KILLED
proof of /export/starexec/sandbox2/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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__NATSFROM(z0) -> c(MARK(z0))
   A__NATSFROM(z0) -> c1
   A__FST(pair(z0, z1)) -> c2(MARK(z0))
   A__FST(z0) -> c3
   A__SND(pair(z0, z1)) -> c4(MARK(z1))
   A__SND(z0) -> c5
   A__SPLITAT(0, z0) -> c6(MARK(z0))
   A__SPLITAT(s(z0), cons(z1, z2)) -> c7(A__U(a__splitAt(mark(z0), mark(z2)), z0, z1, z2), A__SPLITAT(mark(z0), mark(z2)), MARK(z0))
   A__SPLITAT(s(z0), cons(z1, z2)) -> c8(A__U(a__splitAt(mark(z0), mark(z2)), z0, z1, z2), A__SPLITAT(mark(z0), mark(z2)), MARK(z2))
   A__SPLITAT(z0, z1) -> c9
   A__U(pair(z0, z1), z2, z3, z4) -> c10(MARK(z3))
   A__U(pair(z0, z1), z2, z3, z4) -> c11(MARK(z1))
   A__U(z0, z1, z2, z3) -> c12
   A__HEAD(cons(z0, z1)) -> c13(MARK(z0))
   A__HEAD(z0) -> c14
   A__TAIL(cons(z0, z1)) -> c15(MARK(z1))
   A__TAIL(z0) -> c16
   A__SEL(z0, z1) -> c17(A__HEAD(a__afterNth(mark(z0), mark(z1))), A__AFTERNTH(mark(z0), mark(z1)), MARK(z0))
   A__SEL(z0, z1) -> c18(A__HEAD(a__afterNth(mark(z0), mark(z1))), A__AFTERNTH(mark(z0), mark(z1)), MARK(z1))
   A__SEL(z0, z1) -> c19
   A__TAKE(z0, z1) -> c20(A__FST(a__splitAt(mark(z0), mark(z1))), A__SPLITAT(mark(z0), mark(z1)), MARK(z0))
   A__TAKE(z0, z1) -> c21(A__FST(a__splitAt(mark(z0), mark(z1))), A__SPLITAT(mark(z0), mark(z1)), MARK(z1))
   A__TAKE(z0, z1) -> c22
   A__AFTERNTH(z0, z1) -> c23(A__SND(a__splitAt(mark(z0), mark(z1))), A__SPLITAT(mark(z0), mark(z1)), MARK(z0))
   A__AFTERNTH(z0, z1) -> c24(A__SND(a__splitAt(mark(z0), mark(z1))), A__SPLITAT(mark(z0), mark(z1)), MARK(z1))
   A__AFTERNTH(z0, z1) -> c25
   MARK(natsFrom(z0)) -> c26(A__NATSFROM(mark(z0)), MARK(z0))
   MARK(fst(z0)) -> c27(A__FST(mark(z0)), MARK(z0))
   MARK(snd(z0)) -> c28(A__SND(mark(z0)), MARK(z0))
   MARK(splitAt(z0, z1)) -> c29(A__SPLITAT(mark(z0), mark(z1)), MARK(z0))
   MARK(splitAt(z0, z1)) -> c30(A__SPLITAT(mark(z0), mark(z1)), MARK(z1))
   MARK(u(z0, z1, z2, z3)) -> c31(A__U(mark(z0), z1, z2, z3), MARK(z0))
   MARK(head(z0)) -> c32(A__HEAD(mark(z0)), MARK(z0))
   MARK(tail(z0)) -> c33(A__TAIL(mark(z0)), MARK(z0))
   MARK(sel(z0, z1)) -> c34(A__SEL(mark(z0), mark(z1)), MARK(z0))
   MARK(sel(z0, z1)) -> c35(A__SEL(mark(z0), mark(z1)), MARK(z1))
   MARK(afterNth(z0, z1)) -> c36(A__AFTERNTH(mark(z0), mark(z1)), MARK(z0))
   MARK(afterNth(z0, z1)) -> c37(A__AFTERNTH(mark(z0), mark(z1)), MARK(z1))
   MARK(take(z0, z1)) -> c38(A__TAKE(mark(z0), mark(z1)), MARK(z0))
   MARK(take(z0, z1)) -> c39(A__TAKE(mark(z0), mark(z1)), MARK(z1))
   MARK(cons(z0, z1)) -> c40(MARK(z0))
   MARK(s(z0)) -> c41(MARK(z0))
   MARK(pair(z0, z1)) -> c42(MARK(z0))
   MARK(pair(z0, z1)) -> c43(MARK(z1))
   MARK(0) -> c44
   MARK(nil) -> c45

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

   a__natsFrom(z0) -> cons(mark(z0), natsFrom(s(z0)))
   a__natsFrom(z0) -> natsFrom(z0)
   a__fst(pair(z0, z1)) -> mark(z0)
   a__fst(z0) -> fst(z0)
   a__snd(pair(z0, z1)) -> mark(z1)
   a__snd(z0) -> snd(z0)
   a__splitAt(0, z0) -> pair(nil, mark(z0))
   a__splitAt(s(z0), cons(z1, z2)) -> a__u(a__splitAt(mark(z0), mark(z2)), z0, z1, z2)
   a__splitAt(z0, z1) -> splitAt(z0, z1)
   a__u(pair(z0, z1), z2, z3, z4) -> pair(cons(mark(z3), z0), mark(z1))
   a__u(z0, z1, z2, z3) -> u(z0, z1, z2, z3)
   a__head(cons(z0, z1)) -> mark(z0)
   a__head(z0) -> head(z0)
   a__tail(cons(z0, z1)) -> mark(z1)
   a__tail(z0) -> tail(z0)
   a__sel(z0, z1) -> a__head(a__afterNth(mark(z0), mark(z1)))
   a__sel(z0, z1) -> sel(z0, z1)
   a__take(z0, z1) -> a__fst(a__splitAt(mark(z0), mark(z1)))
   a__take(z0, z1) -> take(z0, z1)
   a__afterNth(z0, z1) -> a__snd(a__splitAt(mark(z0), mark(z1)))
   a__afterNth(z0, z1) -> afterNth(z0, z1)
   mark(natsFrom(z0)) -> a__natsFrom(mark(z0))
   mark(fst(z0)) -> a__fst(mark(z0))
   mark(snd(z0)) -> a__snd(mark(z0))
   mark(splitAt(z0, z1)) -> a__splitAt(mark(z0), mark(z1))
   mark(u(z0, z1, z2, z3)) -> a__u(mark(z0), z1, z2, z3)
   mark(head(z0)) -> a__head(mark(z0))
   mark(tail(z0)) -> a__tail(mark(z0))
   mark(sel(z0, z1)) -> a__sel(mark(z0), mark(z1))
   mark(afterNth(z0, z1)) -> a__afterNth(mark(z0), mark(z1))
   mark(take(z0, z1)) -> a__take(mark(z0), mark(z1))
   mark(cons(z0, z1)) -> cons(mark(z0), z1)
   mark(s(z0)) -> s(mark(z0))
   mark(pair(z0, z1)) -> pair(mark(z0), mark(z1))
   mark(0) -> 0
   mark(nil) -> nil

Rewrite Strategy: INNERMOST