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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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__ZEROS -> c
   A__ZEROS -> c1
   A__U11(tt, z0) -> c2(A__LENGTH(mark(z0)), MARK(z0))
   A__U11(z0, z1) -> c3
   A__U21(tt) -> c4
   A__U21(z0) -> c5
   A__U31(tt, z0, z1, z2) -> c6(MARK(z2))
   A__U31(z0, z1, z2, z3) -> c7
   A__AND(tt, z0) -> c8(MARK(z0))
   A__AND(z0, z1) -> c9
   A__ISNAT(0) -> c10
   A__ISNAT(length(z0)) -> c11(A__ISNATLIST(z0))
   A__ISNAT(s(z0)) -> c12(A__ISNAT(z0))
   A__ISNAT(z0) -> c13
   A__ISNATILIST(z0) -> c14(A__ISNATLIST(z0))
   A__ISNATILIST(zeros) -> c15
   A__ISNATILIST(cons(z0, z1)) -> c16(A__AND(a__isNat(z0), isNatIList(z1)), A__ISNAT(z0))
   A__ISNATILIST(z0) -> c17
   A__ISNATLIST(nil) -> c18
   A__ISNATLIST(cons(z0, z1)) -> c19(A__AND(a__isNat(z0), isNatList(z1)), A__ISNAT(z0))
   A__ISNATLIST(take(z0, z1)) -> c20(A__AND(a__isNat(z0), isNatIList(z1)), A__ISNAT(z0))
   A__ISNATLIST(z0) -> c21
   A__LENGTH(nil) -> c22
   A__LENGTH(cons(z0, z1)) -> c23(A__U11(a__and(a__isNatList(z1), isNat(z0)), z1), A__AND(a__isNatList(z1), isNat(z0)), A__ISNATLIST(z1))
   A__LENGTH(z0) -> c24
   A__TAKE(0, z0) -> c25(A__U21(a__isNatIList(z0)), A__ISNATILIST(z0))
   A__TAKE(s(z0), cons(z1, z2)) -> c26(A__U31(a__and(a__isNatIList(z2), and(isNat(z0), isNat(z1))), z2, z0, z1), A__AND(a__isNatIList(z2), and(isNat(z0), isNat(z1))), A__ISNATILIST(z2))
   A__TAKE(z0, z1) -> c27
   MARK(zeros) -> c28(A__ZEROS)
   MARK(U11(z0, z1)) -> c29(A__U11(mark(z0), z1), MARK(z0))
   MARK(length(z0)) -> c30(A__LENGTH(mark(z0)), MARK(z0))
   MARK(U21(z0)) -> c31(A__U21(mark(z0)), MARK(z0))
   MARK(U31(z0, z1, z2, z3)) -> c32(A__U31(mark(z0), z1, z2, z3), MARK(z0))
   MARK(take(z0, z1)) -> c33(A__TAKE(mark(z0), mark(z1)), MARK(z0))
   MARK(take(z0, z1)) -> c34(A__TAKE(mark(z0), mark(z1)), MARK(z1))
   MARK(and(z0, z1)) -> c35(A__AND(mark(z0), z1), MARK(z0))
   MARK(isNat(z0)) -> c36(A__ISNAT(z0))
   MARK(isNatList(z0)) -> c37(A__ISNATLIST(z0))
   MARK(isNatIList(z0)) -> c38(A__ISNATILIST(z0))
   MARK(cons(z0, z1)) -> c39(MARK(z0))
   MARK(0) -> c40
   MARK(tt) -> c41
   MARK(s(z0)) -> c42(MARK(z0))
   MARK(nil) -> c43

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

   a__zeros -> cons(0, zeros)
   a__zeros -> zeros
   a__U11(tt, z0) -> s(a__length(mark(z0)))
   a__U11(z0, z1) -> U11(z0, z1)
   a__U21(tt) -> nil
   a__U21(z0) -> U21(z0)
   a__U31(tt, z0, z1, z2) -> cons(mark(z2), take(z1, z0))
   a__U31(z0, z1, z2, z3) -> U31(z0, z1, z2, z3)
   a__and(tt, z0) -> mark(z0)
   a__and(z0, z1) -> and(z0, z1)
   a__isNat(0) -> tt
   a__isNat(length(z0)) -> a__isNatList(z0)
   a__isNat(s(z0)) -> a__isNat(z0)
   a__isNat(z0) -> isNat(z0)
   a__isNatIList(z0) -> a__isNatList(z0)
   a__isNatIList(zeros) -> tt
   a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1))
   a__isNatIList(z0) -> isNatIList(z0)
   a__isNatList(nil) -> tt
   a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1))
   a__isNatList(take(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1))
   a__isNatList(z0) -> isNatList(z0)
   a__length(nil) -> 0
   a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1)
   a__length(z0) -> length(z0)
   a__take(0, z0) -> a__U21(a__isNatIList(z0))
   a__take(s(z0), cons(z1, z2)) -> a__U31(a__and(a__isNatIList(z2), and(isNat(z0), isNat(z1))), z2, z0, z1)
   a__take(z0, z1) -> take(z0, z1)
   mark(zeros) -> a__zeros
   mark(U11(z0, z1)) -> a__U11(mark(z0), z1)
   mark(length(z0)) -> a__length(mark(z0))
   mark(U21(z0)) -> a__U21(mark(z0))
   mark(U31(z0, z1, z2, z3)) -> a__U31(mark(z0), z1, z2, z3)
   mark(take(z0, z1)) -> a__take(mark(z0), mark(z1))
   mark(and(z0, z1)) -> a__and(mark(z0), z1)
   mark(isNat(z0)) -> a__isNat(z0)
   mark(isNatList(z0)) -> a__isNatList(z0)
   mark(isNatIList(z0)) -> a__isNatIList(z0)
   mark(cons(z0, z1)) -> cons(mark(z0), z1)
   mark(0) -> 0
   mark(tt) -> tt
   mark(s(z0)) -> s(mark(z0))
   mark(nil) -> nil

Rewrite Strategy: INNERMOST