/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 228 ms]
(4) CpxRelTRS
(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(6) TRS for Loop Detection
(7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
(8) BEST
    (9) proven lower bound
        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
        (11) BOUNDS(n^1, INF)
    (12) TRS for Loop Detection


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(0)
Obligation:
The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   a__f(a, b, X) -> a__f(mark(X), X, mark(X))
   a__c -> a
   a__c -> b
   mark(f(X1, X2, X3)) -> a__f(mark(X1), X2, mark(X3))
   mark(c) -> a__c
   mark(a) -> a
   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) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID))
The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem:

   encArg(a) -> a
   encArg(b) -> b
   encArg(f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(c) -> c
   encArg(cons_a__f(x_1, x_2, x_3)) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_a__c) -> a__c
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__f(x_1, x_2, x_3) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_a -> a
   encode_b -> b
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_a__c -> a__c
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_c -> c


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

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

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

   encArg(a) -> a
   encArg(b) -> b
   encArg(f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(c) -> c
   encArg(cons_a__f(x_1, x_2, x_3)) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_a__c) -> a__c
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__f(x_1, x_2, x_3) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_a -> a
   encode_b -> b
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_a__c -> a__c
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_c -> c

Rewrite Strategy: INNERMOST
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(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(4)
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__f(a, b, X) -> a__f(mark(X), X, mark(X))
   a__c -> a
   a__c -> b
   mark(f(X1, X2, X3)) -> a__f(mark(X1), X2, mark(X3))
   mark(c) -> a__c
   mark(a) -> a
   mark(b) -> b
   a__f(X1, X2, X3) -> f(X1, X2, X3)
   a__c -> c

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

   encArg(a) -> a
   encArg(b) -> b
   encArg(f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(c) -> c
   encArg(cons_a__f(x_1, x_2, x_3)) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_a__c) -> a__c
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__f(x_1, x_2, x_3) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_a -> a
   encode_b -> b
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_a__c -> a__c
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_c -> c

Rewrite Strategy: INNERMOST
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(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(6)
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__f(a, b, X) -> a__f(mark(X), X, mark(X))
   a__c -> a
   a__c -> b
   mark(f(X1, X2, X3)) -> a__f(mark(X1), X2, mark(X3))
   mark(c) -> a__c
   mark(a) -> a
   mark(b) -> b
   a__f(X1, X2, X3) -> f(X1, X2, X3)
   a__c -> c

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

   encArg(a) -> a
   encArg(b) -> b
   encArg(f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(c) -> c
   encArg(cons_a__f(x_1, x_2, x_3)) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_a__c) -> a__c
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__f(x_1, x_2, x_3) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_a -> a
   encode_b -> b
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_a__c -> a__c
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_c -> c

Rewrite Strategy: INNERMOST
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(7) 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(mark(X1), X2, mark(X3))

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

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

The result substitution is [ ].




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(8)
Complex Obligation (BEST)

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

(9)
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__f(a, b, X) -> a__f(mark(X), X, mark(X))
   a__c -> a
   a__c -> b
   mark(f(X1, X2, X3)) -> a__f(mark(X1), X2, mark(X3))
   mark(c) -> a__c
   mark(a) -> a
   mark(b) -> b
   a__f(X1, X2, X3) -> f(X1, X2, X3)
   a__c -> c

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

   encArg(a) -> a
   encArg(b) -> b
   encArg(f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(c) -> c
   encArg(cons_a__f(x_1, x_2, x_3)) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_a__c) -> a__c
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__f(x_1, x_2, x_3) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_a -> a
   encode_b -> b
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_a__c -> a__c
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_c -> c

Rewrite Strategy: INNERMOST
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(10) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(11)
BOUNDS(n^1, INF)

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

(12)
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__f(a, b, X) -> a__f(mark(X), X, mark(X))
   a__c -> a
   a__c -> b
   mark(f(X1, X2, X3)) -> a__f(mark(X1), X2, mark(X3))
   mark(c) -> a__c
   mark(a) -> a
   mark(b) -> b
   a__f(X1, X2, X3) -> f(X1, X2, X3)
   a__c -> c

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

   encArg(a) -> a
   encArg(b) -> b
   encArg(f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(c) -> c
   encArg(cons_a__f(x_1, x_2, x_3)) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_a__c) -> a__c
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__f(x_1, x_2, x_3) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_a -> a
   encode_b -> b
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_a__c -> a__c
   encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_c -> c

Rewrite Strategy: INNERMOST