WORST_CASE(Omega(n^1),?)
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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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(n^1, INF).

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 518 ms]
(2) CpxRelTRS
(3) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(4) TRS for Loop Detection
(5) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
(6) BEST
    (7) proven lower bound
        (8) LowerBoundPropagationProof [FINISHED, 0 ms]
        (9) BOUNDS(n^1, INF)
    (10) TRS for Loop Detection


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(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   REV1(0, nil) -> c
   REV1(s(z0), nil) -> c1
   REV1(z0, cons(z1, z2)) -> c2(REV1(z1, z2))
   REV(nil) -> c3
   REV(cons(z0, z1)) -> c4(REV1(z0, z1))
   REV(cons(z0, z1)) -> c5(REV2(z0, z1))
   REV2(z0, nil) -> c6
   REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, rev(rev2(z1, z2)))), REV(rev2(z1, z2)), REV2(z1, z2))

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

   rev1(0, nil) -> 0
   rev1(s(z0), nil) -> s(z0)
   rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
   rev(nil) -> nil
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2))))

Rewrite Strategy: INNERMOST
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(1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
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(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   REV1(0, nil) -> c
   REV1(s(z0), nil) -> c1
   REV1(z0, cons(z1, z2)) -> c2(REV1(z1, z2))
   REV(nil) -> c3
   REV(cons(z0, z1)) -> c4(REV1(z0, z1))
   REV(cons(z0, z1)) -> c5(REV2(z0, z1))
   REV2(z0, nil) -> c6
   REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, rev(rev2(z1, z2)))), REV(rev2(z1, z2)), REV2(z1, z2))

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

   rev1(0, nil) -> 0
   rev1(s(z0), nil) -> s(z0)
   rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
   rev(nil) -> nil
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2))))

Rewrite Strategy: INNERMOST
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(3) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(4)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   REV1(0, nil) -> c
   REV1(s(z0), nil) -> c1
   REV1(z0, cons(z1, z2)) -> c2(REV1(z1, z2))
   REV(nil) -> c3
   REV(cons(z0, z1)) -> c4(REV1(z0, z1))
   REV(cons(z0, z1)) -> c5(REV2(z0, z1))
   REV2(z0, nil) -> c6
   REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, rev(rev2(z1, z2)))), REV(rev2(z1, z2)), REV2(z1, z2))

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

   rev1(0, nil) -> 0
   rev1(s(z0), nil) -> s(z0)
   rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
   rev(nil) -> nil
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2))))

Rewrite Strategy: INNERMOST
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(5) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

REV1(z0, cons(z1, z2)) ->^+ c2(REV1(z1, z2))

gives rise to a decreasing loop by considering the right hand sides subterm at position [0].

The pumping substitution is [z2 / cons(z1, z2)].

The result substitution is [z0 / z1].




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(6)
Complex Obligation (BEST)

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(7)
Obligation:
Proved the lower bound n^1 for the following obligation:

The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   REV1(0, nil) -> c
   REV1(s(z0), nil) -> c1
   REV1(z0, cons(z1, z2)) -> c2(REV1(z1, z2))
   REV(nil) -> c3
   REV(cons(z0, z1)) -> c4(REV1(z0, z1))
   REV(cons(z0, z1)) -> c5(REV2(z0, z1))
   REV2(z0, nil) -> c6
   REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, rev(rev2(z1, z2)))), REV(rev2(z1, z2)), REV2(z1, z2))

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

   rev1(0, nil) -> 0
   rev1(s(z0), nil) -> s(z0)
   rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
   rev(nil) -> nil
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2))))

Rewrite Strategy: INNERMOST
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(8) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(9)
BOUNDS(n^1, INF)

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(10)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   REV1(0, nil) -> c
   REV1(s(z0), nil) -> c1
   REV1(z0, cons(z1, z2)) -> c2(REV1(z1, z2))
   REV(nil) -> c3
   REV(cons(z0, z1)) -> c4(REV1(z0, z1))
   REV(cons(z0, z1)) -> c5(REV2(z0, z1))
   REV2(z0, nil) -> c6
   REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, rev(rev2(z1, z2)))), REV(rev2(z1, z2)), REV2(z1, z2))

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

   rev1(0, nil) -> 0
   rev1(s(z0), nil) -> s(z0)
   rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
   rev(nil) -> nil
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2))))

Rewrite Strategy: INNERMOST