396.43/291.58 WORST_CASE(Omega(n^1), ?) 396.43/291.59 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 396.43/291.59 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 396.43/291.59 396.43/291.59 396.43/291.59 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 396.43/291.59 396.43/291.59 (0) CpxTRS 396.43/291.59 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 396.43/291.59 (2) TRS for Loop Detection 396.43/291.59 (3) DecreasingLoopProof [LOWER BOUND(ID), 211 ms] 396.43/291.59 (4) BEST 396.43/291.59 (5) proven lower bound 396.43/291.59 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 396.43/291.59 (7) BOUNDS(n^1, INF) 396.43/291.59 (8) TRS for Loop Detection 396.43/291.59 396.43/291.59 396.43/291.59 ---------------------------------------- 396.43/291.59 396.43/291.59 (0) 396.43/291.59 Obligation: 396.43/291.59 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 396.43/291.59 396.43/291.59 396.43/291.59 The TRS R consists of the following rules: 396.43/291.59 396.43/291.59 U101(tt, N, XS) -> fst(splitAt(activate(N), activate(XS))) 396.43/291.59 U11(tt, N, XS) -> snd(splitAt(activate(N), activate(XS))) 396.43/291.59 U21(tt, X) -> activate(X) 396.43/291.59 U31(tt, N) -> activate(N) 396.43/291.59 U41(tt, N) -> cons(activate(N), n__natsFrom(s(activate(N)))) 396.43/291.59 U51(tt, N, XS) -> head(afterNth(activate(N), activate(XS))) 396.43/291.59 U61(tt, Y) -> activate(Y) 396.43/291.59 U71(tt, XS) -> pair(nil, activate(XS)) 396.43/291.59 U81(tt, N, X, XS) -> U82(splitAt(activate(N), activate(XS)), activate(X)) 396.43/291.59 U82(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS) 396.43/291.59 U91(tt, XS) -> activate(XS) 396.43/291.59 afterNth(N, XS) -> U11(and(isNatural(N), n__isLNat(XS)), N, XS) 396.43/291.59 and(tt, X) -> activate(X) 396.43/291.59 fst(pair(X, Y)) -> U21(and(isLNat(X), n__isLNat(Y)), X) 396.43/291.59 head(cons(N, XS)) -> U31(and(isNatural(N), n__isLNat(activate(XS))), N) 396.43/291.59 isLNat(n__nil) -> tt 396.43/291.59 isLNat(n__afterNth(V1, V2)) -> and(isNatural(activate(V1)), n__isLNat(activate(V2))) 396.43/291.59 isLNat(n__cons(V1, V2)) -> and(isNatural(activate(V1)), n__isLNat(activate(V2))) 396.43/291.59 isLNat(n__fst(V1)) -> isPLNat(activate(V1)) 396.43/291.59 isLNat(n__natsFrom(V1)) -> isNatural(activate(V1)) 396.43/291.59 isLNat(n__snd(V1)) -> isPLNat(activate(V1)) 396.43/291.59 isLNat(n__tail(V1)) -> isLNat(activate(V1)) 396.43/291.59 isLNat(n__take(V1, V2)) -> and(isNatural(activate(V1)), n__isLNat(activate(V2))) 396.43/291.59 isNatural(n__0) -> tt 396.43/291.59 isNatural(n__head(V1)) -> isLNat(activate(V1)) 396.43/291.59 isNatural(n__s(V1)) -> isNatural(activate(V1)) 396.43/291.59 isNatural(n__sel(V1, V2)) -> and(isNatural(activate(V1)), n__isLNat(activate(V2))) 396.43/291.59 isPLNat(n__pair(V1, V2)) -> and(isLNat(activate(V1)), n__isLNat(activate(V2))) 396.43/291.59 isPLNat(n__splitAt(V1, V2)) -> and(isNatural(activate(V1)), n__isLNat(activate(V2))) 396.43/291.59 natsFrom(N) -> U41(isNatural(N), N) 396.43/291.59 sel(N, XS) -> U51(and(isNatural(N), n__isLNat(XS)), N, XS) 396.43/291.59 snd(pair(X, Y)) -> U61(and(isLNat(X), n__isLNat(Y)), Y) 396.43/291.59 splitAt(0, XS) -> U71(isLNat(XS), XS) 396.43/291.59 splitAt(s(N), cons(X, XS)) -> U81(and(isNatural(N), n__and(isNatural(X), n__isLNat(activate(XS)))), N, X, activate(XS)) 396.43/291.59 tail(cons(N, XS)) -> U91(and(isNatural(N), n__isLNat(activate(XS))), activate(XS)) 396.43/291.59 take(N, XS) -> U101(and(isNatural(N), n__isLNat(XS)), N, XS) 396.43/291.59 natsFrom(X) -> n__natsFrom(X) 396.43/291.59 isLNat(X) -> n__isLNat(X) 396.43/291.59 nil -> n__nil 396.43/291.59 afterNth(X1, X2) -> n__afterNth(X1, X2) 396.43/291.59 cons(X1, X2) -> n__cons(X1, X2) 396.43/291.59 fst(X) -> n__fst(X) 396.43/291.59 snd(X) -> n__snd(X) 396.43/291.59 tail(X) -> n__tail(X) 396.43/291.59 take(X1, X2) -> n__take(X1, X2) 396.43/291.59 0 -> n__0 396.43/291.59 head(X) -> n__head(X) 396.43/291.59 s(X) -> n__s(X) 396.43/291.59 sel(X1, X2) -> n__sel(X1, X2) 396.43/291.59 pair(X1, X2) -> n__pair(X1, X2) 396.43/291.59 splitAt(X1, X2) -> n__splitAt(X1, X2) 396.43/291.59 and(X1, X2) -> n__and(X1, X2) 396.43/291.59 activate(n__natsFrom(X)) -> natsFrom(X) 396.43/291.59 activate(n__isLNat(X)) -> isLNat(X) 396.43/291.59 activate(n__nil) -> nil 396.43/291.59 activate(n__afterNth(X1, X2)) -> afterNth(X1, X2) 396.43/291.59 activate(n__cons(X1, X2)) -> cons(X1, X2) 396.43/291.59 activate(n__fst(X)) -> fst(X) 396.43/291.59 activate(n__snd(X)) -> snd(X) 396.43/291.59 activate(n__tail(X)) -> tail(X) 396.43/291.59 activate(n__take(X1, X2)) -> take(X1, X2) 396.43/291.59 activate(n__0) -> 0 396.43/291.59 activate(n__head(X)) -> head(X) 396.43/291.59 activate(n__s(X)) -> s(X) 396.43/291.59 activate(n__sel(X1, X2)) -> sel(X1, X2) 396.43/291.59 activate(n__pair(X1, X2)) -> pair(X1, X2) 396.43/291.59 activate(n__splitAt(X1, X2)) -> splitAt(X1, X2) 396.43/291.59 activate(n__and(X1, X2)) -> and(X1, X2) 396.43/291.59 activate(X) -> X 396.43/291.59 396.43/291.59 S is empty. 396.43/291.59 Rewrite Strategy: FULL 396.43/291.59 ---------------------------------------- 396.43/291.59 396.43/291.59 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 396.43/291.59 Transformed a relative TRS into a decreasing-loop problem. 396.43/291.59 ---------------------------------------- 396.43/291.59 396.43/291.59 (2) 396.43/291.59 Obligation: 396.43/291.59 Analyzing the following TRS for decreasing loops: 396.43/291.59 396.43/291.59 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 396.43/291.59 396.43/291.59 396.43/291.59 The TRS R consists of the following rules: 396.43/291.59 396.43/291.59 U101(tt, N, XS) -> fst(splitAt(activate(N), activate(XS))) 396.43/291.59 U11(tt, N, XS) -> snd(splitAt(activate(N), activate(XS))) 396.43/291.59 U21(tt, X) -> activate(X) 396.43/291.59 U31(tt, N) -> activate(N) 396.43/291.59 U41(tt, N) -> cons(activate(N), n__natsFrom(s(activate(N)))) 396.43/291.59 U51(tt, N, XS) -> head(afterNth(activate(N), activate(XS))) 396.43/291.59 U61(tt, Y) -> activate(Y) 396.43/291.59 U71(tt, XS) -> pair(nil, activate(XS)) 396.43/291.59 U81(tt, N, X, XS) -> U82(splitAt(activate(N), activate(XS)), activate(X)) 396.43/291.59 U82(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS) 396.43/291.59 U91(tt, XS) -> activate(XS) 396.43/291.59 afterNth(N, XS) -> U11(and(isNatural(N), n__isLNat(XS)), N, XS) 396.43/291.59 and(tt, X) -> activate(X) 396.43/291.59 fst(pair(X, Y)) -> U21(and(isLNat(X), n__isLNat(Y)), X) 396.43/291.59 head(cons(N, XS)) -> U31(and(isNatural(N), n__isLNat(activate(XS))), N) 396.43/291.59 isLNat(n__nil) -> tt 396.43/291.59 isLNat(n__afterNth(V1, V2)) -> and(isNatural(activate(V1)), n__isLNat(activate(V2))) 396.43/291.59 isLNat(n__cons(V1, V2)) -> and(isNatural(activate(V1)), n__isLNat(activate(V2))) 396.43/291.59 isLNat(n__fst(V1)) -> isPLNat(activate(V1)) 396.43/291.59 isLNat(n__natsFrom(V1)) -> isNatural(activate(V1)) 396.43/291.59 isLNat(n__snd(V1)) -> isPLNat(activate(V1)) 396.43/291.59 isLNat(n__tail(V1)) -> isLNat(activate(V1)) 396.43/291.59 isLNat(n__take(V1, V2)) -> and(isNatural(activate(V1)), n__isLNat(activate(V2))) 396.43/291.59 isNatural(n__0) -> tt 396.43/291.59 isNatural(n__head(V1)) -> isLNat(activate(V1)) 396.43/291.59 isNatural(n__s(V1)) -> isNatural(activate(V1)) 396.43/291.59 isNatural(n__sel(V1, V2)) -> and(isNatural(activate(V1)), n__isLNat(activate(V2))) 396.43/291.59 isPLNat(n__pair(V1, V2)) -> and(isLNat(activate(V1)), n__isLNat(activate(V2))) 396.43/291.59 isPLNat(n__splitAt(V1, V2)) -> and(isNatural(activate(V1)), n__isLNat(activate(V2))) 396.43/291.59 natsFrom(N) -> U41(isNatural(N), N) 396.43/291.59 sel(N, XS) -> U51(and(isNatural(N), n__isLNat(XS)), N, XS) 396.43/291.59 snd(pair(X, Y)) -> U61(and(isLNat(X), n__isLNat(Y)), Y) 396.43/291.59 splitAt(0, XS) -> U71(isLNat(XS), XS) 396.43/291.59 splitAt(s(N), cons(X, XS)) -> U81(and(isNatural(N), n__and(isNatural(X), n__isLNat(activate(XS)))), N, X, activate(XS)) 396.43/291.59 tail(cons(N, XS)) -> U91(and(isNatural(N), n__isLNat(activate(XS))), activate(XS)) 396.43/291.59 take(N, XS) -> U101(and(isNatural(N), n__isLNat(XS)), N, XS) 396.43/291.59 natsFrom(X) -> n__natsFrom(X) 396.43/291.59 isLNat(X) -> n__isLNat(X) 396.43/291.59 nil -> n__nil 396.43/291.59 afterNth(X1, X2) -> n__afterNth(X1, X2) 396.43/291.59 cons(X1, X2) -> n__cons(X1, X2) 396.43/291.59 fst(X) -> n__fst(X) 396.43/291.59 snd(X) -> n__snd(X) 396.43/291.59 tail(X) -> n__tail(X) 396.43/291.59 take(X1, X2) -> n__take(X1, X2) 396.43/291.59 0 -> n__0 396.43/291.59 head(X) -> n__head(X) 396.43/291.59 s(X) -> n__s(X) 396.43/291.59 sel(X1, X2) -> n__sel(X1, X2) 396.43/291.59 pair(X1, X2) -> n__pair(X1, X2) 396.43/291.59 splitAt(X1, X2) -> n__splitAt(X1, X2) 396.43/291.59 and(X1, X2) -> n__and(X1, X2) 396.43/291.59 activate(n__natsFrom(X)) -> natsFrom(X) 396.43/291.59 activate(n__isLNat(X)) -> isLNat(X) 396.43/291.59 activate(n__nil) -> nil 396.43/291.59 activate(n__afterNth(X1, X2)) -> afterNth(X1, X2) 396.43/291.59 activate(n__cons(X1, X2)) -> cons(X1, X2) 396.43/291.59 activate(n__fst(X)) -> fst(X) 396.43/291.59 activate(n__snd(X)) -> snd(X) 396.43/291.59 activate(n__tail(X)) -> tail(X) 396.43/291.59 activate(n__take(X1, X2)) -> take(X1, X2) 396.43/291.59 activate(n__0) -> 0 396.43/291.59 activate(n__head(X)) -> head(X) 396.43/291.59 activate(n__s(X)) -> s(X) 396.43/291.59 activate(n__sel(X1, X2)) -> sel(X1, X2) 396.43/291.59 activate(n__pair(X1, X2)) -> pair(X1, X2) 396.43/291.59 activate(n__splitAt(X1, X2)) -> splitAt(X1, X2) 396.43/291.59 activate(n__and(X1, X2)) -> and(X1, X2) 396.43/291.59 activate(X) -> X 396.43/291.59 396.43/291.59 S is empty. 396.43/291.59 Rewrite Strategy: FULL 396.43/291.59 ---------------------------------------- 396.43/291.59 396.43/291.59 (3) DecreasingLoopProof (LOWER BOUND(ID)) 396.43/291.59 The following loop(s) give(s) rise to the lower bound Omega(n^1): 396.43/291.59 396.43/291.59 The rewrite sequence 396.43/291.59 396.43/291.59 isLNat(n__natsFrom(n__isLNat(X1_0))) ->^+ isNatural(isLNat(X1_0)) 396.43/291.59 396.43/291.59 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 396.43/291.59 396.43/291.59 The pumping substitution is [X1_0 / n__natsFrom(n__isLNat(X1_0))]. 396.43/291.59 396.43/291.59 The result substitution is [ ]. 396.43/291.59 396.43/291.59 396.43/291.59 396.43/291.59 396.43/291.59 ---------------------------------------- 396.43/291.59 396.43/291.59 (4) 396.43/291.59 Complex Obligation (BEST) 396.43/291.59 396.43/291.59 ---------------------------------------- 396.43/291.59 396.43/291.59 (5) 396.43/291.59 Obligation: 396.43/291.59 Proved the lower bound n^1 for the following obligation: 396.43/291.59 396.43/291.59 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 396.43/291.59 396.43/291.59 396.43/291.59 The TRS R consists of the following rules: 396.43/291.59 396.43/291.59 U101(tt, N, XS) -> fst(splitAt(activate(N), activate(XS))) 396.43/291.59 U11(tt, N, XS) -> snd(splitAt(activate(N), activate(XS))) 396.43/291.59 U21(tt, X) -> activate(X) 396.43/291.59 U31(tt, N) -> activate(N) 396.43/291.59 U41(tt, N) -> cons(activate(N), n__natsFrom(s(activate(N)))) 396.43/291.59 U51(tt, N, XS) -> head(afterNth(activate(N), activate(XS))) 396.43/291.59 U61(tt, Y) -> activate(Y) 396.43/291.59 U71(tt, XS) -> pair(nil, activate(XS)) 396.43/291.59 U81(tt, N, X, XS) -> U82(splitAt(activate(N), activate(XS)), activate(X)) 396.43/291.59 U82(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS) 396.43/291.59 U91(tt, XS) -> activate(XS) 396.43/291.59 afterNth(N, XS) -> U11(and(isNatural(N), n__isLNat(XS)), N, XS) 396.43/291.59 and(tt, X) -> activate(X) 396.43/291.59 fst(pair(X, Y)) -> U21(and(isLNat(X), n__isLNat(Y)), X) 396.43/291.59 head(cons(N, XS)) -> U31(and(isNatural(N), n__isLNat(activate(XS))), N) 396.43/291.59 isLNat(n__nil) -> tt 396.43/291.59 isLNat(n__afterNth(V1, V2)) -> and(isNatural(activate(V1)), n__isLNat(activate(V2))) 396.43/291.59 isLNat(n__cons(V1, V2)) -> and(isNatural(activate(V1)), n__isLNat(activate(V2))) 396.43/291.59 isLNat(n__fst(V1)) -> isPLNat(activate(V1)) 396.43/291.59 isLNat(n__natsFrom(V1)) -> isNatural(activate(V1)) 396.43/291.59 isLNat(n__snd(V1)) -> isPLNat(activate(V1)) 396.43/291.59 isLNat(n__tail(V1)) -> isLNat(activate(V1)) 396.43/291.59 isLNat(n__take(V1, V2)) -> and(isNatural(activate(V1)), n__isLNat(activate(V2))) 396.43/291.59 isNatural(n__0) -> tt 396.43/291.59 isNatural(n__head(V1)) -> isLNat(activate(V1)) 396.43/291.59 isNatural(n__s(V1)) -> isNatural(activate(V1)) 396.43/291.59 isNatural(n__sel(V1, V2)) -> and(isNatural(activate(V1)), n__isLNat(activate(V2))) 396.43/291.59 isPLNat(n__pair(V1, V2)) -> and(isLNat(activate(V1)), n__isLNat(activate(V2))) 396.43/291.59 isPLNat(n__splitAt(V1, V2)) -> and(isNatural(activate(V1)), n__isLNat(activate(V2))) 396.43/291.59 natsFrom(N) -> U41(isNatural(N), N) 396.43/291.59 sel(N, XS) -> U51(and(isNatural(N), n__isLNat(XS)), N, XS) 396.43/291.59 snd(pair(X, Y)) -> U61(and(isLNat(X), n__isLNat(Y)), Y) 396.43/291.59 splitAt(0, XS) -> U71(isLNat(XS), XS) 396.43/291.59 splitAt(s(N), cons(X, XS)) -> U81(and(isNatural(N), n__and(isNatural(X), n__isLNat(activate(XS)))), N, X, activate(XS)) 396.43/291.59 tail(cons(N, XS)) -> U91(and(isNatural(N), n__isLNat(activate(XS))), activate(XS)) 396.43/291.59 take(N, XS) -> U101(and(isNatural(N), n__isLNat(XS)), N, XS) 396.43/291.59 natsFrom(X) -> n__natsFrom(X) 396.43/291.59 isLNat(X) -> n__isLNat(X) 396.43/291.59 nil -> n__nil 396.43/291.59 afterNth(X1, X2) -> n__afterNth(X1, X2) 396.43/291.59 cons(X1, X2) -> n__cons(X1, X2) 396.43/291.59 fst(X) -> n__fst(X) 396.43/291.59 snd(X) -> n__snd(X) 396.43/291.59 tail(X) -> n__tail(X) 396.43/291.59 take(X1, X2) -> n__take(X1, X2) 396.43/291.59 0 -> n__0 396.43/291.59 head(X) -> n__head(X) 396.43/291.59 s(X) -> n__s(X) 396.43/291.59 sel(X1, X2) -> n__sel(X1, X2) 396.43/291.59 pair(X1, X2) -> n__pair(X1, X2) 396.43/291.59 splitAt(X1, X2) -> n__splitAt(X1, X2) 396.43/291.59 and(X1, X2) -> n__and(X1, X2) 396.43/291.59 activate(n__natsFrom(X)) -> natsFrom(X) 396.43/291.59 activate(n__isLNat(X)) -> isLNat(X) 396.43/291.59 activate(n__nil) -> nil 396.43/291.59 activate(n__afterNth(X1, X2)) -> afterNth(X1, X2) 396.43/291.59 activate(n__cons(X1, X2)) -> cons(X1, X2) 396.43/291.59 activate(n__fst(X)) -> fst(X) 396.43/291.59 activate(n__snd(X)) -> snd(X) 396.43/291.59 activate(n__tail(X)) -> tail(X) 396.43/291.59 activate(n__take(X1, X2)) -> take(X1, X2) 396.43/291.59 activate(n__0) -> 0 396.43/291.59 activate(n__head(X)) -> head(X) 396.43/291.59 activate(n__s(X)) -> s(X) 396.43/291.59 activate(n__sel(X1, X2)) -> sel(X1, X2) 396.43/291.59 activate(n__pair(X1, X2)) -> pair(X1, X2) 396.43/291.59 activate(n__splitAt(X1, X2)) -> splitAt(X1, X2) 396.43/291.59 activate(n__and(X1, X2)) -> and(X1, X2) 396.43/291.59 activate(X) -> X 396.43/291.59 396.43/291.59 S is empty. 396.43/291.59 Rewrite Strategy: FULL 396.43/291.59 ---------------------------------------- 396.43/291.59 396.43/291.59 (6) LowerBoundPropagationProof (FINISHED) 396.43/291.59 Propagated lower bound. 396.43/291.59 ---------------------------------------- 396.43/291.59 396.43/291.59 (7) 396.43/291.59 BOUNDS(n^1, INF) 396.43/291.59 396.43/291.59 ---------------------------------------- 396.43/291.59 396.43/291.59 (8) 396.43/291.59 Obligation: 396.43/291.59 Analyzing the following TRS for decreasing loops: 396.43/291.59 396.43/291.59 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 396.43/291.59 396.43/291.59 396.43/291.59 The TRS R consists of the following rules: 396.43/291.59 396.43/291.59 U101(tt, N, XS) -> fst(splitAt(activate(N), activate(XS))) 396.43/291.59 U11(tt, N, XS) -> snd(splitAt(activate(N), activate(XS))) 396.43/291.59 U21(tt, X) -> activate(X) 396.43/291.59 U31(tt, N) -> activate(N) 396.43/291.59 U41(tt, N) -> cons(activate(N), n__natsFrom(s(activate(N)))) 396.43/291.59 U51(tt, N, XS) -> head(afterNth(activate(N), activate(XS))) 396.43/291.59 U61(tt, Y) -> activate(Y) 396.43/291.59 U71(tt, XS) -> pair(nil, activate(XS)) 396.43/291.59 U81(tt, N, X, XS) -> U82(splitAt(activate(N), activate(XS)), activate(X)) 396.43/291.59 U82(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS) 396.43/291.59 U91(tt, XS) -> activate(XS) 396.43/291.59 afterNth(N, XS) -> U11(and(isNatural(N), n__isLNat(XS)), N, XS) 396.43/291.59 and(tt, X) -> activate(X) 396.43/291.59 fst(pair(X, Y)) -> U21(and(isLNat(X), n__isLNat(Y)), X) 396.43/291.59 head(cons(N, XS)) -> U31(and(isNatural(N), n__isLNat(activate(XS))), N) 396.43/291.59 isLNat(n__nil) -> tt 396.43/291.59 isLNat(n__afterNth(V1, V2)) -> and(isNatural(activate(V1)), n__isLNat(activate(V2))) 396.43/291.59 isLNat(n__cons(V1, V2)) -> and(isNatural(activate(V1)), n__isLNat(activate(V2))) 396.43/291.59 isLNat(n__fst(V1)) -> isPLNat(activate(V1)) 396.43/291.59 isLNat(n__natsFrom(V1)) -> isNatural(activate(V1)) 396.43/291.59 isLNat(n__snd(V1)) -> isPLNat(activate(V1)) 396.43/291.59 isLNat(n__tail(V1)) -> isLNat(activate(V1)) 396.43/291.59 isLNat(n__take(V1, V2)) -> and(isNatural(activate(V1)), n__isLNat(activate(V2))) 396.43/291.59 isNatural(n__0) -> tt 396.43/291.59 isNatural(n__head(V1)) -> isLNat(activate(V1)) 396.43/291.59 isNatural(n__s(V1)) -> isNatural(activate(V1)) 396.43/291.59 isNatural(n__sel(V1, V2)) -> and(isNatural(activate(V1)), n__isLNat(activate(V2))) 396.43/291.59 isPLNat(n__pair(V1, V2)) -> and(isLNat(activate(V1)), n__isLNat(activate(V2))) 396.43/291.59 isPLNat(n__splitAt(V1, V2)) -> and(isNatural(activate(V1)), n__isLNat(activate(V2))) 396.43/291.59 natsFrom(N) -> U41(isNatural(N), N) 396.43/291.59 sel(N, XS) -> U51(and(isNatural(N), n__isLNat(XS)), N, XS) 396.43/291.59 snd(pair(X, Y)) -> U61(and(isLNat(X), n__isLNat(Y)), Y) 396.43/291.59 splitAt(0, XS) -> U71(isLNat(XS), XS) 396.43/291.59 splitAt(s(N), cons(X, XS)) -> U81(and(isNatural(N), n__and(isNatural(X), n__isLNat(activate(XS)))), N, X, activate(XS)) 396.43/291.59 tail(cons(N, XS)) -> U91(and(isNatural(N), n__isLNat(activate(XS))), activate(XS)) 396.43/291.59 take(N, XS) -> U101(and(isNatural(N), n__isLNat(XS)), N, XS) 396.43/291.59 natsFrom(X) -> n__natsFrom(X) 396.43/291.59 isLNat(X) -> n__isLNat(X) 396.43/291.59 nil -> n__nil 396.43/291.59 afterNth(X1, X2) -> n__afterNth(X1, X2) 396.43/291.59 cons(X1, X2) -> n__cons(X1, X2) 396.43/291.59 fst(X) -> n__fst(X) 396.43/291.59 snd(X) -> n__snd(X) 396.43/291.59 tail(X) -> n__tail(X) 396.43/291.59 take(X1, X2) -> n__take(X1, X2) 396.43/291.59 0 -> n__0 396.43/291.59 head(X) -> n__head(X) 396.43/291.59 s(X) -> n__s(X) 396.43/291.59 sel(X1, X2) -> n__sel(X1, X2) 396.43/291.59 pair(X1, X2) -> n__pair(X1, X2) 396.43/291.59 splitAt(X1, X2) -> n__splitAt(X1, X2) 396.43/291.59 and(X1, X2) -> n__and(X1, X2) 396.43/291.59 activate(n__natsFrom(X)) -> natsFrom(X) 396.43/291.59 activate(n__isLNat(X)) -> isLNat(X) 396.43/291.59 activate(n__nil) -> nil 396.43/291.59 activate(n__afterNth(X1, X2)) -> afterNth(X1, X2) 396.43/291.59 activate(n__cons(X1, X2)) -> cons(X1, X2) 396.43/291.59 activate(n__fst(X)) -> fst(X) 396.43/291.59 activate(n__snd(X)) -> snd(X) 396.43/291.59 activate(n__tail(X)) -> tail(X) 396.43/291.59 activate(n__take(X1, X2)) -> take(X1, X2) 396.43/291.59 activate(n__0) -> 0 396.43/291.59 activate(n__head(X)) -> head(X) 396.43/291.59 activate(n__s(X)) -> s(X) 396.43/291.59 activate(n__sel(X1, X2)) -> sel(X1, X2) 396.43/291.59 activate(n__pair(X1, X2)) -> pair(X1, X2) 396.43/291.59 activate(n__splitAt(X1, X2)) -> splitAt(X1, X2) 396.43/291.59 activate(n__and(X1, X2)) -> and(X1, X2) 396.43/291.59 activate(X) -> X 396.43/291.59 396.43/291.59 S is empty. 396.43/291.59 Rewrite Strategy: FULL 396.47/291.62 EOF