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


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WORST_CASE(NON_POLY, ?)
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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(INF, INF).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 176 ms]
(4) CpxRelTRS
(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(6) TRS for Loop Detection
(7) InfiniteLowerBoundProof [FINISHED, 0 ms]
(8) BOUNDS(INF, INF)


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


The TRS R consists of the following rules:

   ap(ap(g, x), y) -> y
   ap(f, x) -> ap(f, app(g, x))

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(g) -> g
   encArg(f) -> f
   encArg(app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
   encArg(cons_ap(x_1, x_2)) -> ap(encArg(x_1), encArg(x_2))
   encode_ap(x_1, x_2) -> ap(encArg(x_1), encArg(x_2))
   encode_g -> g
   encode_f -> f
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))


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(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(INF, INF).


The TRS R consists of the following rules:

   ap(ap(g, x), y) -> y
   ap(f, x) -> ap(f, app(g, x))

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

   encArg(g) -> g
   encArg(f) -> f
   encArg(app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
   encArg(cons_ap(x_1, x_2)) -> ap(encArg(x_1), encArg(x_2))
   encode_ap(x_1, x_2) -> ap(encArg(x_1), encArg(x_2))
   encode_g -> g
   encode_f -> f
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))

Rewrite Strategy: INNERMOST
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(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
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(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(INF, INF).


The TRS R consists of the following rules:

   ap(ap(g, x), y) -> y
   ap(f, x) -> ap(f, app(g, x))

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

   encArg(g) -> g
   encArg(f) -> f
   encArg(app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
   encArg(cons_ap(x_1, x_2)) -> ap(encArg(x_1), encArg(x_2))
   encode_ap(x_1, x_2) -> ap(encArg(x_1), encArg(x_2))
   encode_g -> g
   encode_f -> f
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))

Rewrite Strategy: INNERMOST
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(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(6)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   ap(ap(g, x), y) -> y
   ap(f, x) -> ap(f, app(g, x))

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

   encArg(g) -> g
   encArg(f) -> f
   encArg(app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
   encArg(cons_ap(x_1, x_2)) -> ap(encArg(x_1), encArg(x_2))
   encode_ap(x_1, x_2) -> ap(encArg(x_1), encArg(x_2))
   encode_g -> g
   encode_f -> f
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))

Rewrite Strategy: INNERMOST
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(7) InfiniteLowerBoundProof (FINISHED)
The following loop proves infinite runtime complexity:

The rewrite sequence

ap(f, x) ->^+ ap(f, app(g, x))

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

The pumping substitution is [ ].

The result substitution is [x / app(g, x)].




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(8)
BOUNDS(INF, INF)