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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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), 2927 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:

   A__2ND(cons1(z0, cons(z1, z2))) -> c(MARK(z1))
   A__2ND(cons(z0, z1)) -> c1(A__2ND(cons1(mark(z0), mark(z1))), MARK(z0))
   A__2ND(cons(z0, z1)) -> c2(A__2ND(cons1(mark(z0), mark(z1))), MARK(z1))
   A__2ND(z0) -> c3
   A__FROM(z0) -> c4(MARK(z0))
   A__FROM(z0) -> c5
   MARK(2nd(z0)) -> c6(A__2ND(mark(z0)), MARK(z0))
   MARK(from(z0)) -> c7(A__FROM(mark(z0)), MARK(z0))
   MARK(cons(z0, z1)) -> c8(MARK(z0))
   MARK(s(z0)) -> c9(MARK(z0))
   MARK(cons1(z0, z1)) -> c10(MARK(z0))
   MARK(cons1(z0, z1)) -> c11(MARK(z1))

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

   a__2nd(cons1(z0, cons(z1, z2))) -> mark(z1)
   a__2nd(cons(z0, z1)) -> a__2nd(cons1(mark(z0), mark(z1)))
   a__2nd(z0) -> 2nd(z0)
   a__from(z0) -> cons(mark(z0), from(s(z0)))
   a__from(z0) -> from(z0)
   mark(2nd(z0)) -> a__2nd(mark(z0))
   mark(from(z0)) -> a__from(mark(z0))
   mark(cons(z0, z1)) -> cons(mark(z0), z1)
   mark(s(z0)) -> s(mark(z0))
   mark(cons1(z0, z1)) -> cons1(mark(z0), mark(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__2ND(cons1(z0, cons(z1, z2))) -> c(MARK(z1))
   A__2ND(cons(z0, z1)) -> c1(A__2ND(cons1(mark(z0), mark(z1))), MARK(z0))
   A__2ND(cons(z0, z1)) -> c2(A__2ND(cons1(mark(z0), mark(z1))), MARK(z1))
   A__2ND(z0) -> c3
   A__FROM(z0) -> c4(MARK(z0))
   A__FROM(z0) -> c5
   MARK(2nd(z0)) -> c6(A__2ND(mark(z0)), MARK(z0))
   MARK(from(z0)) -> c7(A__FROM(mark(z0)), MARK(z0))
   MARK(cons(z0, z1)) -> c8(MARK(z0))
   MARK(s(z0)) -> c9(MARK(z0))
   MARK(cons1(z0, z1)) -> c10(MARK(z0))
   MARK(cons1(z0, z1)) -> c11(MARK(z1))

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

   a__2nd(cons1(z0, cons(z1, z2))) -> mark(z1)
   a__2nd(cons(z0, z1)) -> a__2nd(cons1(mark(z0), mark(z1)))
   a__2nd(z0) -> 2nd(z0)
   a__from(z0) -> cons(mark(z0), from(s(z0)))
   a__from(z0) -> from(z0)
   mark(2nd(z0)) -> a__2nd(mark(z0))
   mark(from(z0)) -> a__from(mark(z0))
   mark(cons(z0, z1)) -> cons(mark(z0), z1)
   mark(s(z0)) -> s(mark(z0))
   mark(cons1(z0, z1)) -> cons1(mark(z0), mark(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__2ND(cons1(z0, cons(z1, z2))) -> c(MARK(z1))
   A__2ND(cons(z0, z1)) -> c1(A__2ND(cons1(mark(z0), mark(z1))), MARK(z0))
   A__2ND(cons(z0, z1)) -> c2(A__2ND(cons1(mark(z0), mark(z1))), MARK(z1))
   A__2ND(z0) -> c3
   A__FROM(z0) -> c4(MARK(z0))
   A__FROM(z0) -> c5
   MARK(2nd(z0)) -> c6(A__2ND(mark(z0)), MARK(z0))
   MARK(from(z0)) -> c7(A__FROM(mark(z0)), MARK(z0))
   MARK(cons(z0, z1)) -> c8(MARK(z0))
   MARK(s(z0)) -> c9(MARK(z0))
   MARK(cons1(z0, z1)) -> c10(MARK(z0))
   MARK(cons1(z0, z1)) -> c11(MARK(z1))

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

   a__2nd(cons1(z0, cons(z1, z2))) -> mark(z1)
   a__2nd(cons(z0, z1)) -> a__2nd(cons1(mark(z0), mark(z1)))
   a__2nd(z0) -> 2nd(z0)
   a__from(z0) -> cons(mark(z0), from(s(z0)))
   a__from(z0) -> from(z0)
   mark(2nd(z0)) -> a__2nd(mark(z0))
   mark(from(z0)) -> a__from(mark(z0))
   mark(cons(z0, z1)) -> cons(mark(z0), z1)
   mark(s(z0)) -> s(mark(z0))
   mark(cons1(z0, z1)) -> cons1(mark(z0), mark(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(cons1(z0, z1)) ->^+ c11(MARK(z1))

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

The pumping substitution is [z1 / cons1(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__2ND(cons1(z0, cons(z1, z2))) -> c(MARK(z1))
   A__2ND(cons(z0, z1)) -> c1(A__2ND(cons1(mark(z0), mark(z1))), MARK(z0))
   A__2ND(cons(z0, z1)) -> c2(A__2ND(cons1(mark(z0), mark(z1))), MARK(z1))
   A__2ND(z0) -> c3
   A__FROM(z0) -> c4(MARK(z0))
   A__FROM(z0) -> c5
   MARK(2nd(z0)) -> c6(A__2ND(mark(z0)), MARK(z0))
   MARK(from(z0)) -> c7(A__FROM(mark(z0)), MARK(z0))
   MARK(cons(z0, z1)) -> c8(MARK(z0))
   MARK(s(z0)) -> c9(MARK(z0))
   MARK(cons1(z0, z1)) -> c10(MARK(z0))
   MARK(cons1(z0, z1)) -> c11(MARK(z1))

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

   a__2nd(cons1(z0, cons(z1, z2))) -> mark(z1)
   a__2nd(cons(z0, z1)) -> a__2nd(cons1(mark(z0), mark(z1)))
   a__2nd(z0) -> 2nd(z0)
   a__from(z0) -> cons(mark(z0), from(s(z0)))
   a__from(z0) -> from(z0)
   mark(2nd(z0)) -> a__2nd(mark(z0))
   mark(from(z0)) -> a__from(mark(z0))
   mark(cons(z0, z1)) -> cons(mark(z0), z1)
   mark(s(z0)) -> s(mark(z0))
   mark(cons1(z0, z1)) -> cons1(mark(z0), mark(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__2ND(cons1(z0, cons(z1, z2))) -> c(MARK(z1))
   A__2ND(cons(z0, z1)) -> c1(A__2ND(cons1(mark(z0), mark(z1))), MARK(z0))
   A__2ND(cons(z0, z1)) -> c2(A__2ND(cons1(mark(z0), mark(z1))), MARK(z1))
   A__2ND(z0) -> c3
   A__FROM(z0) -> c4(MARK(z0))
   A__FROM(z0) -> c5
   MARK(2nd(z0)) -> c6(A__2ND(mark(z0)), MARK(z0))
   MARK(from(z0)) -> c7(A__FROM(mark(z0)), MARK(z0))
   MARK(cons(z0, z1)) -> c8(MARK(z0))
   MARK(s(z0)) -> c9(MARK(z0))
   MARK(cons1(z0, z1)) -> c10(MARK(z0))
   MARK(cons1(z0, z1)) -> c11(MARK(z1))

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

   a__2nd(cons1(z0, cons(z1, z2))) -> mark(z1)
   a__2nd(cons(z0, z1)) -> a__2nd(cons1(mark(z0), mark(z1)))
   a__2nd(z0) -> 2nd(z0)
   a__from(z0) -> cons(mark(z0), from(s(z0)))
   a__from(z0) -> from(z0)
   mark(2nd(z0)) -> a__2nd(mark(z0))
   mark(from(z0)) -> a__from(mark(z0))
   mark(cons(z0, z1)) -> cons(mark(z0), z1)
   mark(s(z0)) -> s(mark(z0))
   mark(cons1(z0, z1)) -> cons1(mark(z0), mark(z1))

Rewrite Strategy: INNERMOST