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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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

   isLeaf(leaf) -> true
   isLeaf(cons(u, v)) -> false
   left(cons(u, v)) -> u
   right(cons(u, v)) -> v
   concat(leaf, y) -> y
   concat(cons(u, v), y) -> cons(u, concat(v, y))
   less_leaves(u, v) -> if1(isLeaf(u), isLeaf(v), u, v)
   if1(b, true, u, v) -> false
   if1(b, false, u, v) -> if2(b, u, v)
   if2(true, u, v) -> true
   if2(false, u, v) -> less_leaves(concat(left(u), right(u)), concat(left(v), right(v)))

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:

   isLeaf(leaf) -> true
   isLeaf(cons(u, v)) -> false
   left(cons(u, v)) -> u
   right(cons(u, v)) -> v
   concat(leaf, y) -> y
   concat(cons(u, v), y) -> cons(u, concat(v, y))
   less_leaves(u, v) -> if1(isLeaf(u), isLeaf(v), u, v)
   if1(b, true, u, v) -> false
   if1(b, false, u, v) -> if2(b, u, v)
   if2(true, u, v) -> true
   if2(false, u, v) -> less_leaves(concat(left(u), right(u)), concat(left(v), right(v)))

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

concat(cons(u, v), y) ->^+ cons(u, concat(v, y))

gives rise to a decreasing loop by considering the right hand sides subterm at position [1].

The pumping substitution is [v / cons(u, v)].

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:

   isLeaf(leaf) -> true
   isLeaf(cons(u, v)) -> false
   left(cons(u, v)) -> u
   right(cons(u, v)) -> v
   concat(leaf, y) -> y
   concat(cons(u, v), y) -> cons(u, concat(v, y))
   less_leaves(u, v) -> if1(isLeaf(u), isLeaf(v), u, v)
   if1(b, true, u, v) -> false
   if1(b, false, u, v) -> if2(b, u, v)
   if2(true, u, v) -> true
   if2(false, u, v) -> less_leaves(concat(left(u), right(u)), concat(left(v), right(v)))

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:

   isLeaf(leaf) -> true
   isLeaf(cons(u, v)) -> false
   left(cons(u, v)) -> u
   right(cons(u, v)) -> v
   concat(leaf, y) -> y
   concat(cons(u, v), y) -> cons(u, concat(v, y))
   less_leaves(u, v) -> if1(isLeaf(u), isLeaf(v), u, v)
   if1(b, true, u, v) -> false
   if1(b, false, u, v) -> if2(b, u, v)
   if2(true, u, v) -> true
   if2(false, u, v) -> less_leaves(concat(left(u), right(u)), concat(left(v), right(v)))

S is empty.
Rewrite Strategy: FULL