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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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__P(0) -> c
   A__P(s(z0)) -> c1(MARK(z0))
   A__P(z0) -> c2
   A__LEQ(0, z0) -> c3
   A__LEQ(s(z0), 0) -> c4
   A__LEQ(s(z0), s(z1)) -> c5(A__LEQ(mark(z0), mark(z1)), MARK(z0))
   A__LEQ(s(z0), s(z1)) -> c6(A__LEQ(mark(z0), mark(z1)), MARK(z1))
   A__LEQ(z0, z1) -> c7
   A__IF(true, z0, z1) -> c8(MARK(z0))
   A__IF(false, z0, z1) -> c9(MARK(z1))
   A__IF(z0, z1, z2) -> c10
   A__DIFF(z0, z1) -> c11(A__IF(a__leq(mark(z0), mark(z1)), 0, s(diff(p(z0), z1))), A__LEQ(mark(z0), mark(z1)), MARK(z0))
   A__DIFF(z0, z1) -> c12(A__IF(a__leq(mark(z0), mark(z1)), 0, s(diff(p(z0), z1))), A__LEQ(mark(z0), mark(z1)), MARK(z1))
   A__DIFF(z0, z1) -> c13
   MARK(p(z0)) -> c14(A__P(mark(z0)), MARK(z0))
   MARK(leq(z0, z1)) -> c15(A__LEQ(mark(z0), mark(z1)), MARK(z0))
   MARK(leq(z0, z1)) -> c16(A__LEQ(mark(z0), mark(z1)), MARK(z1))
   MARK(if(z0, z1, z2)) -> c17(A__IF(mark(z0), z1, z2), MARK(z0))
   MARK(diff(z0, z1)) -> c18(A__DIFF(mark(z0), mark(z1)), MARK(z0))
   MARK(diff(z0, z1)) -> c19(A__DIFF(mark(z0), mark(z1)), MARK(z1))
   MARK(0) -> c20
   MARK(s(z0)) -> c21(MARK(z0))
   MARK(true) -> c22
   MARK(false) -> c23

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

   a__p(0) -> 0
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   a__leq(0, z0) -> true
   a__leq(s(z0), 0) -> false
   a__leq(s(z0), s(z1)) -> a__leq(mark(z0), mark(z1))
   a__leq(z0, z1) -> leq(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__diff(z0, z1) -> a__if(a__leq(mark(z0), mark(z1)), 0, s(diff(p(z0), z1)))
   a__diff(z0, z1) -> diff(z0, z1)
   mark(p(z0)) -> a__p(mark(z0))
   mark(leq(z0, z1)) -> a__leq(mark(z0), mark(z1))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(diff(z0, z1)) -> a__diff(mark(z0), mark(z1))
   mark(0) -> 0
   mark(s(z0)) -> s(mark(z0))
   mark(true) -> true
   mark(false) -> false

Rewrite Strategy: INNERMOST