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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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

   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   rev(x) -> if(x, eq(0, length(x)), nil, 0, length(x))
   if(x, true, z, c, l) -> z
   if(x, false, z, c, l) -> help(s(c), l, x, z)
   help(c, l, cons(x, y), z) -> if(append(y, cons(x, nil)), ge(c, l), cons(x, z), c, l)
   append(nil, y) -> y
   append(cons(x, y), z) -> cons(x, append(y, z))
   length(nil) -> 0
   length(cons(x, y)) -> s(length(y))

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:

   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   rev(x) -> if(x, eq(0, length(x)), nil, 0, length(x))
   if(x, true, z, c, l) -> z
   if(x, false, z, c, l) -> help(s(c), l, x, z)
   help(c, l, cons(x, y), z) -> if(append(y, cons(x, nil)), ge(c, l), cons(x, z), c, l)
   append(nil, y) -> y
   append(cons(x, y), z) -> cons(x, append(y, z))
   length(nil) -> 0
   length(cons(x, y)) -> s(length(y))

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

length(cons(x, y)) ->^+ s(length(y))

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

The pumping substitution is [y / cons(x, 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:

   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   rev(x) -> if(x, eq(0, length(x)), nil, 0, length(x))
   if(x, true, z, c, l) -> z
   if(x, false, z, c, l) -> help(s(c), l, x, z)
   help(c, l, cons(x, y), z) -> if(append(y, cons(x, nil)), ge(c, l), cons(x, z), c, l)
   append(nil, y) -> y
   append(cons(x, y), z) -> cons(x, append(y, z))
   length(nil) -> 0
   length(cons(x, y)) -> s(length(y))

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:

   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   rev(x) -> if(x, eq(0, length(x)), nil, 0, length(x))
   if(x, true, z, c, l) -> z
   if(x, false, z, c, l) -> help(s(c), l, x, z)
   help(c, l, cons(x, y), z) -> if(append(y, cons(x, nil)), ge(c, l), cons(x, z), c, l)
   append(nil, y) -> y
   append(cons(x, y), z) -> cons(x, append(y, z))
   length(nil) -> 0
   length(cons(x, y)) -> s(length(y))

S is empty.
Rewrite Strategy: FULL