7.73/2.78 WORST_CASE(NON_POLY, ?) 7.95/2.80 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 7.95/2.80 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 7.95/2.80 7.95/2.80 7.95/2.80 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 7.95/2.80 7.95/2.80 (0) CpxTRS 7.95/2.80 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 7.95/2.80 (2) TRS for Loop Detection 7.95/2.80 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 7.95/2.80 (4) BEST 7.95/2.80 (5) proven lower bound 7.95/2.80 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 7.95/2.80 (7) BOUNDS(n^1, INF) 7.95/2.80 (8) TRS for Loop Detection 7.95/2.80 (9) DecreasingLoopProof [FINISHED, 684 ms] 7.95/2.80 (10) BOUNDS(EXP, INF) 7.95/2.80 7.95/2.80 7.95/2.80 ---------------------------------------- 7.95/2.80 7.95/2.80 (0) 7.95/2.80 Obligation: 7.95/2.80 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 7.95/2.80 7.95/2.80 7.95/2.80 The TRS R consists of the following rules: 7.95/2.80 7.95/2.80 U101(tt, V1, V2) -> U102(isNatural(activate(V1)), activate(V2)) 7.95/2.80 U102(tt, V2) -> U103(isLNat(activate(V2))) 7.95/2.80 U103(tt) -> tt 7.95/2.80 U11(tt, N, XS) -> snd(splitAt(activate(N), activate(XS))) 7.95/2.80 U111(tt, V1) -> U112(isLNat(activate(V1))) 7.95/2.80 U112(tt) -> tt 7.95/2.80 U121(tt, V1) -> U122(isNatural(activate(V1))) 7.95/2.80 U122(tt) -> tt 7.95/2.80 U131(tt, V1, V2) -> U132(isNatural(activate(V1)), activate(V2)) 7.95/2.80 U132(tt, V2) -> U133(isLNat(activate(V2))) 7.95/2.80 U133(tt) -> tt 7.95/2.80 U141(tt, V1, V2) -> U142(isLNat(activate(V1)), activate(V2)) 7.95/2.80 U142(tt, V2) -> U143(isLNat(activate(V2))) 7.95/2.80 U143(tt) -> tt 7.95/2.80 U151(tt, V1, V2) -> U152(isNatural(activate(V1)), activate(V2)) 7.95/2.80 U152(tt, V2) -> U153(isLNat(activate(V2))) 7.95/2.80 U153(tt) -> tt 7.95/2.80 U161(tt, N) -> cons(activate(N), n__natsFrom(n__s(activate(N)))) 7.95/2.80 U171(tt, N, XS) -> head(afterNth(activate(N), activate(XS))) 7.95/2.80 U181(tt, Y) -> activate(Y) 7.95/2.80 U191(tt, XS) -> pair(nil, activate(XS)) 7.95/2.80 U201(tt, N, X, XS) -> U202(splitAt(activate(N), activate(XS)), activate(X)) 7.95/2.80 U202(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS) 7.95/2.80 U21(tt, X) -> activate(X) 7.95/2.80 U211(tt, XS) -> activate(XS) 7.95/2.80 U221(tt, N, XS) -> fst(splitAt(activate(N), activate(XS))) 7.95/2.80 U31(tt, N) -> activate(N) 7.95/2.80 U41(tt, V1, V2) -> U42(isNatural(activate(V1)), activate(V2)) 7.95/2.80 U42(tt, V2) -> U43(isLNat(activate(V2))) 7.95/2.80 U43(tt) -> tt 7.95/2.80 U51(tt, V1, V2) -> U52(isNatural(activate(V1)), activate(V2)) 7.95/2.80 U52(tt, V2) -> U53(isLNat(activate(V2))) 7.95/2.80 U53(tt) -> tt 7.95/2.80 U61(tt, V1) -> U62(isPLNat(activate(V1))) 7.95/2.80 U62(tt) -> tt 7.95/2.80 U71(tt, V1) -> U72(isNatural(activate(V1))) 7.95/2.80 U72(tt) -> tt 7.95/2.80 U81(tt, V1) -> U82(isPLNat(activate(V1))) 7.95/2.80 U82(tt) -> tt 7.95/2.80 U91(tt, V1) -> U92(isLNat(activate(V1))) 7.95/2.80 U92(tt) -> tt 7.95/2.80 afterNth(N, XS) -> U11(and(and(isNatural(N), n__isNaturalKind(N)), n__and(n__isLNat(XS), n__isLNatKind(XS))), N, XS) 7.95/2.80 and(tt, X) -> activate(X) 7.95/2.80 fst(pair(X, Y)) -> U21(and(and(isLNat(X), n__isLNatKind(X)), n__and(n__isLNat(Y), n__isLNatKind(Y))), X) 7.95/2.80 head(cons(N, XS)) -> U31(and(and(isNatural(N), n__isNaturalKind(N)), n__and(n__isLNat(activate(XS)), n__isLNatKind(activate(XS)))), N) 7.95/2.80 isLNat(n__nil) -> tt 7.95/2.80 isLNat(n__afterNth(V1, V2)) -> U41(and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2))), activate(V1), activate(V2)) 7.95/2.80 isLNat(n__cons(V1, V2)) -> U51(and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2))), activate(V1), activate(V2)) 7.95/2.80 isLNat(n__fst(V1)) -> U61(isPLNatKind(activate(V1)), activate(V1)) 7.95/2.80 isLNat(n__natsFrom(V1)) -> U71(isNaturalKind(activate(V1)), activate(V1)) 7.95/2.80 isLNat(n__snd(V1)) -> U81(isPLNatKind(activate(V1)), activate(V1)) 7.95/2.80 isLNat(n__tail(V1)) -> U91(isLNatKind(activate(V1)), activate(V1)) 7.95/2.80 isLNat(n__take(V1, V2)) -> U101(and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2))), activate(V1), activate(V2)) 7.95/2.80 isLNatKind(n__nil) -> tt 7.95/2.80 isLNatKind(n__afterNth(V1, V2)) -> and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2))) 7.95/2.80 isLNatKind(n__cons(V1, V2)) -> and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2))) 7.95/2.80 isLNatKind(n__fst(V1)) -> isPLNatKind(activate(V1)) 7.95/2.80 isLNatKind(n__natsFrom(V1)) -> isNaturalKind(activate(V1)) 7.95/2.80 isLNatKind(n__snd(V1)) -> isPLNatKind(activate(V1)) 7.95/2.80 isLNatKind(n__tail(V1)) -> isLNatKind(activate(V1)) 7.95/2.80 isLNatKind(n__take(V1, V2)) -> and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2))) 7.95/2.80 isNatural(n__0) -> tt 7.95/2.80 isNatural(n__head(V1)) -> U111(isLNatKind(activate(V1)), activate(V1)) 7.95/2.80 isNatural(n__s(V1)) -> U121(isNaturalKind(activate(V1)), activate(V1)) 7.95/2.80 isNatural(n__sel(V1, V2)) -> U131(and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2))), activate(V1), activate(V2)) 7.95/2.80 isNaturalKind(n__0) -> tt 7.95/2.80 isNaturalKind(n__head(V1)) -> isLNatKind(activate(V1)) 7.95/2.80 isNaturalKind(n__s(V1)) -> isNaturalKind(activate(V1)) 7.95/2.80 isNaturalKind(n__sel(V1, V2)) -> and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2))) 7.95/2.80 isPLNat(n__pair(V1, V2)) -> U141(and(isLNatKind(activate(V1)), n__isLNatKind(activate(V2))), activate(V1), activate(V2)) 7.95/2.80 isPLNat(n__splitAt(V1, V2)) -> U151(and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2))), activate(V1), activate(V2)) 7.95/2.80 isPLNatKind(n__pair(V1, V2)) -> and(isLNatKind(activate(V1)), n__isLNatKind(activate(V2))) 7.95/2.80 isPLNatKind(n__splitAt(V1, V2)) -> and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2))) 7.95/2.80 natsFrom(N) -> U161(and(isNatural(N), n__isNaturalKind(N)), N) 7.95/2.80 sel(N, XS) -> U171(and(and(isNatural(N), n__isNaturalKind(N)), n__and(n__isLNat(XS), n__isLNatKind(XS))), N, XS) 7.95/2.80 snd(pair(X, Y)) -> U181(and(and(isLNat(X), n__isLNatKind(X)), n__and(n__isLNat(Y), n__isLNatKind(Y))), Y) 7.95/2.80 splitAt(0, XS) -> U191(and(isLNat(XS), n__isLNatKind(XS)), XS) 7.95/2.80 splitAt(s(N), cons(X, XS)) -> U201(and(and(isNatural(N), n__isNaturalKind(N)), n__and(n__and(n__isNatural(X), n__isNaturalKind(X)), n__and(n__isLNat(activate(XS)), n__isLNatKind(activate(XS))))), N, X, activate(XS)) 7.95/2.80 tail(cons(N, XS)) -> U211(and(and(isNatural(N), n__isNaturalKind(N)), n__and(n__isLNat(activate(XS)), n__isLNatKind(activate(XS)))), activate(XS)) 7.95/2.80 take(N, XS) -> U221(and(and(isNatural(N), n__isNaturalKind(N)), n__and(n__isLNat(XS), n__isLNatKind(XS))), N, XS) 7.95/2.80 natsFrom(X) -> n__natsFrom(X) 7.95/2.80 s(X) -> n__s(X) 7.95/2.80 isNaturalKind(X) -> n__isNaturalKind(X) 7.95/2.80 and(X1, X2) -> n__and(X1, X2) 7.95/2.80 isLNat(X) -> n__isLNat(X) 7.95/2.80 isLNatKind(X) -> n__isLNatKind(X) 7.95/2.80 nil -> n__nil 7.95/2.80 afterNth(X1, X2) -> n__afterNth(X1, X2) 7.95/2.80 cons(X1, X2) -> n__cons(X1, X2) 7.95/2.80 fst(X) -> n__fst(X) 7.95/2.80 snd(X) -> n__snd(X) 7.95/2.80 tail(X) -> n__tail(X) 7.95/2.80 take(X1, X2) -> n__take(X1, X2) 7.95/2.80 0 -> n__0 7.95/2.80 head(X) -> n__head(X) 7.95/2.80 sel(X1, X2) -> n__sel(X1, X2) 7.95/2.80 pair(X1, X2) -> n__pair(X1, X2) 7.95/2.80 splitAt(X1, X2) -> n__splitAt(X1, X2) 7.95/2.80 isNatural(X) -> n__isNatural(X) 7.95/2.80 activate(n__natsFrom(X)) -> natsFrom(activate(X)) 7.95/2.80 activate(n__s(X)) -> s(activate(X)) 7.95/2.80 activate(n__isNaturalKind(X)) -> isNaturalKind(X) 7.95/2.80 activate(n__and(X1, X2)) -> and(activate(X1), X2) 7.95/2.80 activate(n__isLNat(X)) -> isLNat(X) 7.95/2.80 activate(n__isLNatKind(X)) -> isLNatKind(X) 7.95/2.80 activate(n__nil) -> nil 7.95/2.80 activate(n__afterNth(X1, X2)) -> afterNth(activate(X1), activate(X2)) 7.95/2.80 activate(n__cons(X1, X2)) -> cons(activate(X1), X2) 7.95/2.80 activate(n__fst(X)) -> fst(activate(X)) 7.95/2.80 activate(n__snd(X)) -> snd(activate(X)) 7.95/2.80 activate(n__tail(X)) -> tail(activate(X)) 7.95/2.80 activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) 7.95/2.80 activate(n__0) -> 0 7.95/2.80 activate(n__head(X)) -> head(activate(X)) 7.95/2.80 activate(n__sel(X1, X2)) -> sel(activate(X1), activate(X2)) 7.95/2.80 activate(n__pair(X1, X2)) -> pair(activate(X1), activate(X2)) 7.95/2.80 activate(n__splitAt(X1, X2)) -> splitAt(activate(X1), activate(X2)) 7.95/2.80 activate(n__isNatural(X)) -> isNatural(X) 7.95/2.80 activate(X) -> X 7.95/2.80 7.95/2.80 S is empty. 7.95/2.80 Rewrite Strategy: FULL 7.95/2.80 ---------------------------------------- 7.95/2.80 7.95/2.80 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 7.95/2.80 Transformed a relative TRS into a decreasing-loop problem. 7.95/2.80 ---------------------------------------- 7.95/2.80 7.95/2.80 (2) 7.95/2.80 Obligation: 7.95/2.80 Analyzing the following TRS for decreasing loops: 7.95/2.80 7.95/2.80 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 7.95/2.80 7.95/2.80 7.95/2.80 The TRS R consists of the following rules: 7.95/2.80 7.95/2.80 U101(tt, V1, V2) -> U102(isNatural(activate(V1)), activate(V2)) 7.95/2.80 U102(tt, V2) -> U103(isLNat(activate(V2))) 7.95/2.80 U103(tt) -> tt 7.95/2.80 U11(tt, N, XS) -> snd(splitAt(activate(N), activate(XS))) 7.95/2.80 U111(tt, V1) -> U112(isLNat(activate(V1))) 7.95/2.80 U112(tt) -> tt 7.95/2.80 U121(tt, V1) -> U122(isNatural(activate(V1))) 7.95/2.80 U122(tt) -> tt 7.95/2.80 U131(tt, V1, V2) -> U132(isNatural(activate(V1)), activate(V2)) 7.95/2.80 U132(tt, V2) -> U133(isLNat(activate(V2))) 7.95/2.80 U133(tt) -> tt 7.95/2.80 U141(tt, V1, V2) -> U142(isLNat(activate(V1)), activate(V2)) 7.95/2.80 U142(tt, V2) -> U143(isLNat(activate(V2))) 7.95/2.80 U143(tt) -> tt 7.95/2.80 U151(tt, V1, V2) -> U152(isNatural(activate(V1)), activate(V2)) 7.95/2.80 U152(tt, V2) -> U153(isLNat(activate(V2))) 7.95/2.80 U153(tt) -> tt 7.95/2.80 U161(tt, N) -> cons(activate(N), n__natsFrom(n__s(activate(N)))) 7.95/2.80 U171(tt, N, XS) -> head(afterNth(activate(N), activate(XS))) 7.95/2.80 U181(tt, Y) -> activate(Y) 7.95/2.80 U191(tt, XS) -> pair(nil, activate(XS)) 7.95/2.80 U201(tt, N, X, XS) -> U202(splitAt(activate(N), activate(XS)), activate(X)) 7.95/2.80 U202(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS) 7.95/2.80 U21(tt, X) -> activate(X) 7.95/2.80 U211(tt, XS) -> activate(XS) 7.95/2.80 U221(tt, N, XS) -> fst(splitAt(activate(N), activate(XS))) 7.95/2.80 U31(tt, N) -> activate(N) 7.95/2.80 U41(tt, V1, V2) -> U42(isNatural(activate(V1)), activate(V2)) 7.95/2.80 U42(tt, V2) -> U43(isLNat(activate(V2))) 7.95/2.80 U43(tt) -> tt 7.95/2.80 U51(tt, V1, V2) -> U52(isNatural(activate(V1)), activate(V2)) 7.95/2.80 U52(tt, V2) -> U53(isLNat(activate(V2))) 7.95/2.80 U53(tt) -> tt 7.95/2.80 U61(tt, V1) -> U62(isPLNat(activate(V1))) 7.95/2.80 U62(tt) -> tt 7.95/2.80 U71(tt, V1) -> U72(isNatural(activate(V1))) 7.95/2.80 U72(tt) -> tt 7.95/2.80 U81(tt, V1) -> U82(isPLNat(activate(V1))) 7.95/2.80 U82(tt) -> tt 7.95/2.80 U91(tt, V1) -> U92(isLNat(activate(V1))) 7.95/2.80 U92(tt) -> tt 7.95/2.80 afterNth(N, XS) -> U11(and(and(isNatural(N), n__isNaturalKind(N)), n__and(n__isLNat(XS), n__isLNatKind(XS))), N, XS) 7.95/2.80 and(tt, X) -> activate(X) 7.95/2.80 fst(pair(X, Y)) -> U21(and(and(isLNat(X), n__isLNatKind(X)), n__and(n__isLNat(Y), n__isLNatKind(Y))), X) 7.95/2.80 head(cons(N, XS)) -> U31(and(and(isNatural(N), n__isNaturalKind(N)), n__and(n__isLNat(activate(XS)), n__isLNatKind(activate(XS)))), N) 7.95/2.80 isLNat(n__nil) -> tt 7.95/2.80 isLNat(n__afterNth(V1, V2)) -> U41(and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2))), activate(V1), activate(V2)) 7.95/2.80 isLNat(n__cons(V1, V2)) -> U51(and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2))), activate(V1), activate(V2)) 7.95/2.80 isLNat(n__fst(V1)) -> U61(isPLNatKind(activate(V1)), activate(V1)) 7.95/2.80 isLNat(n__natsFrom(V1)) -> U71(isNaturalKind(activate(V1)), activate(V1)) 7.95/2.80 isLNat(n__snd(V1)) -> U81(isPLNatKind(activate(V1)), activate(V1)) 7.95/2.80 isLNat(n__tail(V1)) -> U91(isLNatKind(activate(V1)), activate(V1)) 7.95/2.80 isLNat(n__take(V1, V2)) -> U101(and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2))), activate(V1), activate(V2)) 7.95/2.80 isLNatKind(n__nil) -> tt 7.95/2.80 isLNatKind(n__afterNth(V1, V2)) -> and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2))) 7.95/2.80 isLNatKind(n__cons(V1, V2)) -> and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2))) 7.95/2.80 isLNatKind(n__fst(V1)) -> isPLNatKind(activate(V1)) 7.95/2.80 isLNatKind(n__natsFrom(V1)) -> isNaturalKind(activate(V1)) 7.95/2.80 isLNatKind(n__snd(V1)) -> isPLNatKind(activate(V1)) 7.95/2.80 isLNatKind(n__tail(V1)) -> isLNatKind(activate(V1)) 7.95/2.80 isLNatKind(n__take(V1, V2)) -> and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2))) 7.95/2.80 isNatural(n__0) -> tt 7.95/2.80 isNatural(n__head(V1)) -> U111(isLNatKind(activate(V1)), activate(V1)) 7.95/2.80 isNatural(n__s(V1)) -> U121(isNaturalKind(activate(V1)), activate(V1)) 7.95/2.80 isNatural(n__sel(V1, V2)) -> U131(and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2))), activate(V1), activate(V2)) 7.95/2.80 isNaturalKind(n__0) -> tt 7.95/2.80 isNaturalKind(n__head(V1)) -> isLNatKind(activate(V1)) 7.95/2.80 isNaturalKind(n__s(V1)) -> isNaturalKind(activate(V1)) 7.95/2.80 isNaturalKind(n__sel(V1, V2)) -> and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2))) 7.95/2.80 isPLNat(n__pair(V1, V2)) -> U141(and(isLNatKind(activate(V1)), n__isLNatKind(activate(V2))), activate(V1), activate(V2)) 7.95/2.80 isPLNat(n__splitAt(V1, V2)) -> U151(and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2))), activate(V1), activate(V2)) 7.95/2.80 isPLNatKind(n__pair(V1, V2)) -> and(isLNatKind(activate(V1)), n__isLNatKind(activate(V2))) 7.95/2.80 isPLNatKind(n__splitAt(V1, V2)) -> and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2))) 7.95/2.80 natsFrom(N) -> U161(and(isNatural(N), n__isNaturalKind(N)), N) 7.95/2.80 sel(N, XS) -> U171(and(and(isNatural(N), n__isNaturalKind(N)), n__and(n__isLNat(XS), n__isLNatKind(XS))), N, XS) 7.95/2.80 snd(pair(X, Y)) -> U181(and(and(isLNat(X), n__isLNatKind(X)), n__and(n__isLNat(Y), n__isLNatKind(Y))), Y) 7.95/2.80 splitAt(0, XS) -> U191(and(isLNat(XS), n__isLNatKind(XS)), XS) 7.95/2.80 splitAt(s(N), cons(X, XS)) -> U201(and(and(isNatural(N), n__isNaturalKind(N)), n__and(n__and(n__isNatural(X), n__isNaturalKind(X)), n__and(n__isLNat(activate(XS)), n__isLNatKind(activate(XS))))), N, X, activate(XS)) 7.95/2.80 tail(cons(N, XS)) -> U211(and(and(isNatural(N), n__isNaturalKind(N)), n__and(n__isLNat(activate(XS)), n__isLNatKind(activate(XS)))), activate(XS)) 7.95/2.80 take(N, XS) -> U221(and(and(isNatural(N), n__isNaturalKind(N)), n__and(n__isLNat(XS), n__isLNatKind(XS))), N, XS) 7.95/2.80 natsFrom(X) -> n__natsFrom(X) 7.95/2.80 s(X) -> n__s(X) 7.95/2.80 isNaturalKind(X) -> n__isNaturalKind(X) 7.95/2.80 and(X1, X2) -> n__and(X1, X2) 7.95/2.80 isLNat(X) -> n__isLNat(X) 7.95/2.80 isLNatKind(X) -> n__isLNatKind(X) 7.95/2.80 nil -> n__nil 7.95/2.80 afterNth(X1, X2) -> n__afterNth(X1, X2) 7.95/2.80 cons(X1, X2) -> n__cons(X1, X2) 7.95/2.80 fst(X) -> n__fst(X) 7.95/2.80 snd(X) -> n__snd(X) 7.95/2.80 tail(X) -> n__tail(X) 7.95/2.80 take(X1, X2) -> n__take(X1, X2) 7.95/2.80 0 -> n__0 7.95/2.80 head(X) -> n__head(X) 7.95/2.80 sel(X1, X2) -> n__sel(X1, X2) 7.95/2.80 pair(X1, X2) -> n__pair(X1, X2) 7.95/2.80 splitAt(X1, X2) -> n__splitAt(X1, X2) 7.95/2.80 isNatural(X) -> n__isNatural(X) 7.95/2.80 activate(n__natsFrom(X)) -> natsFrom(activate(X)) 7.95/2.80 activate(n__s(X)) -> s(activate(X)) 7.95/2.80 activate(n__isNaturalKind(X)) -> isNaturalKind(X) 7.95/2.80 activate(n__and(X1, X2)) -> and(activate(X1), X2) 7.95/2.80 activate(n__isLNat(X)) -> isLNat(X) 7.95/2.80 activate(n__isLNatKind(X)) -> isLNatKind(X) 7.95/2.80 activate(n__nil) -> nil 7.95/2.80 activate(n__afterNth(X1, X2)) -> afterNth(activate(X1), activate(X2)) 7.95/2.80 activate(n__cons(X1, X2)) -> cons(activate(X1), X2) 7.95/2.80 activate(n__fst(X)) -> fst(activate(X)) 7.95/2.80 activate(n__snd(X)) -> snd(activate(X)) 7.95/2.80 activate(n__tail(X)) -> tail(activate(X)) 7.95/2.80 activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) 7.95/2.80 activate(n__0) -> 0 7.95/2.80 activate(n__head(X)) -> head(activate(X)) 7.95/2.80 activate(n__sel(X1, X2)) -> sel(activate(X1), activate(X2)) 7.95/2.80 activate(n__pair(X1, X2)) -> pair(activate(X1), activate(X2)) 7.95/2.80 activate(n__splitAt(X1, X2)) -> splitAt(activate(X1), activate(X2)) 7.95/2.80 activate(n__isNatural(X)) -> isNatural(X) 7.95/2.80 activate(X) -> X 7.95/2.80 7.95/2.80 S is empty. 7.95/2.80 Rewrite Strategy: FULL 7.95/2.80 ---------------------------------------- 7.95/2.80 7.95/2.80 (3) DecreasingLoopProof (LOWER BOUND(ID)) 7.95/2.80 The following loop(s) give(s) rise to the lower bound Omega(n^1): 7.95/2.80 7.95/2.80 The rewrite sequence 7.95/2.80 7.95/2.80 activate(n__s(X)) ->^+ s(activate(X)) 7.95/2.80 7.95/2.80 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 7.95/2.80 7.95/2.80 The pumping substitution is [X / n__s(X)]. 7.95/2.80 7.95/2.80 The result substitution is [ ]. 7.95/2.80 7.95/2.80 7.95/2.80 7.95/2.80 7.95/2.80 ---------------------------------------- 7.95/2.80 7.95/2.80 (4) 7.95/2.80 Complex Obligation (BEST) 7.95/2.80 7.95/2.80 ---------------------------------------- 7.95/2.80 7.95/2.80 (5) 7.95/2.80 Obligation: 7.95/2.80 Proved the lower bound n^1 for the following obligation: 7.95/2.80 7.95/2.80 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 7.95/2.80 7.95/2.80 7.95/2.80 The TRS R consists of the following rules: 7.95/2.80 7.95/2.80 U101(tt, V1, V2) -> U102(isNatural(activate(V1)), activate(V2)) 7.95/2.80 U102(tt, V2) -> U103(isLNat(activate(V2))) 7.95/2.80 U103(tt) -> tt 7.95/2.80 U11(tt, N, XS) -> snd(splitAt(activate(N), activate(XS))) 7.95/2.80 U111(tt, V1) -> U112(isLNat(activate(V1))) 7.95/2.80 U112(tt) -> tt 7.95/2.80 U121(tt, V1) -> U122(isNatural(activate(V1))) 7.95/2.80 U122(tt) -> tt 7.95/2.80 U131(tt, V1, V2) -> U132(isNatural(activate(V1)), activate(V2)) 7.95/2.80 U132(tt, V2) -> U133(isLNat(activate(V2))) 7.95/2.80 U133(tt) -> tt 7.95/2.80 U141(tt, V1, V2) -> U142(isLNat(activate(V1)), activate(V2)) 7.95/2.80 U142(tt, V2) -> U143(isLNat(activate(V2))) 7.95/2.80 U143(tt) -> tt 7.95/2.80 U151(tt, V1, V2) -> U152(isNatural(activate(V1)), activate(V2)) 7.95/2.80 U152(tt, V2) -> U153(isLNat(activate(V2))) 7.95/2.80 U153(tt) -> tt 7.95/2.80 U161(tt, N) -> cons(activate(N), n__natsFrom(n__s(activate(N)))) 7.95/2.80 U171(tt, N, XS) -> head(afterNth(activate(N), activate(XS))) 7.95/2.80 U181(tt, Y) -> activate(Y) 7.95/2.80 U191(tt, XS) -> pair(nil, activate(XS)) 7.95/2.80 U201(tt, N, X, XS) -> U202(splitAt(activate(N), activate(XS)), activate(X)) 7.95/2.80 U202(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS) 7.95/2.80 U21(tt, X) -> activate(X) 7.95/2.80 U211(tt, XS) -> activate(XS) 7.95/2.80 U221(tt, N, XS) -> fst(splitAt(activate(N), activate(XS))) 7.95/2.80 U31(tt, N) -> activate(N) 7.95/2.80 U41(tt, V1, V2) -> U42(isNatural(activate(V1)), activate(V2)) 7.95/2.80 U42(tt, V2) -> U43(isLNat(activate(V2))) 7.95/2.80 U43(tt) -> tt 7.95/2.80 U51(tt, V1, V2) -> U52(isNatural(activate(V1)), activate(V2)) 7.95/2.80 U52(tt, V2) -> U53(isLNat(activate(V2))) 7.95/2.80 U53(tt) -> tt 7.95/2.80 U61(tt, V1) -> U62(isPLNat(activate(V1))) 7.95/2.80 U62(tt) -> tt 7.95/2.80 U71(tt, V1) -> U72(isNatural(activate(V1))) 7.95/2.80 U72(tt) -> tt 7.95/2.80 U81(tt, V1) -> U82(isPLNat(activate(V1))) 7.95/2.80 U82(tt) -> tt 7.95/2.80 U91(tt, V1) -> U92(isLNat(activate(V1))) 7.95/2.80 U92(tt) -> tt 7.95/2.80 afterNth(N, XS) -> U11(and(and(isNatural(N), n__isNaturalKind(N)), n__and(n__isLNat(XS), n__isLNatKind(XS))), N, XS) 7.95/2.80 and(tt, X) -> activate(X) 7.95/2.80 fst(pair(X, Y)) -> U21(and(and(isLNat(X), n__isLNatKind(X)), n__and(n__isLNat(Y), n__isLNatKind(Y))), X) 7.95/2.80 head(cons(N, XS)) -> U31(and(and(isNatural(N), n__isNaturalKind(N)), n__and(n__isLNat(activate(XS)), n__isLNatKind(activate(XS)))), N) 7.95/2.80 isLNat(n__nil) -> tt 7.95/2.80 isLNat(n__afterNth(V1, V2)) -> U41(and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2))), activate(V1), activate(V2)) 7.95/2.80 isLNat(n__cons(V1, V2)) -> U51(and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2))), activate(V1), activate(V2)) 7.95/2.80 isLNat(n__fst(V1)) -> U61(isPLNatKind(activate(V1)), activate(V1)) 7.95/2.80 isLNat(n__natsFrom(V1)) -> U71(isNaturalKind(activate(V1)), activate(V1)) 7.95/2.80 isLNat(n__snd(V1)) -> U81(isPLNatKind(activate(V1)), activate(V1)) 7.95/2.80 isLNat(n__tail(V1)) -> U91(isLNatKind(activate(V1)), activate(V1)) 7.95/2.80 isLNat(n__take(V1, V2)) -> U101(and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2))), activate(V1), activate(V2)) 7.95/2.80 isLNatKind(n__nil) -> tt 7.95/2.80 isLNatKind(n__afterNth(V1, V2)) -> and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2))) 7.95/2.80 isLNatKind(n__cons(V1, V2)) -> and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2))) 7.95/2.80 isLNatKind(n__fst(V1)) -> isPLNatKind(activate(V1)) 7.95/2.80 isLNatKind(n__natsFrom(V1)) -> isNaturalKind(activate(V1)) 7.95/2.80 isLNatKind(n__snd(V1)) -> isPLNatKind(activate(V1)) 7.95/2.80 isLNatKind(n__tail(V1)) -> isLNatKind(activate(V1)) 7.95/2.80 isLNatKind(n__take(V1, V2)) -> and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2))) 7.95/2.80 isNatural(n__0) -> tt 7.95/2.80 isNatural(n__head(V1)) -> U111(isLNatKind(activate(V1)), activate(V1)) 7.95/2.80 isNatural(n__s(V1)) -> U121(isNaturalKind(activate(V1)), activate(V1)) 7.95/2.80 isNatural(n__sel(V1, V2)) -> U131(and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2))), activate(V1), activate(V2)) 7.95/2.80 isNaturalKind(n__0) -> tt 7.95/2.80 isNaturalKind(n__head(V1)) -> isLNatKind(activate(V1)) 7.95/2.80 isNaturalKind(n__s(V1)) -> isNaturalKind(activate(V1)) 7.95/2.80 isNaturalKind(n__sel(V1, V2)) -> and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2))) 7.95/2.80 isPLNat(n__pair(V1, V2)) -> U141(and(isLNatKind(activate(V1)), n__isLNatKind(activate(V2))), activate(V1), activate(V2)) 7.95/2.80 isPLNat(n__splitAt(V1, V2)) -> U151(and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2))), activate(V1), activate(V2)) 7.95/2.80 isPLNatKind(n__pair(V1, V2)) -> and(isLNatKind(activate(V1)), n__isLNatKind(activate(V2))) 7.95/2.80 isPLNatKind(n__splitAt(V1, V2)) -> and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2))) 7.95/2.80 natsFrom(N) -> U161(and(isNatural(N), n__isNaturalKind(N)), N) 7.95/2.80 sel(N, XS) -> U171(and(and(isNatural(N), n__isNaturalKind(N)), n__and(n__isLNat(XS), n__isLNatKind(XS))), N, XS) 7.95/2.80 snd(pair(X, Y)) -> U181(and(and(isLNat(X), n__isLNatKind(X)), n__and(n__isLNat(Y), n__isLNatKind(Y))), Y) 7.95/2.80 splitAt(0, XS) -> U191(and(isLNat(XS), n__isLNatKind(XS)), XS) 7.95/2.80 splitAt(s(N), cons(X, XS)) -> U201(and(and(isNatural(N), n__isNaturalKind(N)), n__and(n__and(n__isNatural(X), n__isNaturalKind(X)), n__and(n__isLNat(activate(XS)), n__isLNatKind(activate(XS))))), N, X, activate(XS)) 7.95/2.80 tail(cons(N, XS)) -> U211(and(and(isNatural(N), n__isNaturalKind(N)), n__and(n__isLNat(activate(XS)), n__isLNatKind(activate(XS)))), activate(XS)) 7.95/2.80 take(N, XS) -> U221(and(and(isNatural(N), n__isNaturalKind(N)), n__and(n__isLNat(XS), n__isLNatKind(XS))), N, XS) 7.95/2.80 natsFrom(X) -> n__natsFrom(X) 7.95/2.80 s(X) -> n__s(X) 7.95/2.80 isNaturalKind(X) -> n__isNaturalKind(X) 7.95/2.80 and(X1, X2) -> n__and(X1, X2) 7.95/2.80 isLNat(X) -> n__isLNat(X) 7.95/2.80 isLNatKind(X) -> n__isLNatKind(X) 7.95/2.80 nil -> n__nil 7.95/2.80 afterNth(X1, X2) -> n__afterNth(X1, X2) 7.95/2.80 cons(X1, X2) -> n__cons(X1, X2) 7.95/2.80 fst(X) -> n__fst(X) 7.95/2.80 snd(X) -> n__snd(X) 7.95/2.80 tail(X) -> n__tail(X) 7.95/2.80 take(X1, X2) -> n__take(X1, X2) 7.95/2.80 0 -> n__0 7.95/2.80 head(X) -> n__head(X) 7.95/2.80 sel(X1, X2) -> n__sel(X1, X2) 7.95/2.80 pair(X1, X2) -> n__pair(X1, X2) 7.95/2.80 splitAt(X1, X2) -> n__splitAt(X1, X2) 7.95/2.80 isNatural(X) -> n__isNatural(X) 7.95/2.80 activate(n__natsFrom(X)) -> natsFrom(activate(X)) 7.95/2.80 activate(n__s(X)) -> s(activate(X)) 7.95/2.80 activate(n__isNaturalKind(X)) -> isNaturalKind(X) 7.95/2.80 activate(n__and(X1, X2)) -> and(activate(X1), X2) 7.95/2.80 activate(n__isLNat(X)) -> isLNat(X) 7.95/2.80 activate(n__isLNatKind(X)) -> isLNatKind(X) 7.95/2.80 activate(n__nil) -> nil 7.95/2.80 activate(n__afterNth(X1, X2)) -> afterNth(activate(X1), activate(X2)) 7.95/2.80 activate(n__cons(X1, X2)) -> cons(activate(X1), X2) 7.95/2.80 activate(n__fst(X)) -> fst(activate(X)) 7.95/2.80 activate(n__snd(X)) -> snd(activate(X)) 7.95/2.80 activate(n__tail(X)) -> tail(activate(X)) 7.95/2.80 activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) 7.95/2.80 activate(n__0) -> 0 7.95/2.80 activate(n__head(X)) -> head(activate(X)) 7.95/2.80 activate(n__sel(X1, X2)) -> sel(activate(X1), activate(X2)) 7.95/2.80 activate(n__pair(X1, X2)) -> pair(activate(X1), activate(X2)) 7.95/2.80 activate(n__splitAt(X1, X2)) -> splitAt(activate(X1), activate(X2)) 7.95/2.80 activate(n__isNatural(X)) -> isNatural(X) 7.95/2.80 activate(X) -> X 7.95/2.80 7.95/2.80 S is empty. 7.95/2.80 Rewrite Strategy: FULL 7.95/2.80 ---------------------------------------- 7.95/2.80 7.95/2.80 (6) LowerBoundPropagationProof (FINISHED) 7.95/2.80 Propagated lower bound. 7.95/2.80 ---------------------------------------- 7.95/2.80 7.95/2.80 (7) 7.95/2.80 BOUNDS(n^1, INF) 7.95/2.80 7.95/2.80 ---------------------------------------- 7.95/2.80 7.95/2.80 (8) 7.95/2.80 Obligation: 7.95/2.80 Analyzing the following TRS for decreasing loops: 7.95/2.80 7.95/2.80 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). 7.95/2.80 7.95/2.80 7.95/2.80 The TRS R consists of the following rules: 7.95/2.80 7.95/2.80 U101(tt, V1, V2) -> U102(isNatural(activate(V1)), activate(V2)) 7.95/2.80 U102(tt, V2) -> U103(isLNat(activate(V2))) 7.95/2.80 U103(tt) -> tt 7.95/2.80 U11(tt, N, XS) -> snd(splitAt(activate(N), activate(XS))) 7.95/2.80 U111(tt, V1) -> U112(isLNat(activate(V1))) 7.95/2.80 U112(tt) -> tt 7.95/2.80 U121(tt, V1) -> U122(isNatural(activate(V1))) 7.95/2.80 U122(tt) -> tt 7.95/2.80 U131(tt, V1, V2) -> U132(isNatural(activate(V1)), activate(V2)) 7.95/2.80 U132(tt, V2) -> U133(isLNat(activate(V2))) 7.95/2.80 U133(tt) -> tt 7.95/2.80 U141(tt, V1, V2) -> U142(isLNat(activate(V1)), activate(V2)) 7.95/2.80 U142(tt, V2) -> U143(isLNat(activate(V2))) 7.95/2.80 U143(tt) -> tt 7.95/2.80 U151(tt, V1, V2) -> U152(isNatural(activate(V1)), activate(V2)) 7.95/2.80 U152(tt, V2) -> U153(isLNat(activate(V2))) 7.95/2.80 U153(tt) -> tt 7.95/2.80 U161(tt, N) -> cons(activate(N), n__natsFrom(n__s(activate(N)))) 7.95/2.80 U171(tt, N, XS) -> head(afterNth(activate(N), activate(XS))) 7.95/2.80 U181(tt, Y) -> activate(Y) 7.95/2.80 U191(tt, XS) -> pair(nil, activate(XS)) 7.95/2.80 U201(tt, N, X, XS) -> U202(splitAt(activate(N), activate(XS)), activate(X)) 7.95/2.80 U202(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS) 7.95/2.80 U21(tt, X) -> activate(X) 7.95/2.80 U211(tt, XS) -> activate(XS) 7.95/2.80 U221(tt, N, XS) -> fst(splitAt(activate(N), activate(XS))) 7.95/2.80 U31(tt, N) -> activate(N) 7.95/2.80 U41(tt, V1, V2) -> U42(isNatural(activate(V1)), activate(V2)) 7.95/2.80 U42(tt, V2) -> U43(isLNat(activate(V2))) 7.95/2.80 U43(tt) -> tt 7.95/2.80 U51(tt, V1, V2) -> U52(isNatural(activate(V1)), activate(V2)) 7.95/2.80 U52(tt, V2) -> U53(isLNat(activate(V2))) 7.95/2.80 U53(tt) -> tt 7.95/2.80 U61(tt, V1) -> U62(isPLNat(activate(V1))) 7.95/2.80 U62(tt) -> tt 7.95/2.80 U71(tt, V1) -> U72(isNatural(activate(V1))) 7.95/2.80 U72(tt) -> tt 7.95/2.80 U81(tt, V1) -> U82(isPLNat(activate(V1))) 7.95/2.80 U82(tt) -> tt 7.95/2.80 U91(tt, V1) -> U92(isLNat(activate(V1))) 7.95/2.80 U92(tt) -> tt 7.95/2.80 afterNth(N, XS) -> U11(and(and(isNatural(N), n__isNaturalKind(N)), n__and(n__isLNat(XS), n__isLNatKind(XS))), N, XS) 7.95/2.80 and(tt, X) -> activate(X) 7.95/2.80 fst(pair(X, Y)) -> U21(and(and(isLNat(X), n__isLNatKind(X)), n__and(n__isLNat(Y), n__isLNatKind(Y))), X) 7.95/2.80 head(cons(N, XS)) -> U31(and(and(isNatural(N), n__isNaturalKind(N)), n__and(n__isLNat(activate(XS)), n__isLNatKind(activate(XS)))), N) 7.95/2.80 isLNat(n__nil) -> tt 7.95/2.80 isLNat(n__afterNth(V1, V2)) -> U41(and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2))), activate(V1), activate(V2)) 7.95/2.80 isLNat(n__cons(V1, V2)) -> U51(and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2))), activate(V1), activate(V2)) 7.95/2.80 isLNat(n__fst(V1)) -> U61(isPLNatKind(activate(V1)), activate(V1)) 7.95/2.80 isLNat(n__natsFrom(V1)) -> U71(isNaturalKind(activate(V1)), activate(V1)) 7.95/2.80 isLNat(n__snd(V1)) -> U81(isPLNatKind(activate(V1)), activate(V1)) 7.95/2.80 isLNat(n__tail(V1)) -> U91(isLNatKind(activate(V1)), activate(V1)) 7.95/2.80 isLNat(n__take(V1, V2)) -> U101(and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2))), activate(V1), activate(V2)) 7.95/2.80 isLNatKind(n__nil) -> tt 7.95/2.80 isLNatKind(n__afterNth(V1, V2)) -> and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2))) 7.95/2.80 isLNatKind(n__cons(V1, V2)) -> and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2))) 7.95/2.80 isLNatKind(n__fst(V1)) -> isPLNatKind(activate(V1)) 7.95/2.80 isLNatKind(n__natsFrom(V1)) -> isNaturalKind(activate(V1)) 7.95/2.80 isLNatKind(n__snd(V1)) -> isPLNatKind(activate(V1)) 7.95/2.80 isLNatKind(n__tail(V1)) -> isLNatKind(activate(V1)) 7.95/2.80 isLNatKind(n__take(V1, V2)) -> and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2))) 7.95/2.80 isNatural(n__0) -> tt 7.95/2.80 isNatural(n__head(V1)) -> U111(isLNatKind(activate(V1)), activate(V1)) 7.95/2.80 isNatural(n__s(V1)) -> U121(isNaturalKind(activate(V1)), activate(V1)) 7.95/2.80 isNatural(n__sel(V1, V2)) -> U131(and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2))), activate(V1), activate(V2)) 7.95/2.80 isNaturalKind(n__0) -> tt 7.95/2.80 isNaturalKind(n__head(V1)) -> isLNatKind(activate(V1)) 7.95/2.80 isNaturalKind(n__s(V1)) -> isNaturalKind(activate(V1)) 7.95/2.80 isNaturalKind(n__sel(V1, V2)) -> and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2))) 7.95/2.80 isPLNat(n__pair(V1, V2)) -> U141(and(isLNatKind(activate(V1)), n__isLNatKind(activate(V2))), activate(V1), activate(V2)) 7.95/2.80 isPLNat(n__splitAt(V1, V2)) -> U151(and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2))), activate(V1), activate(V2)) 7.95/2.80 isPLNatKind(n__pair(V1, V2)) -> and(isLNatKind(activate(V1)), n__isLNatKind(activate(V2))) 7.95/2.80 isPLNatKind(n__splitAt(V1, V2)) -> and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2))) 7.95/2.80 natsFrom(N) -> U161(and(isNatural(N), n__isNaturalKind(N)), N) 7.95/2.80 sel(N, XS) -> U171(and(and(isNatural(N), n__isNaturalKind(N)), n__and(n__isLNat(XS), n__isLNatKind(XS))), N, XS) 7.95/2.80 snd(pair(X, Y)) -> U181(and(and(isLNat(X), n__isLNatKind(X)), n__and(n__isLNat(Y), n__isLNatKind(Y))), Y) 7.95/2.80 splitAt(0, XS) -> U191(and(isLNat(XS), n__isLNatKind(XS)), XS) 7.95/2.80 splitAt(s(N), cons(X, XS)) -> U201(and(and(isNatural(N), n__isNaturalKind(N)), n__and(n__and(n__isNatural(X), n__isNaturalKind(X)), n__and(n__isLNat(activate(XS)), n__isLNatKind(activate(XS))))), N, X, activate(XS)) 7.95/2.80 tail(cons(N, XS)) -> U211(and(and(isNatural(N), n__isNaturalKind(N)), n__and(n__isLNat(activate(XS)), n__isLNatKind(activate(XS)))), activate(XS)) 7.95/2.80 take(N, XS) -> U221(and(and(isNatural(N), n__isNaturalKind(N)), n__and(n__isLNat(XS), n__isLNatKind(XS))), N, XS) 7.95/2.80 natsFrom(X) -> n__natsFrom(X) 7.95/2.80 s(X) -> n__s(X) 7.95/2.80 isNaturalKind(X) -> n__isNaturalKind(X) 7.95/2.80 and(X1, X2) -> n__and(X1, X2) 7.95/2.80 isLNat(X) -> n__isLNat(X) 7.95/2.80 isLNatKind(X) -> n__isLNatKind(X) 7.95/2.80 nil -> n__nil 7.95/2.80 afterNth(X1, X2) -> n__afterNth(X1, X2) 7.95/2.80 cons(X1, X2) -> n__cons(X1, X2) 7.95/2.80 fst(X) -> n__fst(X) 7.95/2.80 snd(X) -> n__snd(X) 7.95/2.80 tail(X) -> n__tail(X) 7.95/2.80 take(X1, X2) -> n__take(X1, X2) 7.95/2.80 0 -> n__0 7.95/2.80 head(X) -> n__head(X) 7.95/2.80 sel(X1, X2) -> n__sel(X1, X2) 7.95/2.80 pair(X1, X2) -> n__pair(X1, X2) 7.95/2.80 splitAt(X1, X2) -> n__splitAt(X1, X2) 7.95/2.80 isNatural(X) -> n__isNatural(X) 7.95/2.80 activate(n__natsFrom(X)) -> natsFrom(activate(X)) 7.95/2.80 activate(n__s(X)) -> s(activate(X)) 7.95/2.80 activate(n__isNaturalKind(X)) -> isNaturalKind(X) 7.95/2.80 activate(n__and(X1, X2)) -> and(activate(X1), X2) 7.95/2.80 activate(n__isLNat(X)) -> isLNat(X) 7.95/2.80 activate(n__isLNatKind(X)) -> isLNatKind(X) 7.95/2.80 activate(n__nil) -> nil 7.95/2.80 activate(n__afterNth(X1, X2)) -> afterNth(activate(X1), activate(X2)) 7.95/2.80 activate(n__cons(X1, X2)) -> cons(activate(X1), X2) 7.95/2.80 activate(n__fst(X)) -> fst(activate(X)) 7.95/2.80 activate(n__snd(X)) -> snd(activate(X)) 7.95/2.80 activate(n__tail(X)) -> tail(activate(X)) 7.95/2.80 activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) 7.95/2.80 activate(n__0) -> 0 7.95/2.80 activate(n__head(X)) -> head(activate(X)) 7.95/2.80 activate(n__sel(X1, X2)) -> sel(activate(X1), activate(X2)) 7.95/2.80 activate(n__pair(X1, X2)) -> pair(activate(X1), activate(X2)) 7.95/2.80 activate(n__splitAt(X1, X2)) -> splitAt(activate(X1), activate(X2)) 7.95/2.80 activate(n__isNatural(X)) -> isNatural(X) 7.95/2.80 activate(X) -> X 7.95/2.80 7.95/2.80 S is empty. 7.95/2.80 Rewrite Strategy: FULL 7.95/2.80 ---------------------------------------- 7.95/2.80 7.95/2.80 (9) DecreasingLoopProof (FINISHED) 7.95/2.80 The following loop(s) give(s) rise to the lower bound EXP: 7.95/2.80 7.95/2.80 The rewrite sequence 7.95/2.80 7.95/2.80 activate(n__isNatural(n__head(V11_0))) ->^+ U111(isLNatKind(activate(V11_0)), activate(V11_0)) 7.95/2.80 7.95/2.80 gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0]. 7.95/2.80 7.95/2.80 The pumping substitution is [V11_0 / n__isNatural(n__head(V11_0))]. 7.95/2.80 7.95/2.80 The result substitution is [ ]. 7.95/2.80 7.95/2.80 7.95/2.80 7.95/2.80 The rewrite sequence 7.95/2.80 7.95/2.80 activate(n__isNatural(n__head(V11_0))) ->^+ U111(isLNatKind(activate(V11_0)), activate(V11_0)) 7.95/2.80 7.95/2.80 gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. 7.95/2.80 7.95/2.80 The pumping substitution is [V11_0 / n__isNatural(n__head(V11_0))]. 7.95/2.80 7.95/2.80 The result substitution is [ ]. 7.95/2.80 7.95/2.80 7.95/2.80 7.95/2.80 7.95/2.80 ---------------------------------------- 7.95/2.80 7.95/2.80 (10) 7.95/2.80 BOUNDS(EXP, INF) 8.23/2.89 EOF