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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 243 ms]
(2) CpxRelTRS
(3) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(4) CdtProblem
    (5) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CdtProblem
    (9) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 38 ms]
    (10) CdtProblem
    (11) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) BOUNDS(1, 1)
(13) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 5 ms]
(14) TRS for Loop Detection
    (15) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (16) BEST
        (17) proven lower bound
            (18) LowerBoundPropagationProof [FINISHED, 0 ms]
            (19) BOUNDS(n^1, INF)
        (20) 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, n^1).


The TRS R consists of the following rules:

   *'(z0, +(z1, z2)) -> c(*'(z0, z1))
   *'(z0, +(z1, z2)) -> c1(*'(z0, z2))
   *'(+(z0, z1), z2) -> c2(*'(z0, z2))
   *'(+(z0, z1), z2) -> c3(*'(z1, z2))
   *'(z0, 1) -> c4
   *'(1, z0) -> c5

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

   *(z0, +(z1, z2)) -> +(*(z0, z1), *(z0, z2))
   *(+(z0, z1), z2) -> +(*(z0, z2), *(z1, z2))
   *(z0, 1) -> z0
   *(1, z0) -> z0

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, n^1).


The TRS R consists of the following rules:

   *'(z0, +(z1, z2)) -> c(*'(z0, z1))
   *'(z0, +(z1, z2)) -> c1(*'(z0, z2))
   *'(+(z0, z1), z2) -> c2(*'(z0, z2))
   *'(+(z0, z1), z2) -> c3(*'(z1, z2))
   *'(z0, 1) -> c4
   *'(1, z0) -> c5

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

   *(z0, +(z1, z2)) -> +(*(z0, z1), *(z0, z2))
   *(+(z0, z1), z2) -> +(*(z0, z2), *(z1, z2))
   *(z0, 1) -> z0
   *(1, z0) -> z0

Rewrite Strategy: INNERMOST
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(3) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS to CDT
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(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, +(z1, z2)) -> +(*(z0, z1), *(z0, z2))
   *(+(z0, z1), z2) -> +(*(z0, z2), *(z1, z2))
   *(z0, 1) -> z0
   *(1, z0) -> z0
   *'(z0, +(z1, z2)) -> c(*'(z0, z1))
   *'(z0, +(z1, z2)) -> c1(*'(z0, z2))
   *'(+(z0, z1), z2) -> c2(*'(z0, z2))
   *'(+(z0, z1), z2) -> c3(*'(z1, z2))
   *'(z0, 1) -> c4
   *'(1, z0) -> c5
Tuples:
   *''(z0, +(z1, z2)) -> c6(*''(z0, z1), *''(z0, z2))
   *''(+(z0, z1), z2) -> c7(*''(z0, z2), *''(z1, z2))
   *''(z0, 1) -> c8
   *''(1, z0) -> c9
   *'''(z0, +(z1, z2)) -> c10(*'''(z0, z1))
   *'''(z0, +(z1, z2)) -> c11(*'''(z0, z2))
   *'''(+(z0, z1), z2) -> c12(*'''(z0, z2))
   *'''(+(z0, z1), z2) -> c13(*'''(z1, z2))
   *'''(z0, 1) -> c14
   *'''(1, z0) -> c15
S tuples:
   *'''(z0, +(z1, z2)) -> c10(*'''(z0, z1))
   *'''(z0, +(z1, z2)) -> c11(*'''(z0, z2))
   *'''(+(z0, z1), z2) -> c12(*'''(z0, z2))
   *'''(+(z0, z1), z2) -> c13(*'''(z1, z2))
   *'''(z0, 1) -> c14
   *'''(1, z0) -> c15
K tuples:none
Defined Rule Symbols:   *'_2, *_2

Defined Pair Symbols:   *''_2, *'''_2

Compound Symbols:   c6_2, c7_2, c8, c9, c10_1, c11_1, c12_1, c13_1, c14, c15


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(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing nodes:
   *'''(z0, 1) -> c14
   *''(z0, 1) -> c8
   *'''(1, z0) -> c15
   *''(1, z0) -> c9

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(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, +(z1, z2)) -> +(*(z0, z1), *(z0, z2))
   *(+(z0, z1), z2) -> +(*(z0, z2), *(z1, z2))
   *(z0, 1) -> z0
   *(1, z0) -> z0
   *'(z0, +(z1, z2)) -> c(*'(z0, z1))
   *'(z0, +(z1, z2)) -> c1(*'(z0, z2))
   *'(+(z0, z1), z2) -> c2(*'(z0, z2))
   *'(+(z0, z1), z2) -> c3(*'(z1, z2))
   *'(z0, 1) -> c4
   *'(1, z0) -> c5
Tuples:
   *''(z0, +(z1, z2)) -> c6(*''(z0, z1), *''(z0, z2))
   *''(+(z0, z1), z2) -> c7(*''(z0, z2), *''(z1, z2))
   *'''(z0, +(z1, z2)) -> c10(*'''(z0, z1))
   *'''(z0, +(z1, z2)) -> c11(*'''(z0, z2))
   *'''(+(z0, z1), z2) -> c12(*'''(z0, z2))
   *'''(+(z0, z1), z2) -> c13(*'''(z1, z2))
S tuples:
   *'''(z0, +(z1, z2)) -> c10(*'''(z0, z1))
   *'''(z0, +(z1, z2)) -> c11(*'''(z0, z2))
   *'''(+(z0, z1), z2) -> c12(*'''(z0, z2))
   *'''(+(z0, z1), z2) -> c13(*'''(z1, z2))
K tuples:none
Defined Rule Symbols:   *'_2, *_2

Defined Pair Symbols:   *''_2, *'''_2

Compound Symbols:   c6_2, c7_2, c10_1, c11_1, c12_1, c13_1


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(7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   *(z0, +(z1, z2)) -> +(*(z0, z1), *(z0, z2))
   *(+(z0, z1), z2) -> +(*(z0, z2), *(z1, z2))
   *(z0, 1) -> z0
   *(1, z0) -> z0
   *'(z0, +(z1, z2)) -> c(*'(z0, z1))
   *'(z0, +(z1, z2)) -> c1(*'(z0, z2))
   *'(+(z0, z1), z2) -> c2(*'(z0, z2))
   *'(+(z0, z1), z2) -> c3(*'(z1, z2))
   *'(z0, 1) -> c4
   *'(1, z0) -> c5

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(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   *''(z0, +(z1, z2)) -> c6(*''(z0, z1), *''(z0, z2))
   *''(+(z0, z1), z2) -> c7(*''(z0, z2), *''(z1, z2))
   *'''(z0, +(z1, z2)) -> c10(*'''(z0, z1))
   *'''(z0, +(z1, z2)) -> c11(*'''(z0, z2))
   *'''(+(z0, z1), z2) -> c12(*'''(z0, z2))
   *'''(+(z0, z1), z2) -> c13(*'''(z1, z2))
S tuples:
   *'''(z0, +(z1, z2)) -> c10(*'''(z0, z1))
   *'''(z0, +(z1, z2)) -> c11(*'''(z0, z2))
   *'''(+(z0, z1), z2) -> c12(*'''(z0, z2))
   *'''(+(z0, z1), z2) -> c13(*'''(z1, z2))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   *''_2, *'''_2

Compound Symbols:   c6_2, c7_2, c10_1, c11_1, c12_1, c13_1


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(9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   *'''(z0, +(z1, z2)) -> c10(*'''(z0, z1))
   *'''(z0, +(z1, z2)) -> c11(*'''(z0, z2))
   *'''(+(z0, z1), z2) -> c12(*'''(z0, z2))
   *'''(+(z0, z1), z2) -> c13(*'''(z1, z2))
We considered the (Usable) Rules:none
And the Tuples:
   *''(z0, +(z1, z2)) -> c6(*''(z0, z1), *''(z0, z2))
   *''(+(z0, z1), z2) -> c7(*''(z0, z2), *''(z1, z2))
   *'''(z0, +(z1, z2)) -> c10(*'''(z0, z1))
   *'''(z0, +(z1, z2)) -> c11(*'''(z0, z2))
   *'''(+(z0, z1), z2) -> c12(*'''(z0, z2))
   *'''(+(z0, z1), z2) -> c13(*'''(z1, z2))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(*''(x_1, x_2)) = 0
   POL(*'''(x_1, x_2)) = x_1 + x_2
   POL(+(x_1, x_2)) = [1] + x_1 + x_2
   POL(c10(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c12(x_1)) = x_1
   POL(c13(x_1)) = x_1
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1, x_2)) = x_1 + x_2

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(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   *''(z0, +(z1, z2)) -> c6(*''(z0, z1), *''(z0, z2))
   *''(+(z0, z1), z2) -> c7(*''(z0, z2), *''(z1, z2))
   *'''(z0, +(z1, z2)) -> c10(*'''(z0, z1))
   *'''(z0, +(z1, z2)) -> c11(*'''(z0, z2))
   *'''(+(z0, z1), z2) -> c12(*'''(z0, z2))
   *'''(+(z0, z1), z2) -> c13(*'''(z1, z2))
S tuples:none
K tuples:
   *'''(z0, +(z1, z2)) -> c10(*'''(z0, z1))
   *'''(z0, +(z1, z2)) -> c11(*'''(z0, z2))
   *'''(+(z0, z1), z2) -> c12(*'''(z0, z2))
   *'''(+(z0, z1), z2) -> c13(*'''(z1, z2))
Defined Rule Symbols:none

Defined Pair Symbols:   *''_2, *'''_2

Compound Symbols:   c6_2, c7_2, c10_1, c11_1, c12_1, c13_1


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(11) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
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(12)
BOUNDS(1, 1)

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(13) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(14)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   *'(z0, +(z1, z2)) -> c(*'(z0, z1))
   *'(z0, +(z1, z2)) -> c1(*'(z0, z2))
   *'(+(z0, z1), z2) -> c2(*'(z0, z2))
   *'(+(z0, z1), z2) -> c3(*'(z1, z2))
   *'(z0, 1) -> c4
   *'(1, z0) -> c5

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

   *(z0, +(z1, z2)) -> +(*(z0, z1), *(z0, z2))
   *(+(z0, z1), z2) -> +(*(z0, z2), *(z1, z2))
   *(z0, 1) -> z0
   *(1, z0) -> z0

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

The rewrite sequence

*'(z0, +(z1, z2)) ->^+ c(*'(z0, z1))

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

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

The result substitution is [ ].




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(16)
Complex Obligation (BEST)

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(17)
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, n^1).


The TRS R consists of the following rules:

   *'(z0, +(z1, z2)) -> c(*'(z0, z1))
   *'(z0, +(z1, z2)) -> c1(*'(z0, z2))
   *'(+(z0, z1), z2) -> c2(*'(z0, z2))
   *'(+(z0, z1), z2) -> c3(*'(z1, z2))
   *'(z0, 1) -> c4
   *'(1, z0) -> c5

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

   *(z0, +(z1, z2)) -> +(*(z0, z1), *(z0, z2))
   *(+(z0, z1), z2) -> +(*(z0, z2), *(z1, z2))
   *(z0, 1) -> z0
   *(1, z0) -> z0

Rewrite Strategy: INNERMOST
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(18) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(19)
BOUNDS(n^1, INF)

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(20)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   *'(z0, +(z1, z2)) -> c(*'(z0, z1))
   *'(z0, +(z1, z2)) -> c1(*'(z0, z2))
   *'(+(z0, z1), z2) -> c2(*'(z0, z2))
   *'(+(z0, z1), z2) -> c3(*'(z1, z2))
   *'(z0, 1) -> c4
   *'(1, z0) -> c5

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

   *(z0, +(z1, z2)) -> +(*(z0, z1), *(z0, z2))
   *(+(z0, z1), z2) -> +(*(z0, z2), *(z1, z2))
   *(z0, 1) -> z0
   *(1, z0) -> z0

Rewrite Strategy: INNERMOST