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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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

   0(#) -> #
   +(x, #) -> x
   +(#, x) -> x
   +(0(x), 0(y)) -> 0(+(x, y))
   +(0(x), 1(y)) -> 1(+(x, y))
   +(1(x), 0(y)) -> 1(+(x, y))
   +(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
   +(x, +(y, z)) -> +(+(x, y), z)
   -(x, #) -> x
   -(#, x) -> #
   -(0(x), 0(y)) -> 0(-(x, y))
   -(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
   -(1(x), 0(y)) -> 1(-(x, y))
   -(1(x), 1(y)) -> 0(-(x, y))
   not(false) -> true
   not(true) -> false
   and(x, true) -> x
   and(x, false) -> false
   if(true, x, y) -> x
   if(false, x, y) -> y
   ge(0(x), 0(y)) -> ge(x, y)
   ge(0(x), 1(y)) -> not(ge(y, x))
   ge(1(x), 0(y)) -> ge(x, y)
   ge(1(x), 1(y)) -> ge(x, y)
   ge(x, #) -> true
   ge(#, 1(x)) -> false
   ge(#, 0(x)) -> ge(#, x)
   val(l(x)) -> x
   val(n(x, y, z)) -> x
   min(l(x)) -> x
   min(n(x, y, z)) -> min(y)
   max(l(x)) -> x
   max(n(x, y, z)) -> max(z)
   bs(l(x)) -> true
   bs(n(x, y, z)) -> and(and(ge(x, max(y)), ge(min(z), x)), and(bs(y), bs(z)))
   size(l(x)) -> 1(#)
   size(n(x, y, z)) -> +(+(size(x), size(y)), 1(#))
   wb(l(x)) -> true
   wb(n(x, y, z)) -> and(if(ge(size(y), size(z)), ge(1(#), -(size(y), size(z))), ge(1(#), -(size(z), size(y)))), and(wb(y), wb(z)))

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:

   0(#) -> #
   +(x, #) -> x
   +(#, x) -> x
   +(0(x), 0(y)) -> 0(+(x, y))
   +(0(x), 1(y)) -> 1(+(x, y))
   +(1(x), 0(y)) -> 1(+(x, y))
   +(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
   +(x, +(y, z)) -> +(+(x, y), z)
   -(x, #) -> x
   -(#, x) -> #
   -(0(x), 0(y)) -> 0(-(x, y))
   -(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
   -(1(x), 0(y)) -> 1(-(x, y))
   -(1(x), 1(y)) -> 0(-(x, y))
   not(false) -> true
   not(true) -> false
   and(x, true) -> x
   and(x, false) -> false
   if(true, x, y) -> x
   if(false, x, y) -> y
   ge(0(x), 0(y)) -> ge(x, y)
   ge(0(x), 1(y)) -> not(ge(y, x))
   ge(1(x), 0(y)) -> ge(x, y)
   ge(1(x), 1(y)) -> ge(x, y)
   ge(x, #) -> true
   ge(#, 1(x)) -> false
   ge(#, 0(x)) -> ge(#, x)
   val(l(x)) -> x
   val(n(x, y, z)) -> x
   min(l(x)) -> x
   min(n(x, y, z)) -> min(y)
   max(l(x)) -> x
   max(n(x, y, z)) -> max(z)
   bs(l(x)) -> true
   bs(n(x, y, z)) -> and(and(ge(x, max(y)), ge(min(z), x)), and(bs(y), bs(z)))
   size(l(x)) -> 1(#)
   size(n(x, y, z)) -> +(+(size(x), size(y)), 1(#))
   wb(l(x)) -> true
   wb(n(x, y, z)) -> and(if(ge(size(y), size(z)), ge(1(#), -(size(y), size(z))), ge(1(#), -(size(z), size(y)))), and(wb(y), wb(z)))

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

max(n(x, y, z)) ->^+ max(z)

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

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

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:

   0(#) -> #
   +(x, #) -> x
   +(#, x) -> x
   +(0(x), 0(y)) -> 0(+(x, y))
   +(0(x), 1(y)) -> 1(+(x, y))
   +(1(x), 0(y)) -> 1(+(x, y))
   +(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
   +(x, +(y, z)) -> +(+(x, y), z)
   -(x, #) -> x
   -(#, x) -> #
   -(0(x), 0(y)) -> 0(-(x, y))
   -(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
   -(1(x), 0(y)) -> 1(-(x, y))
   -(1(x), 1(y)) -> 0(-(x, y))
   not(false) -> true
   not(true) -> false
   and(x, true) -> x
   and(x, false) -> false
   if(true, x, y) -> x
   if(false, x, y) -> y
   ge(0(x), 0(y)) -> ge(x, y)
   ge(0(x), 1(y)) -> not(ge(y, x))
   ge(1(x), 0(y)) -> ge(x, y)
   ge(1(x), 1(y)) -> ge(x, y)
   ge(x, #) -> true
   ge(#, 1(x)) -> false
   ge(#, 0(x)) -> ge(#, x)
   val(l(x)) -> x
   val(n(x, y, z)) -> x
   min(l(x)) -> x
   min(n(x, y, z)) -> min(y)
   max(l(x)) -> x
   max(n(x, y, z)) -> max(z)
   bs(l(x)) -> true
   bs(n(x, y, z)) -> and(and(ge(x, max(y)), ge(min(z), x)), and(bs(y), bs(z)))
   size(l(x)) -> 1(#)
   size(n(x, y, z)) -> +(+(size(x), size(y)), 1(#))
   wb(l(x)) -> true
   wb(n(x, y, z)) -> and(if(ge(size(y), size(z)), ge(1(#), -(size(y), size(z))), ge(1(#), -(size(z), size(y)))), and(wb(y), wb(z)))

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:

   0(#) -> #
   +(x, #) -> x
   +(#, x) -> x
   +(0(x), 0(y)) -> 0(+(x, y))
   +(0(x), 1(y)) -> 1(+(x, y))
   +(1(x), 0(y)) -> 1(+(x, y))
   +(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
   +(x, +(y, z)) -> +(+(x, y), z)
   -(x, #) -> x
   -(#, x) -> #
   -(0(x), 0(y)) -> 0(-(x, y))
   -(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
   -(1(x), 0(y)) -> 1(-(x, y))
   -(1(x), 1(y)) -> 0(-(x, y))
   not(false) -> true
   not(true) -> false
   and(x, true) -> x
   and(x, false) -> false
   if(true, x, y) -> x
   if(false, x, y) -> y
   ge(0(x), 0(y)) -> ge(x, y)
   ge(0(x), 1(y)) -> not(ge(y, x))
   ge(1(x), 0(y)) -> ge(x, y)
   ge(1(x), 1(y)) -> ge(x, y)
   ge(x, #) -> true
   ge(#, 1(x)) -> false
   ge(#, 0(x)) -> ge(#, x)
   val(l(x)) -> x
   val(n(x, y, z)) -> x
   min(l(x)) -> x
   min(n(x, y, z)) -> min(y)
   max(l(x)) -> x
   max(n(x, y, z)) -> max(z)
   bs(l(x)) -> true
   bs(n(x, y, z)) -> and(and(ge(x, max(y)), ge(min(z), x)), and(bs(y), bs(z)))
   size(l(x)) -> 1(#)
   size(n(x, y, z)) -> +(+(size(x), size(y)), 1(#))
   wb(l(x)) -> true
   wb(n(x, y, z)) -> and(if(ge(size(y), size(z)), ge(1(#), -(size(y), size(z))), ge(1(#), -(size(z), size(y)))), and(wb(y), wb(z)))

S is empty.
Rewrite Strategy: FULL