KILLED
proof of /export/starexec/sandbox/benchmark/theBenchmark.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).

(0) CpxRelTRS


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(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   A__U101(tt, z0, z1) -> c(A__FST(a__splitAt(mark(z0), mark(z1))), A__SPLITAT(mark(z0), mark(z1)), MARK(z0))
   A__U101(tt, z0, z1) -> c1(A__FST(a__splitAt(mark(z0), mark(z1))), A__SPLITAT(mark(z0), mark(z1)), MARK(z1))
   A__U101(z0, z1, z2) -> c2
   A__U11(tt, z0, z1) -> c3(A__SND(a__splitAt(mark(z0), mark(z1))), A__SPLITAT(mark(z0), mark(z1)), MARK(z0))
   A__U11(tt, z0, z1) -> c4(A__SND(a__splitAt(mark(z0), mark(z1))), A__SPLITAT(mark(z0), mark(z1)), MARK(z1))
   A__U11(z0, z1, z2) -> c5
   A__U21(tt, z0) -> c6(MARK(z0))
   A__U21(z0, z1) -> c7
   A__U31(tt, z0) -> c8(MARK(z0))
   A__U31(z0, z1) -> c9
   A__U41(tt, z0) -> c10(MARK(z0))
   A__U41(z0, z1) -> c11
   A__U51(tt, z0, z1) -> c12(A__HEAD(a__afterNth(mark(z0), mark(z1))), A__AFTERNTH(mark(z0), mark(z1)), MARK(z0))
   A__U51(tt, z0, z1) -> c13(A__HEAD(a__afterNth(mark(z0), mark(z1))), A__AFTERNTH(mark(z0), mark(z1)), MARK(z1))
   A__U51(z0, z1, z2) -> c14
   A__U61(tt, z0) -> c15(MARK(z0))
   A__U61(z0, z1) -> c16
   A__U71(tt, z0) -> c17(MARK(z0))
   A__U71(z0, z1) -> c18
   A__U81(tt, z0, z1, z2) -> c19(A__U82(a__splitAt(mark(z0), mark(z2)), z1), A__SPLITAT(mark(z0), mark(z2)), MARK(z0))
   A__U81(tt, z0, z1, z2) -> c20(A__U82(a__splitAt(mark(z0), mark(z2)), z1), A__SPLITAT(mark(z0), mark(z2)), MARK(z2))
   A__U81(z0, z1, z2, z3) -> c21
   A__U82(pair(z0, z1), z2) -> c22(MARK(z2))
   A__U82(pair(z0, z1), z2) -> c23(MARK(z1))
   A__U82(z0, z1) -> c24
   A__U91(tt, z0) -> c25(MARK(z0))
   A__U91(z0, z1) -> c26
   A__AFTERNTH(z0, z1) -> c27(A__U11(a__and(a__isNatural(z0), isLNat(z1)), z0, z1), A__AND(a__isNatural(z0), isLNat(z1)), A__ISNATURAL(z0))
   A__AFTERNTH(z0, z1) -> c28
   A__AND(tt, z0) -> c29(MARK(z0))
   A__AND(z0, z1) -> c30
   A__FST(pair(z0, z1)) -> c31(A__U21(a__and(a__isLNat(z0), isLNat(z1)), z0), A__AND(a__isLNat(z0), isLNat(z1)), A__ISLNAT(z0))
   A__FST(z0) -> c32
   A__HEAD(cons(z0, z1)) -> c33(A__U31(a__and(a__isNatural(z0), isLNat(z1)), z0), A__AND(a__isNatural(z0), isLNat(z1)), A__ISNATURAL(z0))
   A__HEAD(z0) -> c34
   A__ISLNAT(nil) -> c35
   A__ISLNAT(afterNth(z0, z1)) -> c36(A__AND(a__isNatural(z0), isLNat(z1)), A__ISNATURAL(z0))
   A__ISLNAT(cons(z0, z1)) -> c37(A__AND(a__isNatural(z0), isLNat(z1)), A__ISNATURAL(z0))
   A__ISLNAT(fst(z0)) -> c38(A__ISPLNAT(z0))
   A__ISLNAT(natsFrom(z0)) -> c39(A__ISNATURAL(z0))
   A__ISLNAT(snd(z0)) -> c40(A__ISPLNAT(z0))
   A__ISLNAT(tail(z0)) -> c41(A__ISLNAT(z0))
   A__ISLNAT(take(z0, z1)) -> c42(A__AND(a__isNatural(z0), isLNat(z1)), A__ISNATURAL(z0))
   A__ISLNAT(z0) -> c43
   A__ISNATURAL(0) -> c44
   A__ISNATURAL(head(z0)) -> c45(A__ISLNAT(z0))
   A__ISNATURAL(s(z0)) -> c46(A__ISNATURAL(z0))
   A__ISNATURAL(sel(z0, z1)) -> c47(A__AND(a__isNatural(z0), isLNat(z1)), A__ISNATURAL(z0))
   A__ISNATURAL(z0) -> c48
   A__ISPLNAT(pair(z0, z1)) -> c49(A__AND(a__isLNat(z0), isLNat(z1)), A__ISLNAT(z0))
   A__ISPLNAT(splitAt(z0, z1)) -> c50(A__AND(a__isNatural(z0), isLNat(z1)), A__ISNATURAL(z0))
   A__ISPLNAT(z0) -> c51
   A__NATSFROM(z0) -> c52(A__U41(a__isNatural(z0), z0), A__ISNATURAL(z0))
   A__NATSFROM(z0) -> c53
   A__SEL(z0, z1) -> c54(A__U51(a__and(a__isNatural(z0), isLNat(z1)), z0, z1), A__AND(a__isNatural(z0), isLNat(z1)), A__ISNATURAL(z0))
   A__SEL(z0, z1) -> c55
   A__SND(pair(z0, z1)) -> c56(A__U61(a__and(a__isLNat(z0), isLNat(z1)), z1), A__AND(a__isLNat(z0), isLNat(z1)), A__ISLNAT(z0))
   A__SND(z0) -> c57
   A__SPLITAT(0, z0) -> c58(A__U71(a__isLNat(z0), z0), A__ISLNAT(z0))
   A__SPLITAT(s(z0), cons(z1, z2)) -> c59(A__U81(a__and(a__isNatural(z0), and(isNatural(z1), isLNat(z2))), z0, z1, z2), A__AND(a__isNatural(z0), and(isNatural(z1), isLNat(z2))), A__ISNATURAL(z0))
   A__SPLITAT(z0, z1) -> c60
   A__TAIL(cons(z0, z1)) -> c61(A__U91(a__and(a__isNatural(z0), isLNat(z1)), z1), A__AND(a__isNatural(z0), isLNat(z1)), A__ISNATURAL(z0))
   A__TAIL(z0) -> c62
   A__TAKE(z0, z1) -> c63(A__U101(a__and(a__isNatural(z0), isLNat(z1)), z0, z1), A__AND(a__isNatural(z0), isLNat(z1)), A__ISNATURAL(z0))
   A__TAKE(z0, z1) -> c64
   MARK(U101(z0, z1, z2)) -> c65(A__U101(mark(z0), z1, z2), MARK(z0))
   MARK(fst(z0)) -> c66(A__FST(mark(z0)), MARK(z0))
   MARK(splitAt(z0, z1)) -> c67(A__SPLITAT(mark(z0), mark(z1)), MARK(z0))
   MARK(splitAt(z0, z1)) -> c68(A__SPLITAT(mark(z0), mark(z1)), MARK(z1))
   MARK(U11(z0, z1, z2)) -> c69(A__U11(mark(z0), z1, z2), MARK(z0))
   MARK(snd(z0)) -> c70(A__SND(mark(z0)), MARK(z0))
   MARK(U21(z0, z1)) -> c71(A__U21(mark(z0), z1), MARK(z0))
   MARK(U31(z0, z1)) -> c72(A__U31(mark(z0), z1), MARK(z0))
   MARK(U41(z0, z1)) -> c73(A__U41(mark(z0), z1), MARK(z0))
   MARK(natsFrom(z0)) -> c74(A__NATSFROM(mark(z0)), MARK(z0))
   MARK(U51(z0, z1, z2)) -> c75(A__U51(mark(z0), z1, z2), MARK(z0))
   MARK(head(z0)) -> c76(A__HEAD(mark(z0)), MARK(z0))
   MARK(afterNth(z0, z1)) -> c77(A__AFTERNTH(mark(z0), mark(z1)), MARK(z0))
   MARK(afterNth(z0, z1)) -> c78(A__AFTERNTH(mark(z0), mark(z1)), MARK(z1))
   MARK(U61(z0, z1)) -> c79(A__U61(mark(z0), z1), MARK(z0))
   MARK(U71(z0, z1)) -> c80(A__U71(mark(z0), z1), MARK(z0))
   MARK(U81(z0, z1, z2, z3)) -> c81(A__U81(mark(z0), z1, z2, z3), MARK(z0))
   MARK(U82(z0, z1)) -> c82(A__U82(mark(z0), z1), MARK(z0))
   MARK(U91(z0, z1)) -> c83(A__U91(mark(z0), z1), MARK(z0))
   MARK(and(z0, z1)) -> c84(A__AND(mark(z0), z1), MARK(z0))
   MARK(isNatural(z0)) -> c85(A__ISNATURAL(z0))
   MARK(isLNat(z0)) -> c86(A__ISLNAT(z0))
   MARK(isPLNat(z0)) -> c87(A__ISPLNAT(z0))
   MARK(tail(z0)) -> c88(A__TAIL(mark(z0)), MARK(z0))
   MARK(take(z0, z1)) -> c89(A__TAKE(mark(z0), mark(z1)), MARK(z0))
   MARK(take(z0, z1)) -> c90(A__TAKE(mark(z0), mark(z1)), MARK(z1))
   MARK(sel(z0, z1)) -> c91(A__SEL(mark(z0), mark(z1)), MARK(z0))
   MARK(sel(z0, z1)) -> c92(A__SEL(mark(z0), mark(z1)), MARK(z1))
   MARK(tt) -> c93
   MARK(cons(z0, z1)) -> c94(MARK(z0))
   MARK(s(z0)) -> c95(MARK(z0))
   MARK(pair(z0, z1)) -> c96(MARK(z0))
   MARK(pair(z0, z1)) -> c97(MARK(z1))
   MARK(nil) -> c98
   MARK(0) -> c99

The (relative) TRS S consists of the following rules:

   a__U101(tt, z0, z1) -> a__fst(a__splitAt(mark(z0), mark(z1)))
   a__U101(z0, z1, z2) -> U101(z0, z1, z2)
   a__U11(tt, z0, z1) -> a__snd(a__splitAt(mark(z0), mark(z1)))
   a__U11(z0, z1, z2) -> U11(z0, z1, z2)
   a__U21(tt, z0) -> mark(z0)
   a__U21(z0, z1) -> U21(z0, z1)
   a__U31(tt, z0) -> mark(z0)
   a__U31(z0, z1) -> U31(z0, z1)
   a__U41(tt, z0) -> cons(mark(z0), natsFrom(s(z0)))
   a__U41(z0, z1) -> U41(z0, z1)
   a__U51(tt, z0, z1) -> a__head(a__afterNth(mark(z0), mark(z1)))
   a__U51(z0, z1, z2) -> U51(z0, z1, z2)
   a__U61(tt, z0) -> mark(z0)
   a__U61(z0, z1) -> U61(z0, z1)
   a__U71(tt, z0) -> pair(nil, mark(z0))
   a__U71(z0, z1) -> U71(z0, z1)
   a__U81(tt, z0, z1, z2) -> a__U82(a__splitAt(mark(z0), mark(z2)), z1)
   a__U81(z0, z1, z2, z3) -> U81(z0, z1, z2, z3)
   a__U82(pair(z0, z1), z2) -> pair(cons(mark(z2), z0), mark(z1))
   a__U82(z0, z1) -> U82(z0, z1)
   a__U91(tt, z0) -> mark(z0)
   a__U91(z0, z1) -> U91(z0, z1)
   a__afterNth(z0, z1) -> a__U11(a__and(a__isNatural(z0), isLNat(z1)), z0, z1)
   a__afterNth(z0, z1) -> afterNth(z0, z1)
   a__and(tt, z0) -> mark(z0)
   a__and(z0, z1) -> and(z0, z1)
   a__fst(pair(z0, z1)) -> a__U21(a__and(a__isLNat(z0), isLNat(z1)), z0)
   a__fst(z0) -> fst(z0)
   a__head(cons(z0, z1)) -> a__U31(a__and(a__isNatural(z0), isLNat(z1)), z0)
   a__head(z0) -> head(z0)
   a__isLNat(nil) -> tt
   a__isLNat(afterNth(z0, z1)) -> a__and(a__isNatural(z0), isLNat(z1))
   a__isLNat(cons(z0, z1)) -> a__and(a__isNatural(z0), isLNat(z1))
   a__isLNat(fst(z0)) -> a__isPLNat(z0)
   a__isLNat(natsFrom(z0)) -> a__isNatural(z0)
   a__isLNat(snd(z0)) -> a__isPLNat(z0)
   a__isLNat(tail(z0)) -> a__isLNat(z0)
   a__isLNat(take(z0, z1)) -> a__and(a__isNatural(z0), isLNat(z1))
   a__isLNat(z0) -> isLNat(z0)
   a__isNatural(0) -> tt
   a__isNatural(head(z0)) -> a__isLNat(z0)
   a__isNatural(s(z0)) -> a__isNatural(z0)
   a__isNatural(sel(z0, z1)) -> a__and(a__isNatural(z0), isLNat(z1))
   a__isNatural(z0) -> isNatural(z0)
   a__isPLNat(pair(z0, z1)) -> a__and(a__isLNat(z0), isLNat(z1))
   a__isPLNat(splitAt(z0, z1)) -> a__and(a__isNatural(z0), isLNat(z1))
   a__isPLNat(z0) -> isPLNat(z0)
   a__natsFrom(z0) -> a__U41(a__isNatural(z0), z0)
   a__natsFrom(z0) -> natsFrom(z0)
   a__sel(z0, z1) -> a__U51(a__and(a__isNatural(z0), isLNat(z1)), z0, z1)
   a__sel(z0, z1) -> sel(z0, z1)
   a__snd(pair(z0, z1)) -> a__U61(a__and(a__isLNat(z0), isLNat(z1)), z1)
   a__snd(z0) -> snd(z0)
   a__splitAt(0, z0) -> a__U71(a__isLNat(z0), z0)
   a__splitAt(s(z0), cons(z1, z2)) -> a__U81(a__and(a__isNatural(z0), and(isNatural(z1), isLNat(z2))), z0, z1, z2)
   a__splitAt(z0, z1) -> splitAt(z0, z1)
   a__tail(cons(z0, z1)) -> a__U91(a__and(a__isNatural(z0), isLNat(z1)), z1)
   a__tail(z0) -> tail(z0)
   a__take(z0, z1) -> a__U101(a__and(a__isNatural(z0), isLNat(z1)), z0, z1)
   a__take(z0, z1) -> take(z0, z1)
   mark(U101(z0, z1, z2)) -> a__U101(mark(z0), z1, z2)
   mark(fst(z0)) -> a__fst(mark(z0))
   mark(splitAt(z0, z1)) -> a__splitAt(mark(z0), mark(z1))
   mark(U11(z0, z1, z2)) -> a__U11(mark(z0), z1, z2)
   mark(snd(z0)) -> a__snd(mark(z0))
   mark(U21(z0, z1)) -> a__U21(mark(z0), z1)
   mark(U31(z0, z1)) -> a__U31(mark(z0), z1)
   mark(U41(z0, z1)) -> a__U41(mark(z0), z1)
   mark(natsFrom(z0)) -> a__natsFrom(mark(z0))
   mark(U51(z0, z1, z2)) -> a__U51(mark(z0), z1, z2)
   mark(head(z0)) -> a__head(mark(z0))
   mark(afterNth(z0, z1)) -> a__afterNth(mark(z0), mark(z1))
   mark(U61(z0, z1)) -> a__U61(mark(z0), z1)
   mark(U71(z0, z1)) -> a__U71(mark(z0), z1)
   mark(U81(z0, z1, z2, z3)) -> a__U81(mark(z0), z1, z2, z3)
   mark(U82(z0, z1)) -> a__U82(mark(z0), z1)
   mark(U91(z0, z1)) -> a__U91(mark(z0), z1)
   mark(and(z0, z1)) -> a__and(mark(z0), z1)
   mark(isNatural(z0)) -> a__isNatural(z0)
   mark(isLNat(z0)) -> a__isLNat(z0)
   mark(isPLNat(z0)) -> a__isPLNat(z0)
   mark(tail(z0)) -> a__tail(mark(z0))
   mark(take(z0, z1)) -> a__take(mark(z0), mark(z1))
   mark(sel(z0, z1)) -> a__sel(mark(z0), mark(z1))
   mark(tt) -> tt
   mark(cons(z0, z1)) -> cons(mark(z0), z1)
   mark(s(z0)) -> s(mark(z0))
   mark(pair(z0, z1)) -> pair(mark(z0), mark(z1))
   mark(nil) -> nil
   mark(0) -> 0

Rewrite Strategy: INNERMOST