 105.90/45.89 105.90/45.89 The pumping substitution is [V1 / n__sel(V1, V2)]. 105.90/45.89 105.90/45.89 The result substitution is [ ]. 105.90/45.89 105.90/45.89 105.90/45.89 105.90/45.89 105.90/45.89 ---------------------------------------- 105.90/45.89 105.90/45.89 (4) 105.90/45.89 Complex Obligation (BEST) 105.90/45.89 105.90/45.89 ---------------------------------------- 105.90/45.89 105.90/45.89 (5) 105.90/45.89 Obligation: 105.90/45.89 Proved the lower bound n^1 for the following obligation: 105.90/45.89 105.90/45.89 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 105.90/45.89 105.90/45.89 105.90/45.89 The TRS R consists of the following rules: 105.90/45.89 105.90/45.89 U101(tt, V2) -> U102(isLNat(activate(V2))) 105.90/45.89 U102(tt) -> tt 105.90/45.89 U11(tt, N, XS) -> U12(isLNat(activate(XS)), activate(N), activate(XS)) 105.90/45.89 U111(tt) -> tt 105.90/45.89 U12(tt, N, XS) -> snd(splitAt(activate(N), activate(XS))) 105.90/45.89 U121(tt) -> tt 105.90/45.89 U131(tt, V2) -> U132(isLNat(activate(V2))) 105.90/45.89 U132(tt) -> tt 105.90/45.89 U141(tt, V2) -> U142(isLNat(activate(V2))) 105.90/45.89 U142(tt) -> tt 105.90/45.89 U151(tt, V2) -> U152(isLNat(activate(V2))) 105.90/45.89 U152(tt) -> tt 105.90/45.89 U161(tt, N) -> cons(activate(N), n__natsFrom(s(activate(N)))) 105.90/45.89 U171(tt, N, XS) -> U172(isLNat(activate(XS)), activate(N), activate(XS)) 105.90/45.89 U172(tt, N, XS) -> head(afterNth(activate(N), activate(XS))) 105.90/45.89 U181(tt, Y) -> U182(isLNat(activate(Y)), activate(Y)) 105.90/45.89 U182(tt, Y) -> activate(Y) 105.90/45.89 U191(tt, XS) -> pair(nil, activate(XS)) 105.90/45.89 U201(tt, N, X, XS) -> U202(isNatural(activate(X)), activate(N), activate(X), activate(XS)) 105.90/45.89 U202(tt, N, X, XS) -> U203(isLNat(activate(XS)), activate(N), activate(X), activate(XS)) 105.90/45.89 U203(tt, N, X, XS) -> U204(splitAt(activate(N), activate(XS)), activate(X)) 105.90/45.89 U204(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS) 105.90/45.89 U21(tt, X, Y) -> U22(isLNat(activate(Y)), activate(X)) 105.90/45.89 U211(tt, XS) -> U212(isLNat(activate(XS)), activate(XS)) 105.90/45.89 U212(tt, XS) -> activate(XS) 105.90/45.89 U22(tt, X) -> activate(X) 105.90/45.89 U221(tt, N, XS) -> U222(isLNat(activate(XS)), activate(N), activate(XS)) 105.90/45.89 U222(tt, N, XS) -> fst(splitAt(activate(N), activate(XS))) 105.90/45.89 U31(tt, N, XS) -> U32(isLNat(activate(XS)), activate(N)) 105.90/45.89 U32(tt, N) -> activate(N) 105.90/45.89 U41(tt, V2) -> U42(isLNat(activate(V2))) 105.90/45.89 U42(tt) -> tt 105.90/45.89 U51(tt, V2) -> U52(isLNat(activate(V2))) 105.90/45.89 U52(tt) -> tt 105.90/45.89 U61(tt) -> tt 105.90/45.89 U71(tt) -> tt 105.90/45.89 U81(tt) -> tt 105.90/45.89 U91(tt) -> tt 105.90/45.89 afterNth(N, XS) -> U11(isNatural(N), N, XS) 105.90/45.89 fst(pair(X, Y)) -> U21(isLNat(X), X, Y) 105.90/45.89 head(cons(N, XS)) -> U31(isNatural(N), N, activate(XS)) 105.90/45.89 isLNat(n__nil) -> tt 105.90/45.89 isLNat(n__afterNth(V1, V2)) -> U41(isNatural(activate(V1)), activate(V2)) 105.90/45.89 isLNat(n__cons(V1, V2)) -> U51(isNatural(activate(V1)), activate(V2)) 105.90/45.89 isLNat(n__fst(V1)) -> U61(isPLNat(activate(V1))) 105.90/45.89 isLNat(n__natsFrom(V1)) -> U71(isNatural(activate(V1))) 105.90/45.89 isLNat(n__snd(V1)) -> U81(isPLNat(activate(V1))) 105.90/45.89 isLNat(n__tail(V1)) -> U91(isLNat(activate(V1))) 105.90/45.89 isLNat(n__take(V1, V2)) -> U101(isNatural(activate(V1)), activate(V2)) 105.90/45.89 isNatural(n__0) -> tt 105.90/45.89 isNatural(n__head(V1)) -> U111(isLNat(activate(V1))) 105.90/45.89 isNatural(n__s(V1)) -> U121(isNatural(activate(V1))) 105.90/45.89 isNatural(n__sel(V1, V2)) -> U131(isNatural(activate(V1)), activate(V2)) 105.90/45.89 isPLNat(n__pair(V1, V2)) -> U141(isLNat(activate(V1)), activate(V2)) 105.90/45.89 isPLNat(n__splitAt(V1, V2)) -> U151(isNatural(activate(V1)), activate(V2)) 105.90/45.89 natsFrom(N) -> U161(isNatural(N), N) 105.90/45.89 sel(N, XS) -> U171(isNatural(N), N, XS) 105.90/45.89 snd(pair(X, Y)) -> U181(isLNat(X), Y) 105.90/45.89 splitAt(0, XS) -> U191(isLNat(XS), XS) 105.90/45.89 splitAt(s(N), cons(X, XS)) -> U201(isNatural(N), N, X, activate(XS)) 105.90/45.89 tail(cons(N, XS)) -> U211(isNatural(N), activate(XS)) 105.90/45.89 take(N, XS) -> U221(isNatural(N), N, XS) 105.90/45.89 natsFrom(X) -> n__natsFrom(X) 105.90/45.89 nil -> n__nil 105.90/45.89 afterNth(X1, X2) -> n__afterNth(X1, X2) 105.90/45.89 cons(X1, X2) -> n__cons(X1, X2) 105.90/45.89 fst(X) -> n__fst(X) 105.90/45.89 snd(X) -> n__snd(X) 105.90/45.89 tail(X) -> n__tail(X) 105.90/45.89 take(X1, X2) -> n__take(X1, X2) 105.90/45.89 0 -> n__0 105.90/45.89 head(X) -> n__head(X) 105.90/45.89 s(X) -> n__s(X) 105.90/45.89 sel(X1, X2) -> n__sel(X1, X2) 105.90/45.89 pair(X1, X2) -> n__pair(X1, X2) 105.90/45.89 splitAt(X1, X2) -> n__splitAt(X1, X2) 105.90/45.89 activate(n__natsFrom(X)) -> natsFrom(X) 105.90/45.89 activate(n__nil) -> nil 105.90/45.89 activate(n__afterNth(X1, X2)) -> afterNth(X1, X2) 105.90/45.89 activate(n__cons(X1, X2)) -> cons(X1, X2) 105.90/45.89 activate(n__fst(X)) -> fst(X) 105.90/45.89 activate(n__snd(X)) -> snd(X) 105.90/45.89 activate(n__tail(X)) -> tail(X) 105.90/45.89 activate(n__take(X1, X2)) -> take(X1, X2) 105.90/45.89 activate(n__0) -> 0 105.90/45.89 activate(n__head(X)) -> head(X) 105.90/45.89 activate(n__s(X)) -> s(X) 105.90/45.89 activate(n__sel(X1, X2)) -> sel(X1, X2) 105.90/45.89 activate(n__pair(X1, X2)) -> pair(X1, X2) 105.90/45.89 activate(n__splitAt(X1, X2)) -> splitAt(X1, X2) 105.90/45.89 activate(X) -> X 105.90/45.89 105.90/45.89 S is empty. 105.90/45.89 Rewrite Strategy: FULL 105.90/45.89 ---------------------------------------- 105.90/45.89 105.90/45.89 (6) LowerBoundPropagationProof (FINISHED) 105.90/45.89 Propagated lower bound. 105.90/45.89 ---------------------------------------- 105.90/45.89 105.90/45.89 (7) 105.90/45.89 BOUNDS(n^1, INF) 105.90/45.89 105.90/45.89 ---------------------------------------- 105.90/45.89 105.90/45.89 (8) 105.90/45.89 Obligation: 105.90/45.89 Analyzing the following TRS for decreasing loops: 105.90/45.89 105.90/45.89 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 105.90/45.89 105.90/45.89 105.90/45.89 The TRS R consists of the following rules: 105.90/45.89 105.90/45.89 U101(tt, V2) -> U102(isLNat(activate(V2))) 105.90/45.89 U102(tt) -> tt 105.90/45.89 U11(tt, N, XS) -> U12(isLNat(activate(XS)), activate(N), activate(XS)) 105.90/45.89 U111(tt) -> tt 105.90/45.89 U12(tt, N, XS) -> snd(splitAt(activate(N), activate(XS))) 105.90/45.89 U121(tt) -> tt 105.90/45.89 U131(tt, V2) -> U132(isLNat(activate(V2))) 105.90/45.89 U132(tt) -> tt 105.90/45.89 U141(tt, V2) -> U142(isLNat(activate(V2))) 105.90/45.89 U142(tt) -> tt 105.90/45.89 U151(tt, V2) -> U152(isLNat(activate(V2))) 105.90/45.89 U152(tt) -> tt 105.90/45.89 U161(tt, N) -> cons(activate(N), n__natsFrom(s(activate(N)))) 105.90/45.89 U171(tt, N, XS) -> U172(isLNat(activate(XS)), activate(N), activate(XS)) 105.90/45.89 U172(tt, N, XS) -> head(afterNth(activate(N), activate(XS))) 105.90/45.89 U181(tt, Y) -> U182(isLNat(activate(Y)), activate(Y)) 105.90/45.89 U182(tt, Y) -> activate(Y) 105.90/45.89 U191(tt, XS) -> pair(nil, activate(XS)) 105.90/45.89 U201(tt, N, X, XS) -> U202(isNatural(activate(X)), activate(N), activate(X), activate(XS)) 105.90/45.89 U202(tt, N, X, XS) -> U203(isLNat(activate(XS)), activate(N), activate(X), activate(XS)) 105.90/45.89 U203(tt, N, X, XS) -> U204(splitAt(activate(N), activate(XS)), activate(X)) 105.90/45.89 U204(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS) 105.90/45.89 U21(tt, X, Y) -> U22(isLNat(activate(Y)), activate(X)) 105.90/45.89 U211(tt, XS) -> U212(isLNat(activate(XS)), activate(XS)) 105.90/45.89 U212(tt, XS) -> activate(XS) 105.90/45.89 U22(tt, X) -> activate(X) 105.90/45.89 U221(tt, N, XS) -> U222(isLNat(activate(XS)), activate(N), activate(XS)) 105.90/45.89 U222(tt, N, XS) -> fst(splitAt(activate(N), activate(XS))) 105.90/45.89 U31(tt, N, XS) -> U32(isLNat(activate(XS)), activate(N)) 105.90/45.89 U32(tt, N) -> activate(N) 105.90/45.89 U41(tt, V2) -> U42(isLNat(activate(V2))) 105.90/45.89 U42(tt) -> tt 105.90/45.89 U51(tt, V2) -> U52(isLNat(activate(V2))) 105.90/45.89 U52(tt) -> tt 105.90/45.89 U61(tt) -> tt 105.90/45.89 U71(tt) -> tt 105.90/45.89 U81(tt) -> tt 105.90/45.89 U91(tt) -> tt 105.90/45.89 afterNth(N, XS) -> U11(isNatural(N), N, XS) 105.90/45.89 fst(pair(X, Y)) -> U21(isLNat(X), X, Y) 105.90/45.89 head(cons(N, XS)) -> U31(isNatural(N), N, activate(XS)) 105.90/45.89 isLNat(n__nil) -> tt 105.90/45.89 isLNat(n__afterNth(V1, V2)) -> U41(isNatural(activate(V1)), activate(V2)) 105.90/45.89 isLNat(n__cons(V1, V2)) -> U51(isNatural(activate(V1)), activate(V2)) 105.90/45.89 isLNat(n__fst(V1)) -> U61(isPLNat(activate(V1))) 105.90/45.89 isLNat(n__natsFrom(V1)) -> U71(isNatural(activate(V1))) 105.90/45.89 isLNat(n__snd(V1)) -> U81(isPLNat(activate(V1))) 105.90/45.89 isLNat(n__tail(V1)) -> U91(isLNat(activate(V1))) 105.90/45.89 isLNat(n__take(V1, V2)) -> U101(isNatural(activate(V1)), activate(V2)) 105.90/45.89 isNatural(n__0) -> tt 105.90/45.89 isNatural(n__head(V1)) -> U111(isLNat(activate(V1))) 105.90/45.89 isNatural(n__s(V1)) -> U121(isNatural(activate(V1))) 105.90/45.89 isNatural(n__sel(V1, V2)) -> U131(isNatural(activate(V1)), activate(V2)) 105.90/45.89 isPLNat(n__pair(V1, V2)) -> U141(isLNat(activate(V1)), activate(V2)) 105.90/45.89 isPLNat(n__splitAt(V1, V2)) -> U151(isNatural(activate(V1)), activate(V2)) 105.90/45.89 natsFrom(N) -> U161(isNatural(N), N) 105.90/45.89 sel(N, XS) -> U171(isNatural(N), N, XS) 105.90/45.89 snd(pair(X, Y)) -> U181(isLNat(X), Y) 105.90/45.89 splitAt(0, XS) -> U191(isLNat(XS), XS) 105.90/45.89 splitAt(s(N), cons(X, XS)) -> U201(isNatural(N), N, X, activate(XS)) 105.90/45.89 tail(cons(N, XS)) -> U211(isNatural(N), activate(XS)) 105.90/45.89 take(N, XS) -> U221(isNatural(N), N, XS) 105.90/45.89 natsFrom(X) -> n__natsFrom(X) 105.90/45.89 nil -> n__nil 105.90/45.89 afterNth(X1, X2) -> n__afterNth(X1, X2) 105.90/45.89 cons(X1, X2) -> n__cons(X1, X2) 105.90/45.89 fst(X) -> n__fst(X) 105.90/45.89 snd(X) -> n__snd(X) 105.90/45.89 tail(X) -> n__tail(X) 105.90/45.89 take(X1, X2) -> n__take(X1, X2) 105.90/45.89 0 -> n__0 105.90/45.89 head(X) -> n__head(X) 105.90/45.89 s(X) -> n__s(X) 105.90/45.89 sel(X1, X2) -> n__sel(X1, X2) 105.90/45.89 pair(X1, X2) -> n__pair(X1, X2) 105.90/45.89 splitAt(X1, X2) -> n__splitAt(X1, X2) 105.90/45.89 activate(n__natsFrom(X)) -> natsFrom(X) 105.90/45.89 activate(n__nil) -> nil 105.90/45.89 activate(n__afterNth(X1, X2)) -> afterNth(X1, X2) 105.90/45.89 activate(n__cons(X1, X2)) -> cons(X1, X2) 105.90/45.89 activate(n__fst(X)) -> fst(X) 105.90/45.89 activate(n__snd(X)) -> snd(X) 105.90/45.89 activate(n__tail(X)) -> tail(X) 105.90/45.89 activate(n__take(X1, X2)) -> take(X1, X2) 105.90/45.89 activate(n__0) -> 0 105.90/45.89 activate(n__head(X)) -> head(X) 105.90/45.89 activate(n__s(X)) -> s(X) 105.90/45.89 activate(n__sel(X1, X2)) -> sel(X1, X2) 105.90/45.89 activate(n__pair(X1, X2)) -> pair(X1, X2) 105.90/45.89 activate(n__splitAt(X1, X2)) -> splitAt(X1, X2) 105.90/45.89 activate(X) -> X 105.90/45.89 105.90/45.89 S is empty. 105.90/45.89 Rewrite Strategy: FULL 105.90/45.89 ---------------------------------------- 105.90/45.89 105.90/45.89 (9) DecreasingLoopProof (FINISHED) 105.90/45.89 The following loop(s) give(s) rise to the lower bound EXP: 105.90/45.89 105.90/45.89 The rewrite sequence 105.90/45.89 105.90/45.89 activate(n__take(n__0, X2)) ->^+ U222(isLNat(activate(X2)), activate(n__0), activate(X2)) 105.90/45.89 105.90/45.89 gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0]. 105.90/45.89 105.90/45.89 The pumping substitution is [X2 / n__take(n__0, X2)]. 105.90/45.89 105.90/45.89 The result substitution is [ ]. 105.90/45.89 105.90/45.89 105.90/45.89 105.90/45.89 The rewrite sequence 105.90/45.89 105.90/45.89 activate(n__take(n__0, X2)) ->^+ U222(isLNat(activate(X2)), activate(n__0), activate(X2)) 105.90/45.89 105.90/45.89 gives rise to a decreasing loop by considering the right hand sides subterm at position [2]. 105.90/45.89 105.90/45.89 The pumping substitution is [X2 / n__take(n__0, X2)]. 105.90/45.89 105.90/45.89 The result substitution is [ ]. 105.90/45.89 105.90/45.89 105.90/45.89 105.90/45.89 105.90/45.89 ---------------------------------------- 105.90/45.89 105.90/45.89 (10) 105.90/45.89 BOUNDS(EXP, INF) 105.95/45.91 EOF