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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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

(0) CpxTRS
(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(2) TRS for Loop Detection
(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
(4) BEST
    (5) proven lower bound
        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
        (7) BOUNDS(n^1, INF)
    (8) TRS for Loop Detection
        (9) DecreasingLoopProof [FINISHED, 25 ms]
        (10) BOUNDS(EXP, INF)


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


The TRS R consists of the following rules:

   f(S(x'), S(x)) -> h(g(x', S(x)), f(S(S(S(x'))), x))
   g(S(x), S(x')) -> h(f(S(x), S(x')), g(x, S(S(S(x')))))
   h(0, S(x)) -> h(0, x)
   h(0, 0) -> 0
   g(S(x), 0) -> 0
   f(S(x), 0) -> 0
   h(S(x), x2) -> h(x, x2)
   g(0, x2) -> 0
   f(0, x2) -> 0

S is empty.
Rewrite Strategy: INNERMOST
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(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(2)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   f(S(x'), S(x)) -> h(g(x', S(x)), f(S(S(S(x'))), x))
   g(S(x), S(x')) -> h(f(S(x), S(x')), g(x, S(S(S(x')))))
   h(0, S(x)) -> h(0, x)
   h(0, 0) -> 0
   g(S(x), 0) -> 0
   f(S(x), 0) -> 0
   h(S(x), x2) -> h(x, x2)
   g(0, x2) -> 0
   f(0, x2) -> 0

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

The rewrite sequence

f(S(x'), S(x)) ->^+ h(g(x', S(x)), f(S(S(S(x'))), x))

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

The pumping substitution is [x / S(x)].

The result substitution is [x' / S(S(x'))].




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(4)
Complex Obligation (BEST)

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(5)
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(EXP, INF).


The TRS R consists of the following rules:

   f(S(x'), S(x)) -> h(g(x', S(x)), f(S(S(S(x'))), x))
   g(S(x), S(x')) -> h(f(S(x), S(x')), g(x, S(S(S(x')))))
   h(0, S(x)) -> h(0, x)
   h(0, 0) -> 0
   g(S(x), 0) -> 0
   f(S(x), 0) -> 0
   h(S(x), x2) -> h(x, x2)
   g(0, x2) -> 0
   f(0, x2) -> 0

S is empty.
Rewrite Strategy: INNERMOST
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(6) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(7)
BOUNDS(n^1, INF)

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(8)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   f(S(x'), S(x)) -> h(g(x', S(x)), f(S(S(S(x'))), x))
   g(S(x), S(x')) -> h(f(S(x), S(x')), g(x, S(S(S(x')))))
   h(0, S(x)) -> h(0, x)
   h(0, 0) -> 0
   g(S(x), 0) -> 0
   f(S(x), 0) -> 0
   h(S(x), x2) -> h(x, x2)
   g(0, x2) -> 0
   f(0, x2) -> 0

S is empty.
Rewrite Strategy: INNERMOST
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(9) DecreasingLoopProof (FINISHED)
The following loop(s) give(s) rise to the lower bound EXP:

The rewrite sequence

g(S(x), S(x')) ->^+ h(h(g(x, S(x')), f(S(S(S(x))), x')), g(x, S(S(S(x')))))

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

The pumping substitution is [x / S(x)].

The result substitution is [ ].



The rewrite sequence

g(S(x), S(x')) ->^+ h(h(g(x, S(x')), f(S(S(S(x))), x')), g(x, S(S(S(x')))))

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

The pumping substitution is [x / S(x)].

The result substitution is [x' / S(S(x'))].




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(10)
BOUNDS(EXP, INF)