/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
    (3) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CdtProblem
    (5) CdtRhsSimplificationProcessorProof [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)), 25 ms]
    (10) CdtProblem
    (11) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) BOUNDS(1, 1)
(13) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 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 CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   a__f(b, X, c) -> a__f(X, a__c, X)
   a__c -> b
   mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
   mark(c) -> a__c
   mark(b) -> b
   a__f(X1, X2, X3) -> f(X1, X2, X3)
   a__c -> c

S is empty.
Rewrite Strategy: INNERMOST
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(1) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS to CDT
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(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__f(b, z0, c) -> a__f(z0, a__c, z0)
   a__f(z0, z1, z2) -> f(z0, z1, z2)
   a__c -> b
   a__c -> c
   mark(f(z0, z1, z2)) -> a__f(z0, mark(z1), z2)
   mark(c) -> a__c
   mark(b) -> b
Tuples:
   A__F(b, z0, c) -> c1(A__F(z0, a__c, z0), A__C)
   A__F(z0, z1, z2) -> c2
   A__C -> c3
   A__C -> c4
   MARK(f(z0, z1, z2)) -> c5(A__F(z0, mark(z1), z2), MARK(z1))
   MARK(c) -> c6(A__C)
   MARK(b) -> c7
S tuples:
   A__F(b, z0, c) -> c1(A__F(z0, a__c, z0), A__C)
   A__F(z0, z1, z2) -> c2
   A__C -> c3
   A__C -> c4
   MARK(f(z0, z1, z2)) -> c5(A__F(z0, mark(z1), z2), MARK(z1))
   MARK(c) -> c6(A__C)
   MARK(b) -> c7
K tuples:none
Defined Rule Symbols:   a__f_3, a__c, mark_1

Defined Pair Symbols:   A__F_3, A__C, MARK_1

Compound Symbols:   c1_2, c2, c3, c4, c5_2, c6_1, c7


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(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 6 trailing nodes:
   A__C -> c3
   A__F(b, z0, c) -> c1(A__F(z0, a__c, z0), A__C)
   MARK(c) -> c6(A__C)
   A__C -> c4
   A__F(z0, z1, z2) -> c2
   MARK(b) -> c7

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(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__f(b, z0, c) -> a__f(z0, a__c, z0)
   a__f(z0, z1, z2) -> f(z0, z1, z2)
   a__c -> b
   a__c -> c
   mark(f(z0, z1, z2)) -> a__f(z0, mark(z1), z2)
   mark(c) -> a__c
   mark(b) -> b
Tuples:
   MARK(f(z0, z1, z2)) -> c5(A__F(z0, mark(z1), z2), MARK(z1))
S tuples:
   MARK(f(z0, z1, z2)) -> c5(A__F(z0, mark(z1), z2), MARK(z1))
K tuples:none
Defined Rule Symbols:   a__f_3, a__c, mark_1

Defined Pair Symbols:   MARK_1

Compound Symbols:   c5_2


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(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
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(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__f(b, z0, c) -> a__f(z0, a__c, z0)
   a__f(z0, z1, z2) -> f(z0, z1, z2)
   a__c -> b
   a__c -> c
   mark(f(z0, z1, z2)) -> a__f(z0, mark(z1), z2)
   mark(c) -> a__c
   mark(b) -> b
Tuples:
   MARK(f(z0, z1, z2)) -> c5(MARK(z1))
S tuples:
   MARK(f(z0, z1, z2)) -> c5(MARK(z1))
K tuples:none
Defined Rule Symbols:   a__f_3, a__c, mark_1

Defined Pair Symbols:   MARK_1

Compound Symbols:   c5_1


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(7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   a__f(b, z0, c) -> a__f(z0, a__c, z0)
   a__f(z0, z1, z2) -> f(z0, z1, z2)
   a__c -> b
   a__c -> c
   mark(f(z0, z1, z2)) -> a__f(z0, mark(z1), z2)
   mark(c) -> a__c
   mark(b) -> b

----------------------------------------

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   MARK(f(z0, z1, z2)) -> c5(MARK(z1))
S tuples:
   MARK(f(z0, z1, z2)) -> c5(MARK(z1))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   MARK_1

Compound Symbols:   c5_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.
   MARK(f(z0, z1, z2)) -> c5(MARK(z1))
We considered the (Usable) Rules:none
And the Tuples:
   MARK(f(z0, z1, z2)) -> c5(MARK(z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(MARK(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(f(x_1, x_2, x_3)) = [1] + x_2

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(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   MARK(f(z0, z1, z2)) -> c5(MARK(z1))
S tuples:none
K tuples:
   MARK(f(z0, z1, z2)) -> c5(MARK(z1))
Defined Rule Symbols:none

Defined Pair Symbols:   MARK_1

Compound Symbols:   c5_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 CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   a__f(b, X, c) -> a__f(X, a__c, X)
   a__c -> b
   mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
   mark(c) -> a__c
   mark(b) -> b
   a__f(X1, X2, X3) -> f(X1, X2, X3)
   a__c -> c

S is empty.
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

mark(f(X1, X2, X3)) ->^+ a__f(X1, mark(X2), X3)

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

The pumping substitution is [X2 / f(X1, X2, X3)].

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 CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   a__f(b, X, c) -> a__f(X, a__c, X)
   a__c -> b
   mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
   mark(c) -> a__c
   mark(b) -> b
   a__f(X1, X2, X3) -> f(X1, X2, X3)
   a__c -> c

S is empty.
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 CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   a__f(b, X, c) -> a__f(X, a__c, X)
   a__c -> b
   mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
   mark(c) -> a__c
   mark(b) -> b
   a__f(X1, X2, X3) -> f(X1, X2, X3)
   a__c -> c

S is empty.
Rewrite Strategy: INNERMOST