/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(NON_POLY, ?)
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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(EXP, INF).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 258 ms]
(4) CpxRelTRS
(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(6) TRS for Loop Detection
(7) DecreasingLoopProof [FINISHED, 0 ms]
(8) BOUNDS(EXP, INF)


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


The TRS R consists of the following rules:

   merge(nil, y) -> y
   merge(x, nil) -> x
   merge(.(x, y), .(u, v)) -> if(<(x, u), .(x, merge(y, .(u, v))), .(u, merge(.(x, y), v)))
   ++(nil, y) -> y
   ++(.(x, y), z) -> .(x, ++(y, z))
   if(true, x, y) -> x
   if(false, x, y) -> 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(nil) -> nil
   encArg(.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
   encArg(<(x_1, x_2)) -> <(encArg(x_1), encArg(x_2))
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_merge(x_1, x_2)) -> merge(encArg(x_1), encArg(x_2))
   encArg(cons_++(x_1, x_2)) -> ++(encArg(x_1), encArg(x_2))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_merge(x_1, x_2) -> merge(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_<(x_1, x_2) -> <(encArg(x_1), encArg(x_2))
   encode_++(x_1, x_2) -> ++(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_false -> false


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


The TRS R consists of the following rules:

   merge(nil, y) -> y
   merge(x, nil) -> x
   merge(.(x, y), .(u, v)) -> if(<(x, u), .(x, merge(y, .(u, v))), .(u, merge(.(x, y), v)))
   ++(nil, y) -> y
   ++(.(x, y), z) -> .(x, ++(y, z))
   if(true, x, y) -> x
   if(false, x, y) -> x

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

   encArg(nil) -> nil
   encArg(.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
   encArg(<(x_1, x_2)) -> <(encArg(x_1), encArg(x_2))
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_merge(x_1, x_2)) -> merge(encArg(x_1), encArg(x_2))
   encArg(cons_++(x_1, x_2)) -> ++(encArg(x_1), encArg(x_2))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_merge(x_1, x_2) -> merge(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_<(x_1, x_2) -> <(encArg(x_1), encArg(x_2))
   encode_++(x_1, x_2) -> ++(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_false -> false

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(EXP, INF).


The TRS R consists of the following rules:

   merge(nil, y) -> y
   merge(x, nil) -> x
   merge(.(x, y), .(u, v)) -> if(<(x, u), .(x, merge(y, .(u, v))), .(u, merge(.(x, y), v)))
   ++(nil, y) -> y
   ++(.(x, y), z) -> .(x, ++(y, z))
   if(true, x, y) -> x
   if(false, x, y) -> x

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

   encArg(nil) -> nil
   encArg(.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
   encArg(<(x_1, x_2)) -> <(encArg(x_1), encArg(x_2))
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_merge(x_1, x_2)) -> merge(encArg(x_1), encArg(x_2))
   encArg(cons_++(x_1, x_2)) -> ++(encArg(x_1), encArg(x_2))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_merge(x_1, x_2) -> merge(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_<(x_1, x_2) -> <(encArg(x_1), encArg(x_2))
   encode_++(x_1, x_2) -> ++(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_false -> false

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(EXP, INF).


The TRS R consists of the following rules:

   merge(nil, y) -> y
   merge(x, nil) -> x
   merge(.(x, y), .(u, v)) -> if(<(x, u), .(x, merge(y, .(u, v))), .(u, merge(.(x, y), v)))
   ++(nil, y) -> y
   ++(.(x, y), z) -> .(x, ++(y, z))
   if(true, x, y) -> x
   if(false, x, y) -> x

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

   encArg(nil) -> nil
   encArg(.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
   encArg(<(x_1, x_2)) -> <(encArg(x_1), encArg(x_2))
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_merge(x_1, x_2)) -> merge(encArg(x_1), encArg(x_2))
   encArg(cons_++(x_1, x_2)) -> ++(encArg(x_1), encArg(x_2))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_merge(x_1, x_2) -> merge(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_<(x_1, x_2) -> <(encArg(x_1), encArg(x_2))
   encode_++(x_1, x_2) -> ++(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_false -> false

Rewrite Strategy: INNERMOST
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(7) DecreasingLoopProof (FINISHED)
The following loop(s) give(s) rise to the lower bound EXP:

The rewrite sequence

merge(.(x, y), .(u, v)) ->^+ if(<(x, u), .(x, merge(y, .(u, v))), .(u, merge(.(x, y), v)))

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

The pumping substitution is [y / .(x, y)].

The result substitution is [ ].



The rewrite sequence

merge(.(x, y), .(u, v)) ->^+ if(<(x, u), .(x, merge(y, .(u, v))), .(u, merge(.(x, y), v)))

gives rise to a decreasing loop by considering the right hand sides subterm at position [2,1].

The pumping substitution is [v / .(u, v)].

The result substitution is [ ].




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(8)
BOUNDS(EXP, INF)