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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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), 203 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__U11(tt, z0, z1) -> c(A__U12(tt, z0, z1))
   A__U11(z0, z1, z2) -> c1
   A__U12(tt, z0, z1) -> c2(A__PLUS(mark(z1), mark(z0)), MARK(z1))
   A__U12(tt, z0, z1) -> c3(A__PLUS(mark(z1), mark(z0)), MARK(z0))
   A__U12(z0, z1, z2) -> c4
   A__PLUS(z0, 0) -> c5(MARK(z0))
   A__PLUS(z0, s(z1)) -> c6(A__U11(tt, z1, z0))
   A__PLUS(z0, z1) -> c7
   MARK(U11(z0, z1, z2)) -> c8(A__U11(mark(z0), z1, z2), MARK(z0))
   MARK(U12(z0, z1, z2)) -> c9(A__U12(mark(z0), z1, z2), MARK(z0))
   MARK(plus(z0, z1)) -> c10(A__PLUS(mark(z0), mark(z1)), MARK(z0))
   MARK(plus(z0, z1)) -> c11(A__PLUS(mark(z0), mark(z1)), MARK(z1))
   MARK(tt) -> c12
   MARK(s(z0)) -> c13(MARK(z0))
   MARK(0) -> c14

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

   a__U11(tt, z0, z1) -> a__U12(tt, z0, z1)
   a__U11(z0, z1, z2) -> U11(z0, z1, z2)
   a__U12(tt, z0, z1) -> s(a__plus(mark(z1), mark(z0)))
   a__U12(z0, z1, z2) -> U12(z0, z1, z2)
   a__plus(z0, 0) -> mark(z0)
   a__plus(z0, s(z1)) -> a__U11(tt, z1, z0)
   a__plus(z0, z1) -> plus(z0, z1)
   mark(U11(z0, z1, z2)) -> a__U11(mark(z0), z1, z2)
   mark(U12(z0, z1, z2)) -> a__U12(mark(z0), z1, z2)
   mark(plus(z0, z1)) -> a__plus(mark(z0), mark(z1))
   mark(tt) -> tt
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0

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__U11(tt, z0, z1) -> c(A__U12(tt, z0, z1))
   A__U11(z0, z1, z2) -> c1
   A__U12(tt, z0, z1) -> c2(A__PLUS(mark(z1), mark(z0)), MARK(z1))
   A__U12(tt, z0, z1) -> c3(A__PLUS(mark(z1), mark(z0)), MARK(z0))
   A__U12(z0, z1, z2) -> c4
   A__PLUS(z0, 0) -> c5(MARK(z0))
   A__PLUS(z0, s(z1)) -> c6(A__U11(tt, z1, z0))
   A__PLUS(z0, z1) -> c7
   MARK(U11(z0, z1, z2)) -> c8(A__U11(mark(z0), z1, z2), MARK(z0))
   MARK(U12(z0, z1, z2)) -> c9(A__U12(mark(z0), z1, z2), MARK(z0))
   MARK(plus(z0, z1)) -> c10(A__PLUS(mark(z0), mark(z1)), MARK(z0))
   MARK(plus(z0, z1)) -> c11(A__PLUS(mark(z0), mark(z1)), MARK(z1))
   MARK(tt) -> c12
   MARK(s(z0)) -> c13(MARK(z0))
   MARK(0) -> c14

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

   a__U11(tt, z0, z1) -> a__U12(tt, z0, z1)
   a__U11(z0, z1, z2) -> U11(z0, z1, z2)
   a__U12(tt, z0, z1) -> s(a__plus(mark(z1), mark(z0)))
   a__U12(z0, z1, z2) -> U12(z0, z1, z2)
   a__plus(z0, 0) -> mark(z0)
   a__plus(z0, s(z1)) -> a__U11(tt, z1, z0)
   a__plus(z0, z1) -> plus(z0, z1)
   mark(U11(z0, z1, z2)) -> a__U11(mark(z0), z1, z2)
   mark(U12(z0, z1, z2)) -> a__U12(mark(z0), z1, z2)
   mark(plus(z0, z1)) -> a__plus(mark(z0), mark(z1))
   mark(tt) -> tt
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0

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__U11(tt, z0, z1) -> c(A__U12(tt, z0, z1))
   A__U11(z0, z1, z2) -> c1
   A__U12(tt, z0, z1) -> c2(A__PLUS(mark(z1), mark(z0)), MARK(z1))
   A__U12(tt, z0, z1) -> c3(A__PLUS(mark(z1), mark(z0)), MARK(z0))
   A__U12(z0, z1, z2) -> c4
   A__PLUS(z0, 0) -> c5(MARK(z0))
   A__PLUS(z0, s(z1)) -> c6(A__U11(tt, z1, z0))
   A__PLUS(z0, z1) -> c7
   MARK(U11(z0, z1, z2)) -> c8(A__U11(mark(z0), z1, z2), MARK(z0))
   MARK(U12(z0, z1, z2)) -> c9(A__U12(mark(z0), z1, z2), MARK(z0))
   MARK(plus(z0, z1)) -> c10(A__PLUS(mark(z0), mark(z1)), MARK(z0))
   MARK(plus(z0, z1)) -> c11(A__PLUS(mark(z0), mark(z1)), MARK(z1))
   MARK(tt) -> c12
   MARK(s(z0)) -> c13(MARK(z0))
   MARK(0) -> c14

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

   a__U11(tt, z0, z1) -> a__U12(tt, z0, z1)
   a__U11(z0, z1, z2) -> U11(z0, z1, z2)
   a__U12(tt, z0, z1) -> s(a__plus(mark(z1), mark(z0)))
   a__U12(z0, z1, z2) -> U12(z0, z1, z2)
   a__plus(z0, 0) -> mark(z0)
   a__plus(z0, s(z1)) -> a__U11(tt, z1, z0)
   a__plus(z0, z1) -> plus(z0, z1)
   mark(U11(z0, z1, z2)) -> a__U11(mark(z0), z1, z2)
   mark(U12(z0, z1, z2)) -> a__U12(mark(z0), z1, z2)
   mark(plus(z0, z1)) -> a__plus(mark(z0), mark(z1))
   mark(tt) -> tt
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0

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(U12(z0, z1, z2)) ->^+ c9(A__U12(mark(z0), z1, z2), MARK(z0))

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

The pumping substitution is [z0 / U12(z0, z1, z2)].

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__U11(tt, z0, z1) -> c(A__U12(tt, z0, z1))
   A__U11(z0, z1, z2) -> c1
   A__U12(tt, z0, z1) -> c2(A__PLUS(mark(z1), mark(z0)), MARK(z1))
   A__U12(tt, z0, z1) -> c3(A__PLUS(mark(z1), mark(z0)), MARK(z0))
   A__U12(z0, z1, z2) -> c4
   A__PLUS(z0, 0) -> c5(MARK(z0))
   A__PLUS(z0, s(z1)) -> c6(A__U11(tt, z1, z0))
   A__PLUS(z0, z1) -> c7
   MARK(U11(z0, z1, z2)) -> c8(A__U11(mark(z0), z1, z2), MARK(z0))
   MARK(U12(z0, z1, z2)) -> c9(A__U12(mark(z0), z1, z2), MARK(z0))
   MARK(plus(z0, z1)) -> c10(A__PLUS(mark(z0), mark(z1)), MARK(z0))
   MARK(plus(z0, z1)) -> c11(A__PLUS(mark(z0), mark(z1)), MARK(z1))
   MARK(tt) -> c12
   MARK(s(z0)) -> c13(MARK(z0))
   MARK(0) -> c14

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

   a__U11(tt, z0, z1) -> a__U12(tt, z0, z1)
   a__U11(z0, z1, z2) -> U11(z0, z1, z2)
   a__U12(tt, z0, z1) -> s(a__plus(mark(z1), mark(z0)))
   a__U12(z0, z1, z2) -> U12(z0, z1, z2)
   a__plus(z0, 0) -> mark(z0)
   a__plus(z0, s(z1)) -> a__U11(tt, z1, z0)
   a__plus(z0, z1) -> plus(z0, z1)
   mark(U11(z0, z1, z2)) -> a__U11(mark(z0), z1, z2)
   mark(U12(z0, z1, z2)) -> a__U12(mark(z0), z1, z2)
   mark(plus(z0, z1)) -> a__plus(mark(z0), mark(z1))
   mark(tt) -> tt
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0

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__U11(tt, z0, z1) -> c(A__U12(tt, z0, z1))
   A__U11(z0, z1, z2) -> c1
   A__U12(tt, z0, z1) -> c2(A__PLUS(mark(z1), mark(z0)), MARK(z1))
   A__U12(tt, z0, z1) -> c3(A__PLUS(mark(z1), mark(z0)), MARK(z0))
   A__U12(z0, z1, z2) -> c4
   A__PLUS(z0, 0) -> c5(MARK(z0))
   A__PLUS(z0, s(z1)) -> c6(A__U11(tt, z1, z0))
   A__PLUS(z0, z1) -> c7
   MARK(U11(z0, z1, z2)) -> c8(A__U11(mark(z0), z1, z2), MARK(z0))
   MARK(U12(z0, z1, z2)) -> c9(A__U12(mark(z0), z1, z2), MARK(z0))
   MARK(plus(z0, z1)) -> c10(A__PLUS(mark(z0), mark(z1)), MARK(z0))
   MARK(plus(z0, z1)) -> c11(A__PLUS(mark(z0), mark(z1)), MARK(z1))
   MARK(tt) -> c12
   MARK(s(z0)) -> c13(MARK(z0))
   MARK(0) -> c14

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

   a__U11(tt, z0, z1) -> a__U12(tt, z0, z1)
   a__U11(z0, z1, z2) -> U11(z0, z1, z2)
   a__U12(tt, z0, z1) -> s(a__plus(mark(z1), mark(z0)))
   a__U12(z0, z1, z2) -> U12(z0, z1, z2)
   a__plus(z0, 0) -> mark(z0)
   a__plus(z0, s(z1)) -> a__U11(tt, z1, z0)
   a__plus(z0, z1) -> plus(z0, z1)
   mark(U11(z0, z1, z2)) -> a__U11(mark(z0), z1, z2)
   mark(U12(z0, z1, z2)) -> a__U12(mark(z0), z1, z2)
   mark(plus(z0, z1)) -> a__plus(mark(z0), mark(z1))
   mark(tt) -> tt
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0

Rewrite Strategy: INNERMOST