/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(Omega(n^1), ?)
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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


The TRS R consists of the following rules:

   even(0) -> true
   even(s(0)) -> false
   even(s(s(x))) -> even(x)
   half(0) -> 0
   half(s(s(x))) -> s(half(x))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   times(0, y) -> 0
   times(s(x), y) -> if_times(even(s(x)), s(x), y)
   if_times(true, s(x), y) -> plus(times(half(s(x)), y), times(half(s(x)), y))
   if_times(false, s(x), y) -> plus(y, times(x, y))

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(n^1, INF).


The TRS R consists of the following rules:

   even(0) -> true
   even(s(0)) -> false
   even(s(s(x))) -> even(x)
   half(0) -> 0
   half(s(s(x))) -> s(half(x))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   times(0, y) -> 0
   times(s(x), y) -> if_times(even(s(x)), s(x), y)
   if_times(true, s(x), y) -> plus(times(half(s(x)), y), times(half(s(x)), y))
   if_times(false, s(x), y) -> plus(y, times(x, y))

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

even(s(s(x))) ->^+ even(x)

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

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

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


The TRS R consists of the following rules:

   even(0) -> true
   even(s(0)) -> false
   even(s(s(x))) -> even(x)
   half(0) -> 0
   half(s(s(x))) -> s(half(x))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   times(0, y) -> 0
   times(s(x), y) -> if_times(even(s(x)), s(x), y)
   if_times(true, s(x), y) -> plus(times(half(s(x)), y), times(half(s(x)), y))
   if_times(false, s(x), y) -> plus(y, times(x, y))

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(n^1, INF).


The TRS R consists of the following rules:

   even(0) -> true
   even(s(0)) -> false
   even(s(s(x))) -> even(x)
   half(0) -> 0
   half(s(s(x))) -> s(half(x))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   times(0, y) -> 0
   times(s(x), y) -> if_times(even(s(x)), s(x), y)
   if_times(true, s(x), y) -> plus(times(half(s(x)), y), times(half(s(x)), y))
   if_times(false, s(x), y) -> plus(y, times(x, y))

S is empty.
Rewrite Strategy: INNERMOST