/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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:

   average(x, y) -> if(ge(x, y), x, y)
   if(true, x, y) -> averIter(y, x, y)
   if(false, x, y) -> averIter(x, y, x)
   averIter(x, y, z) -> ifIter(ge(x, y), x, y, z)
   ifIter(true, x, y, z) -> z
   ifIter(false, x, y, z) -> averIter(plus(x, s(s(s(0)))), plus(y, s(0)), plus(z, s(0)))
   append(nil, y) -> y
   append(cons(n, x), y) -> cons(n, app(x, y))
   low(n, nil) -> nil
   low(n, cons(m, x)) -> if_low(ge(m, n), n, cons(m, x))
   if_low(false, n, cons(m, x)) -> cons(m, low(n, x))
   if_low(true, n, cons(m, x)) -> low(n, x)
   high(n, nil) -> nil
   high(n, cons(m, x)) -> if_high(ge(m, n), n, cons(m, x))
   if_high(false, n, cons(m, x)) -> high(n, x)
   if_high(true, n, cons(m, x)) -> cons(average(m, m), high(n, x))
   quicksort(x) -> ifquick(isempty(x), x)
   ifquick(true, x) -> nil
   ifquick(false, x) -> append(quicksort(low(head(x), tail(x))), cons(tail(x), quicksort(high(head(x), tail(x)))))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   isempty(nil) -> true
   isempty(cons(n, x)) -> false
   head(nil) -> error
   head(cons(n, x)) -> n
   tail(nil) -> nil
   tail(cons(n, x)) -> x
   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   a -> b
   a -> c

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:

   average(x, y) -> if(ge(x, y), x, y)
   if(true, x, y) -> averIter(y, x, y)
   if(false, x, y) -> averIter(x, y, x)
   averIter(x, y, z) -> ifIter(ge(x, y), x, y, z)
   ifIter(true, x, y, z) -> z
   ifIter(false, x, y, z) -> averIter(plus(x, s(s(s(0)))), plus(y, s(0)), plus(z, s(0)))
   append(nil, y) -> y
   append(cons(n, x), y) -> cons(n, app(x, y))
   low(n, nil) -> nil
   low(n, cons(m, x)) -> if_low(ge(m, n), n, cons(m, x))
   if_low(false, n, cons(m, x)) -> cons(m, low(n, x))
   if_low(true, n, cons(m, x)) -> low(n, x)
   high(n, nil) -> nil
   high(n, cons(m, x)) -> if_high(ge(m, n), n, cons(m, x))
   if_high(false, n, cons(m, x)) -> high(n, x)
   if_high(true, n, cons(m, x)) -> cons(average(m, m), high(n, x))
   quicksort(x) -> ifquick(isempty(x), x)
   ifquick(true, x) -> nil
   ifquick(false, x) -> append(quicksort(low(head(x), tail(x))), cons(tail(x), quicksort(high(head(x), tail(x)))))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   isempty(nil) -> true
   isempty(cons(n, x)) -> false
   head(nil) -> error
   head(cons(n, x)) -> n
   tail(nil) -> nil
   tail(cons(n, x)) -> x
   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   a -> b
   a -> c

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

plus(s(x), y) ->^+ s(plus(x, y))

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

The pumping substitution is [x / 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 (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   average(x, y) -> if(ge(x, y), x, y)
   if(true, x, y) -> averIter(y, x, y)
   if(false, x, y) -> averIter(x, y, x)
   averIter(x, y, z) -> ifIter(ge(x, y), x, y, z)
   ifIter(true, x, y, z) -> z
   ifIter(false, x, y, z) -> averIter(plus(x, s(s(s(0)))), plus(y, s(0)), plus(z, s(0)))
   append(nil, y) -> y
   append(cons(n, x), y) -> cons(n, app(x, y))
   low(n, nil) -> nil
   low(n, cons(m, x)) -> if_low(ge(m, n), n, cons(m, x))
   if_low(false, n, cons(m, x)) -> cons(m, low(n, x))
   if_low(true, n, cons(m, x)) -> low(n, x)
   high(n, nil) -> nil
   high(n, cons(m, x)) -> if_high(ge(m, n), n, cons(m, x))
   if_high(false, n, cons(m, x)) -> high(n, x)
   if_high(true, n, cons(m, x)) -> cons(average(m, m), high(n, x))
   quicksort(x) -> ifquick(isempty(x), x)
   ifquick(true, x) -> nil
   ifquick(false, x) -> append(quicksort(low(head(x), tail(x))), cons(tail(x), quicksort(high(head(x), tail(x)))))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   isempty(nil) -> true
   isempty(cons(n, x)) -> false
   head(nil) -> error
   head(cons(n, x)) -> n
   tail(nil) -> nil
   tail(cons(n, x)) -> x
   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   a -> b
   a -> c

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:

   average(x, y) -> if(ge(x, y), x, y)
   if(true, x, y) -> averIter(y, x, y)
   if(false, x, y) -> averIter(x, y, x)
   averIter(x, y, z) -> ifIter(ge(x, y), x, y, z)
   ifIter(true, x, y, z) -> z
   ifIter(false, x, y, z) -> averIter(plus(x, s(s(s(0)))), plus(y, s(0)), plus(z, s(0)))
   append(nil, y) -> y
   append(cons(n, x), y) -> cons(n, app(x, y))
   low(n, nil) -> nil
   low(n, cons(m, x)) -> if_low(ge(m, n), n, cons(m, x))
   if_low(false, n, cons(m, x)) -> cons(m, low(n, x))
   if_low(true, n, cons(m, x)) -> low(n, x)
   high(n, nil) -> nil
   high(n, cons(m, x)) -> if_high(ge(m, n), n, cons(m, x))
   if_high(false, n, cons(m, x)) -> high(n, x)
   if_high(true, n, cons(m, x)) -> cons(average(m, m), high(n, x))
   quicksort(x) -> ifquick(isempty(x), x)
   ifquick(true, x) -> nil
   ifquick(false, x) -> append(quicksort(low(head(x), tail(x))), cons(tail(x), quicksort(high(head(x), tail(x)))))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   isempty(nil) -> true
   isempty(cons(n, x)) -> false
   head(nil) -> error
   head(cons(n, x)) -> n
   tail(nil) -> nil
   tail(cons(n, x)) -> x
   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   a -> b
   a -> c

S is empty.
Rewrite Strategy: FULL