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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 742 ms]
(2) CpxRelTRS
(3) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(4) TRS for Loop Detection
(5) DecreasingLoopProof [LOWER BOUND(ID), 8 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:

   A__FIRST(0, z0) -> c
   A__FIRST(s(z0), cons(z1, z2)) -> c1(MARK(z1))
   A__FIRST(z0, z1) -> c2
   A__FROM(z0) -> c3(MARK(z0))
   A__FROM(z0) -> c4
   MARK(first(z0, z1)) -> c5(A__FIRST(mark(z0), mark(z1)), MARK(z0))
   MARK(first(z0, z1)) -> c6(A__FIRST(mark(z0), mark(z1)), MARK(z1))
   MARK(from(z0)) -> c7(A__FROM(mark(z0)), MARK(z0))
   MARK(0) -> c8
   MARK(nil) -> c9
   MARK(s(z0)) -> c10(MARK(z0))
   MARK(cons(z0, z1)) -> c11(MARK(z0))

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

   a__first(0, z0) -> nil
   a__first(s(z0), cons(z1, z2)) -> cons(mark(z1), first(z0, z2))
   a__first(z0, z1) -> first(z0, z1)
   a__from(z0) -> cons(mark(z0), from(s(z0)))
   a__from(z0) -> from(z0)
   mark(first(z0, z1)) -> a__first(mark(z0), mark(z1))
   mark(from(z0)) -> a__from(mark(z0))
   mark(0) -> 0
   mark(nil) -> nil
   mark(s(z0)) -> s(mark(z0))
   mark(cons(z0, z1)) -> cons(mark(z0), z1)

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:

   A__FIRST(0, z0) -> c
   A__FIRST(s(z0), cons(z1, z2)) -> c1(MARK(z1))
   A__FIRST(z0, z1) -> c2
   A__FROM(z0) -> c3(MARK(z0))
   A__FROM(z0) -> c4
   MARK(first(z0, z1)) -> c5(A__FIRST(mark(z0), mark(z1)), MARK(z0))
   MARK(first(z0, z1)) -> c6(A__FIRST(mark(z0), mark(z1)), MARK(z1))
   MARK(from(z0)) -> c7(A__FROM(mark(z0)), MARK(z0))
   MARK(0) -> c8
   MARK(nil) -> c9
   MARK(s(z0)) -> c10(MARK(z0))
   MARK(cons(z0, z1)) -> c11(MARK(z0))

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

   a__first(0, z0) -> nil
   a__first(s(z0), cons(z1, z2)) -> cons(mark(z1), first(z0, z2))
   a__first(z0, z1) -> first(z0, z1)
   a__from(z0) -> cons(mark(z0), from(s(z0)))
   a__from(z0) -> from(z0)
   mark(first(z0, z1)) -> a__first(mark(z0), mark(z1))
   mark(from(z0)) -> a__from(mark(z0))
   mark(0) -> 0
   mark(nil) -> nil
   mark(s(z0)) -> s(mark(z0))
   mark(cons(z0, z1)) -> cons(mark(z0), z1)

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:

   A__FIRST(0, z0) -> c
   A__FIRST(s(z0), cons(z1, z2)) -> c1(MARK(z1))
   A__FIRST(z0, z1) -> c2
   A__FROM(z0) -> c3(MARK(z0))
   A__FROM(z0) -> c4
   MARK(first(z0, z1)) -> c5(A__FIRST(mark(z0), mark(z1)), MARK(z0))
   MARK(first(z0, z1)) -> c6(A__FIRST(mark(z0), mark(z1)), MARK(z1))
   MARK(from(z0)) -> c7(A__FROM(mark(z0)), MARK(z0))
   MARK(0) -> c8
   MARK(nil) -> c9
   MARK(s(z0)) -> c10(MARK(z0))
   MARK(cons(z0, z1)) -> c11(MARK(z0))

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

   a__first(0, z0) -> nil
   a__first(s(z0), cons(z1, z2)) -> cons(mark(z1), first(z0, z2))
   a__first(z0, z1) -> first(z0, z1)
   a__from(z0) -> cons(mark(z0), from(s(z0)))
   a__from(z0) -> from(z0)
   mark(first(z0, z1)) -> a__first(mark(z0), mark(z1))
   mark(from(z0)) -> a__from(mark(z0))
   mark(0) -> 0
   mark(nil) -> nil
   mark(s(z0)) -> s(mark(z0))
   mark(cons(z0, z1)) -> cons(mark(z0), z1)

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

MARK(first(z0, z1)) ->^+ c6(A__FIRST(mark(z0), mark(z1)), MARK(z1))

gives rise to a decreasing loop by considering the right hand sides subterm at position [1].

The pumping substitution is [z1 / first(z0, z1)].

The result substitution is [ ].




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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:

   A__FIRST(0, z0) -> c
   A__FIRST(s(z0), cons(z1, z2)) -> c1(MARK(z1))
   A__FIRST(z0, z1) -> c2
   A__FROM(z0) -> c3(MARK(z0))
   A__FROM(z0) -> c4
   MARK(first(z0, z1)) -> c5(A__FIRST(mark(z0), mark(z1)), MARK(z0))
   MARK(first(z0, z1)) -> c6(A__FIRST(mark(z0), mark(z1)), MARK(z1))
   MARK(from(z0)) -> c7(A__FROM(mark(z0)), MARK(z0))
   MARK(0) -> c8
   MARK(nil) -> c9
   MARK(s(z0)) -> c10(MARK(z0))
   MARK(cons(z0, z1)) -> c11(MARK(z0))

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

   a__first(0, z0) -> nil
   a__first(s(z0), cons(z1, z2)) -> cons(mark(z1), first(z0, z2))
   a__first(z0, z1) -> first(z0, z1)
   a__from(z0) -> cons(mark(z0), from(s(z0)))
   a__from(z0) -> from(z0)
   mark(first(z0, z1)) -> a__first(mark(z0), mark(z1))
   mark(from(z0)) -> a__from(mark(z0))
   mark(0) -> 0
   mark(nil) -> nil
   mark(s(z0)) -> s(mark(z0))
   mark(cons(z0, z1)) -> cons(mark(z0), z1)

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:

   A__FIRST(0, z0) -> c
   A__FIRST(s(z0), cons(z1, z2)) -> c1(MARK(z1))
   A__FIRST(z0, z1) -> c2
   A__FROM(z0) -> c3(MARK(z0))
   A__FROM(z0) -> c4
   MARK(first(z0, z1)) -> c5(A__FIRST(mark(z0), mark(z1)), MARK(z0))
   MARK(first(z0, z1)) -> c6(A__FIRST(mark(z0), mark(z1)), MARK(z1))
   MARK(from(z0)) -> c7(A__FROM(mark(z0)), MARK(z0))
   MARK(0) -> c8
   MARK(nil) -> c9
   MARK(s(z0)) -> c10(MARK(z0))
   MARK(cons(z0, z1)) -> c11(MARK(z0))

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

   a__first(0, z0) -> nil
   a__first(s(z0), cons(z1, z2)) -> cons(mark(z1), first(z0, z2))
   a__first(z0, z1) -> first(z0, z1)
   a__from(z0) -> cons(mark(z0), from(s(z0)))
   a__from(z0) -> from(z0)
   mark(first(z0, z1)) -> a__first(mark(z0), mark(z1))
   mark(from(z0)) -> a__from(mark(z0))
   mark(0) -> 0
   mark(nil) -> nil
   mark(s(z0)) -> s(mark(z0))
   mark(cons(z0, z1)) -> cons(mark(z0), z1)

Rewrite Strategy: INNERMOST