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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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


----------------------------------------

(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__U11(tt, z0, z1) -> c(A__U12(tt, z0, z1))
   A__U11(z0, z1, z2) -> c1
   A__U12(tt, z0, z1) -> c2(A__SND(a__splitAt(mark(z0), mark(z1))), A__SPLITAT(mark(z0), mark(z1)), MARK(z0))
   A__U12(tt, z0, z1) -> c3(A__SND(a__splitAt(mark(z0), mark(z1))), A__SPLITAT(mark(z0), mark(z1)), MARK(z1))
   A__U12(z0, z1, z2) -> c4
   A__U21(tt, z0) -> c5(A__U22(tt, z0))
   A__U21(z0, z1) -> c6
   A__U22(tt, z0) -> c7(MARK(z0))
   A__U22(z0, z1) -> c8
   A__U31(tt, z0) -> c9(A__U32(tt, z0))
   A__U31(z0, z1) -> c10
   A__U32(tt, z0) -> c11(MARK(z0))
   A__U32(z0, z1) -> c12
   A__U41(tt, z0, z1) -> c13(A__U42(tt, z0, z1))
   A__U41(z0, z1, z2) -> c14
   A__U42(tt, z0, z1) -> c15(A__HEAD(a__afterNth(mark(z0), mark(z1))), A__AFTERNTH(mark(z0), mark(z1)), MARK(z0))
   A__U42(tt, z0, z1) -> c16(A__HEAD(a__afterNth(mark(z0), mark(z1))), A__AFTERNTH(mark(z0), mark(z1)), MARK(z1))
   A__U42(z0, z1, z2) -> c17
   A__U51(tt, z0) -> c18(A__U52(tt, z0))
   A__U51(z0, z1) -> c19
   A__U52(tt, z0) -> c20(MARK(z0))
   A__U52(z0, z1) -> c21
   A__U61(tt, z0, z1, z2) -> c22(A__U62(tt, z0, z1, z2))
   A__U61(z0, z1, z2, z3) -> c23
   A__U62(tt, z0, z1, z2) -> c24(A__U63(tt, z0, z1, z2))
   A__U62(z0, z1, z2, z3) -> c25
   A__U63(tt, z0, z1, z2) -> c26(A__U64(a__splitAt(mark(z0), mark(z2)), z1), A__SPLITAT(mark(z0), mark(z2)), MARK(z0))
   A__U63(tt, z0, z1, z2) -> c27(A__U64(a__splitAt(mark(z0), mark(z2)), z1), A__SPLITAT(mark(z0), mark(z2)), MARK(z2))
   A__U63(z0, z1, z2, z3) -> c28
   A__U64(pair(z0, z1), z2) -> c29(MARK(z2))
   A__U64(pair(z0, z1), z2) -> c30(MARK(z1))
   A__U64(z0, z1) -> c31
   A__U71(tt, z0) -> c32(A__U72(tt, z0))
   A__U71(z0, z1) -> c33
   A__U72(tt, z0) -> c34(MARK(z0))
   A__U72(z0, z1) -> c35
   A__U81(tt, z0, z1) -> c36(A__U82(tt, z0, z1))
   A__U81(z0, z1, z2) -> c37
   A__U82(tt, z0, z1) -> c38(A__FST(a__splitAt(mark(z0), mark(z1))), A__SPLITAT(mark(z0), mark(z1)), MARK(z0))
   A__U82(tt, z0, z1) -> c39(A__FST(a__splitAt(mark(z0), mark(z1))), A__SPLITAT(mark(z0), mark(z1)), MARK(z1))
   A__U82(z0, z1, z2) -> c40
   A__AFTERNTH(z0, z1) -> c41(A__U11(tt, z0, z1))
   A__AFTERNTH(z0, z1) -> c42
   A__FST(pair(z0, z1)) -> c43(A__U21(tt, z0))
   A__FST(z0) -> c44
   A__HEAD(cons(z0, z1)) -> c45(A__U31(tt, z0))
   A__HEAD(z0) -> c46
   A__NATSFROM(z0) -> c47(MARK(z0))
   A__NATSFROM(z0) -> c48
   A__SEL(z0, z1) -> c49(A__U41(tt, z0, z1))
   A__SEL(z0, z1) -> c50
   A__SND(pair(z0, z1)) -> c51(A__U51(tt, z1))
   A__SND(z0) -> c52
   A__SPLITAT(0, z0) -> c53(MARK(z0))
   A__SPLITAT(s(z0), cons(z1, z2)) -> c54(A__U61(tt, z0, z1, z2))
   A__SPLITAT(z0, z1) -> c55
   A__TAIL(cons(z0, z1)) -> c56(A__U71(tt, z1))
   A__TAIL(z0) -> c57
   A__TAKE(z0, z1) -> c58(A__U81(tt, z0, z1))
   A__TAKE(z0, z1) -> c59
   MARK(U11(z0, z1, z2)) -> c60(A__U11(mark(z0), z1, z2), MARK(z0))
   MARK(U12(z0, z1, z2)) -> c61(A__U12(mark(z0), z1, z2), MARK(z0))
   MARK(snd(z0)) -> c62(A__SND(mark(z0)), MARK(z0))
   MARK(splitAt(z0, z1)) -> c63(A__SPLITAT(mark(z0), mark(z1)), MARK(z0))
   MARK(splitAt(z0, z1)) -> c64(A__SPLITAT(mark(z0), mark(z1)), MARK(z1))
   MARK(U21(z0, z1)) -> c65(A__U21(mark(z0), z1), MARK(z0))
   MARK(U22(z0, z1)) -> c66(A__U22(mark(z0), z1), MARK(z0))
   MARK(U31(z0, z1)) -> c67(A__U31(mark(z0), z1), MARK(z0))
   MARK(U32(z0, z1)) -> c68(A__U32(mark(z0), z1), MARK(z0))
   MARK(U41(z0, z1, z2)) -> c69(A__U41(mark(z0), z1, z2), MARK(z0))
   MARK(U42(z0, z1, z2)) -> c70(A__U42(mark(z0), z1, z2), MARK(z0))
   MARK(head(z0)) -> c71(A__HEAD(mark(z0)), MARK(z0))
   MARK(afterNth(z0, z1)) -> c72(A__AFTERNTH(mark(z0), mark(z1)), MARK(z0))
   MARK(afterNth(z0, z1)) -> c73(A__AFTERNTH(mark(z0), mark(z1)), MARK(z1))
   MARK(U51(z0, z1)) -> c74(A__U51(mark(z0), z1), MARK(z0))
   MARK(U52(z0, z1)) -> c75(A__U52(mark(z0), z1), MARK(z0))
   MARK(U61(z0, z1, z2, z3)) -> c76(A__U61(mark(z0), z1, z2, z3), MARK(z0))
   MARK(U62(z0, z1, z2, z3)) -> c77(A__U62(mark(z0), z1, z2, z3), MARK(z0))
   MARK(U63(z0, z1, z2, z3)) -> c78(A__U63(mark(z0), z1, z2, z3), MARK(z0))
   MARK(U64(z0, z1)) -> c79(A__U64(mark(z0), z1), MARK(z0))
   MARK(U71(z0, z1)) -> c80(A__U71(mark(z0), z1), MARK(z0))
   MARK(U72(z0, z1)) -> c81(A__U72(mark(z0), z1), MARK(z0))
   MARK(U81(z0, z1, z2)) -> c82(A__U81(mark(z0), z1, z2), MARK(z0))
   MARK(U82(z0, z1, z2)) -> c83(A__U82(mark(z0), z1, z2), MARK(z0))
   MARK(fst(z0)) -> c84(A__FST(mark(z0)), MARK(z0))
   MARK(natsFrom(z0)) -> c85(A__NATSFROM(mark(z0)), MARK(z0))
   MARK(sel(z0, z1)) -> c86(A__SEL(mark(z0), mark(z1)), MARK(z0))
   MARK(sel(z0, z1)) -> c87(A__SEL(mark(z0), mark(z1)), MARK(z1))
   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(tt) -> c91
   MARK(pair(z0, z1)) -> c92(MARK(z0))
   MARK(pair(z0, z1)) -> c93(MARK(z1))
   MARK(cons(z0, z1)) -> c94(MARK(z0))
   MARK(s(z0)) -> c95(MARK(z0))
   MARK(0) -> c96
   MARK(nil) -> c97

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

   a__U11(tt, z0, z1) -> a__U12(tt, z0, z1)
   a__U11(z0, z1, z2) -> U11(z0, z1, z2)
   a__U12(tt, z0, z1) -> a__snd(a__splitAt(mark(z0), mark(z1)))
   a__U12(z0, z1, z2) -> U12(z0, z1, z2)
   a__U21(tt, z0) -> a__U22(tt, z0)
   a__U21(z0, z1) -> U21(z0, z1)
   a__U22(tt, z0) -> mark(z0)
   a__U22(z0, z1) -> U22(z0, z1)
   a__U31(tt, z0) -> a__U32(tt, z0)
   a__U31(z0, z1) -> U31(z0, z1)
   a__U32(tt, z0) -> mark(z0)
   a__U32(z0, z1) -> U32(z0, z1)
   a__U41(tt, z0, z1) -> a__U42(tt, z0, z1)
   a__U41(z0, z1, z2) -> U41(z0, z1, z2)
   a__U42(tt, z0, z1) -> a__head(a__afterNth(mark(z0), mark(z1)))
   a__U42(z0, z1, z2) -> U42(z0, z1, z2)
   a__U51(tt, z0) -> a__U52(tt, z0)
   a__U51(z0, z1) -> U51(z0, z1)
   a__U52(tt, z0) -> mark(z0)
   a__U52(z0, z1) -> U52(z0, z1)
   a__U61(tt, z0, z1, z2) -> a__U62(tt, z0, z1, z2)
   a__U61(z0, z1, z2, z3) -> U61(z0, z1, z2, z3)
   a__U62(tt, z0, z1, z2) -> a__U63(tt, z0, z1, z2)
   a__U62(z0, z1, z2, z3) -> U62(z0, z1, z2, z3)
   a__U63(tt, z0, z1, z2) -> a__U64(a__splitAt(mark(z0), mark(z2)), z1)
   a__U63(z0, z1, z2, z3) -> U63(z0, z1, z2, z3)
   a__U64(pair(z0, z1), z2) -> pair(cons(mark(z2), z0), mark(z1))
   a__U64(z0, z1) -> U64(z0, z1)
   a__U71(tt, z0) -> a__U72(tt, z0)
   a__U71(z0, z1) -> U71(z0, z1)
   a__U72(tt, z0) -> mark(z0)
   a__U72(z0, z1) -> U72(z0, z1)
   a__U81(tt, z0, z1) -> a__U82(tt, z0, z1)
   a__U81(z0, z1, z2) -> U81(z0, z1, z2)
   a__U82(tt, z0, z1) -> a__fst(a__splitAt(mark(z0), mark(z1)))
   a__U82(z0, z1, z2) -> U82(z0, z1, z2)
   a__afterNth(z0, z1) -> a__U11(tt, z0, z1)
   a__afterNth(z0, z1) -> afterNth(z0, z1)
   a__fst(pair(z0, z1)) -> a__U21(tt, z0)
   a__fst(z0) -> fst(z0)
   a__head(cons(z0, z1)) -> a__U31(tt, z0)
   a__head(z0) -> head(z0)
   a__natsFrom(z0) -> cons(mark(z0), natsFrom(s(z0)))
   a__natsFrom(z0) -> natsFrom(z0)
   a__sel(z0, z1) -> a__U41(tt, z0, z1)
   a__sel(z0, z1) -> sel(z0, z1)
   a__snd(pair(z0, z1)) -> a__U51(tt, z1)
   a__snd(z0) -> snd(z0)
   a__splitAt(0, z0) -> pair(nil, mark(z0))
   a__splitAt(s(z0), cons(z1, z2)) -> a__U61(tt, z0, z1, z2)
   a__splitAt(z0, z1) -> splitAt(z0, z1)
   a__tail(cons(z0, z1)) -> a__U71(tt, z1)
   a__tail(z0) -> tail(z0)
   a__take(z0, z1) -> a__U81(tt, z0, z1)
   a__take(z0, z1) -> take(z0, z1)
   mark(U11(z0, z1, z2)) -> a__U11(mark(z0), z1, z2)
   mark(U12(z0, z1, z2)) -> a__U12(mark(z0), z1, z2)
   mark(snd(z0)) -> a__snd(mark(z0))
   mark(splitAt(z0, z1)) -> a__splitAt(mark(z0), mark(z1))
   mark(U21(z0, z1)) -> a__U21(mark(z0), z1)
   mark(U22(z0, z1)) -> a__U22(mark(z0), z1)
   mark(U31(z0, z1)) -> a__U31(mark(z0), z1)
   mark(U32(z0, z1)) -> a__U32(mark(z0), z1)
   mark(U41(z0, z1, z2)) -> a__U41(mark(z0), z1, z2)
   mark(U42(z0, z1, z2)) -> a__U42(mark(z0), z1, z2)
   mark(head(z0)) -> a__head(mark(z0))
   mark(afterNth(z0, z1)) -> a__afterNth(mark(z0), mark(z1))
   mark(U51(z0, z1)) -> a__U51(mark(z0), z1)
   mark(U52(z0, z1)) -> a__U52(mark(z0), z1)
   mark(U61(z0, z1, z2, z3)) -> a__U61(mark(z0), z1, z2, z3)
   mark(U62(z0, z1, z2, z3)) -> a__U62(mark(z0), z1, z2, z3)
   mark(U63(z0, z1, z2, z3)) -> a__U63(mark(z0), z1, z2, z3)
   mark(U64(z0, z1)) -> a__U64(mark(z0), z1)
   mark(U71(z0, z1)) -> a__U71(mark(z0), z1)
   mark(U72(z0, z1)) -> a__U72(mark(z0), z1)
   mark(U81(z0, z1, z2)) -> a__U81(mark(z0), z1, z2)
   mark(U82(z0, z1, z2)) -> a__U82(mark(z0), z1, z2)
   mark(fst(z0)) -> a__fst(mark(z0))
   mark(natsFrom(z0)) -> a__natsFrom(mark(z0))
   mark(sel(z0, z1)) -> a__sel(mark(z0), mark(z1))
   mark(tail(z0)) -> a__tail(mark(z0))
   mark(take(z0, z1)) -> a__take(mark(z0), mark(z1))
   mark(tt) -> tt
   mark(pair(z0, z1)) -> pair(mark(z0), mark(z1))
   mark(cons(z0, z1)) -> cons(mark(z0), z1)
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(nil) -> nil

Rewrite Strategy: INNERMOST