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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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


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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__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), A__ZERO(mark(z0)), MARK(z0))
   A__FACT(z0) -> c1
   A__ADD(0, z0) -> c2(MARK(z0))
   A__ADD(s(z0), z1) -> c3(A__ADD(mark(z0), mark(z1)), MARK(z0))
   A__ADD(s(z0), z1) -> c4(A__ADD(mark(z0), mark(z1)), MARK(z1))
   A__ADD(z0, z1) -> c5
   A__PROD(0, z0) -> c6
   A__PROD(s(z0), z1) -> c7(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), MARK(z1))
   A__PROD(s(z0), z1) -> c8(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z0))
   A__PROD(s(z0), z1) -> c9(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z1))
   A__PROD(z0, z1) -> c10
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__IF(z0, z1, z2) -> c13
   A__ZERO(0) -> c14
   A__ZERO(s(z0)) -> c15
   A__ZERO(z0) -> c16
   A__P(s(z0)) -> c17(MARK(z0))
   A__P(z0) -> c18
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(if(z0, z1, z2)) -> c20(A__IF(mark(z0), z1, z2), MARK(z0))
   MARK(zero(z0)) -> c21(A__ZERO(mark(z0)), MARK(z0))
   MARK(prod(z0, z1)) -> c22(A__PROD(mark(z0), mark(z1)), MARK(z0))
   MARK(prod(z0, z1)) -> c23(A__PROD(mark(z0), mark(z1)), MARK(z1))
   MARK(p(z0)) -> c24(A__P(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c25(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(0) -> c28
   MARK(true) -> c29
   MARK(false) -> c30

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

   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(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__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false

Rewrite Strategy: INNERMOST