/export/starexec/sandbox/solver/bin/starexec_run_complexity /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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, 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


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(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   f(true, x, y) -> f(and(gt(x, y), gt(y, s(s(0)))), plus(s(0), x), double(y))
   gt(0, v) -> false
   gt(s(u), 0) -> true
   gt(s(u), s(v)) -> gt(u, v)
   and(x, true) -> x
   and(x, false) -> false
   plus(n, 0) -> n
   plus(n, s(m)) -> s(plus(n, m))
   double(0) -> 0
   double(s(x)) -> s(s(double(x)))

S is empty.
Rewrite Strategy: FULL
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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 (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   f(true, x, y) -> f(and(gt(x, y), gt(y, s(s(0)))), plus(s(0), x), double(y))
   gt(0, v) -> false
   gt(s(u), 0) -> true
   gt(s(u), s(v)) -> gt(u, v)
   and(x, true) -> x
   and(x, false) -> false
   plus(n, 0) -> n
   plus(n, s(m)) -> s(plus(n, m))
   double(0) -> 0
   double(s(x)) -> s(s(double(x)))

S is empty.
Rewrite Strategy: FULL
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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

gt(s(u), s(v)) ->^+ gt(u, v)

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

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

The result substitution is [ ].




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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 (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   f(true, x, y) -> f(and(gt(x, y), gt(y, s(s(0)))), plus(s(0), x), double(y))
   gt(0, v) -> false
   gt(s(u), 0) -> true
   gt(s(u), s(v)) -> gt(u, v)
   and(x, true) -> x
   and(x, false) -> false
   plus(n, 0) -> n
   plus(n, s(m)) -> s(plus(n, m))
   double(0) -> 0
   double(s(x)) -> s(s(double(x)))

S is empty.
Rewrite Strategy: FULL
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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 (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   f(true, x, y) -> f(and(gt(x, y), gt(y, s(s(0)))), plus(s(0), x), double(y))
   gt(0, v) -> false
   gt(s(u), 0) -> true
   gt(s(u), s(v)) -> gt(u, v)
   and(x, true) -> x
   and(x, false) -> false
   plus(n, 0) -> n
   plus(n, s(m)) -> s(plus(n, m))
   double(0) -> 0
   double(s(x)) -> s(s(double(x)))

S is empty.
Rewrite Strategy: FULL