WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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:

   div(x, y) -> div2(x, y, 0)
   div2(x, y, i) -> if1(le(y, 0), le(y, x), x, y, plus(i, 0), inc(i))
   if1(true, b, x, y, i, j) -> divZeroError
   if1(false, b, x, y, i, j) -> if2(b, x, y, i, j)
   if2(true, x, y, i, j) -> div2(minus(x, y), y, j)
   if2(false, x, y, i, j) -> i
   inc(0) -> 0
   inc(s(i)) -> s(inc(i))
   le(s(x), 0) -> false
   le(0, y) -> true
   le(s(x), s(y)) -> le(x, y)
   minus(x, 0) -> x
   minus(0, y) -> 0
   minus(s(x), s(y)) -> minus(x, y)
   plus(x, y) -> plusIter(x, y, 0)
   plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z)
   ifPlus(true, x, y, z) -> y
   ifPlus(false, x, y, z) -> plusIter(x, s(y), s(z))
   a -> c
   a -> d

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:

   div(x, y) -> div2(x, y, 0)
   div2(x, y, i) -> if1(le(y, 0), le(y, x), x, y, plus(i, 0), inc(i))
   if1(true, b, x, y, i, j) -> divZeroError
   if1(false, b, x, y, i, j) -> if2(b, x, y, i, j)
   if2(true, x, y, i, j) -> div2(minus(x, y), y, j)
   if2(false, x, y, i, j) -> i
   inc(0) -> 0
   inc(s(i)) -> s(inc(i))
   le(s(x), 0) -> false
   le(0, y) -> true
   le(s(x), s(y)) -> le(x, y)
   minus(x, 0) -> x
   minus(0, y) -> 0
   minus(s(x), s(y)) -> minus(x, y)
   plus(x, y) -> plusIter(x, y, 0)
   plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z)
   ifPlus(true, x, y, z) -> y
   ifPlus(false, x, y, z) -> plusIter(x, s(y), s(z))
   a -> c
   a -> d

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

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

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

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

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:

   div(x, y) -> div2(x, y, 0)
   div2(x, y, i) -> if1(le(y, 0), le(y, x), x, y, plus(i, 0), inc(i))
   if1(true, b, x, y, i, j) -> divZeroError
   if1(false, b, x, y, i, j) -> if2(b, x, y, i, j)
   if2(true, x, y, i, j) -> div2(minus(x, y), y, j)
   if2(false, x, y, i, j) -> i
   inc(0) -> 0
   inc(s(i)) -> s(inc(i))
   le(s(x), 0) -> false
   le(0, y) -> true
   le(s(x), s(y)) -> le(x, y)
   minus(x, 0) -> x
   minus(0, y) -> 0
   minus(s(x), s(y)) -> minus(x, y)
   plus(x, y) -> plusIter(x, y, 0)
   plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z)
   ifPlus(true, x, y, z) -> y
   ifPlus(false, x, y, z) -> plusIter(x, s(y), s(z))
   a -> c
   a -> d

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:

   div(x, y) -> div2(x, y, 0)
   div2(x, y, i) -> if1(le(y, 0), le(y, x), x, y, plus(i, 0), inc(i))
   if1(true, b, x, y, i, j) -> divZeroError
   if1(false, b, x, y, i, j) -> if2(b, x, y, i, j)
   if2(true, x, y, i, j) -> div2(minus(x, y), y, j)
   if2(false, x, y, i, j) -> i
   inc(0) -> 0
   inc(s(i)) -> s(inc(i))
   le(s(x), 0) -> false
   le(0, y) -> true
   le(s(x), s(y)) -> le(x, y)
   minus(x, 0) -> x
   minus(0, y) -> 0
   minus(s(x), s(y)) -> minus(x, y)
   plus(x, y) -> plusIter(x, y, 0)
   plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z)
   ifPlus(true, x, y, z) -> y
   ifPlus(false, x, y, z) -> plusIter(x, s(y), s(z))
   a -> c
   a -> d

S is empty.
Rewrite Strategy: FULL