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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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


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

(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__FROM(z0) -> c(MARK(z0))
   A__FROM(z0) -> c1
   A__SEL(0, cons(z0, z1)) -> c2(MARK(z0))
   A__SEL(s(z0), cons(z1, z2)) -> c3(A__SEL(mark(z0), mark(z2)), MARK(z0))
   A__SEL(s(z0), cons(z1, z2)) -> c4(A__SEL(mark(z0), mark(z2)), MARK(z2))
   A__SEL(z0, z1) -> c5
   A__MINUS(z0, 0) -> c6
   A__MINUS(s(z0), s(z1)) -> c7(A__MINUS(mark(z0), mark(z1)), MARK(z0))
   A__MINUS(s(z0), s(z1)) -> c8(A__MINUS(mark(z0), mark(z1)), MARK(z1))
   A__MINUS(z0, z1) -> c9
   A__QUOT(0, s(z0)) -> c10
   A__QUOT(s(z0), s(z1)) -> c11(A__QUOT(a__minus(mark(z0), mark(z1)), s(mark(z1))), A__MINUS(mark(z0), mark(z1)), MARK(z0))
   A__QUOT(s(z0), s(z1)) -> c12(A__QUOT(a__minus(mark(z0), mark(z1)), s(mark(z1))), A__MINUS(mark(z0), mark(z1)), MARK(z1))
   A__QUOT(s(z0), s(z1)) -> c13(A__QUOT(a__minus(mark(z0), mark(z1)), s(mark(z1))), MARK(z1))
   A__QUOT(z0, z1) -> c14
   A__ZWQUOT(z0, nil) -> c15
   A__ZWQUOT(nil, z0) -> c16
   A__ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c17(A__QUOT(mark(z0), mark(z2)), MARK(z0))
   A__ZWQUOT(cons(z0, z1), cons(z2, z3)) -> c18(A__QUOT(mark(z0), mark(z2)), MARK(z2))
   A__ZWQUOT(z0, z1) -> c19
   MARK(from(z0)) -> c20(A__FROM(mark(z0)), MARK(z0))
   MARK(sel(z0, z1)) -> c21(A__SEL(mark(z0), mark(z1)), MARK(z0))
   MARK(sel(z0, z1)) -> c22(A__SEL(mark(z0), mark(z1)), MARK(z1))
   MARK(minus(z0, z1)) -> c23(A__MINUS(mark(z0), mark(z1)), MARK(z0))
   MARK(minus(z0, z1)) -> c24(A__MINUS(mark(z0), mark(z1)), MARK(z1))
   MARK(quot(z0, z1)) -> c25(A__QUOT(mark(z0), mark(z1)), MARK(z0))
   MARK(quot(z0, z1)) -> c26(A__QUOT(mark(z0), mark(z1)), MARK(z1))
   MARK(zWquot(z0, z1)) -> c27(A__ZWQUOT(mark(z0), mark(z1)), MARK(z0))
   MARK(zWquot(z0, z1)) -> c28(A__ZWQUOT(mark(z0), mark(z1)), MARK(z1))
   MARK(cons(z0, z1)) -> c29(MARK(z0))
   MARK(s(z0)) -> c30(MARK(z0))
   MARK(0) -> c31
   MARK(nil) -> c32

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

   a__from(z0) -> cons(mark(z0), from(s(z0)))
   a__from(z0) -> from(z0)
   a__sel(0, cons(z0, z1)) -> mark(z0)
   a__sel(s(z0), cons(z1, z2)) -> a__sel(mark(z0), mark(z2))
   a__sel(z0, z1) -> sel(z0, z1)
   a__minus(z0, 0) -> 0
   a__minus(s(z0), s(z1)) -> a__minus(mark(z0), mark(z1))
   a__minus(z0, z1) -> minus(z0, z1)
   a__quot(0, s(z0)) -> 0
   a__quot(s(z0), s(z1)) -> s(a__quot(a__minus(mark(z0), mark(z1)), s(mark(z1))))
   a__quot(z0, z1) -> quot(z0, z1)
   a__zWquot(z0, nil) -> nil
   a__zWquot(nil, z0) -> nil
   a__zWquot(cons(z0, z1), cons(z2, z3)) -> cons(a__quot(mark(z0), mark(z2)), zWquot(z1, z3))
   a__zWquot(z0, z1) -> zWquot(z0, z1)
   mark(from(z0)) -> a__from(mark(z0))
   mark(sel(z0, z1)) -> a__sel(mark(z0), mark(z1))
   mark(minus(z0, z1)) -> a__minus(mark(z0), mark(z1))
   mark(quot(z0, z1)) -> a__quot(mark(z0), mark(z1))
   mark(zWquot(z0, z1)) -> a__zWquot(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