/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), O(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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, n^1).

(0) CpxTRS
(1) DependencyGraphProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
    (4) CpxTRS
    (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
    (6) CpxTRS
    (7) CpxTrsMatchBoundsTAProof [FINISHED, 16 ms]
    (8) BOUNDS(1, n^1)
(9) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(10) TRS for Loop Detection
    (11) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (12) BEST
        (13) proven lower bound
            (14) LowerBoundPropagationProof [FINISHED, 0 ms]
            (15) BOUNDS(n^1, INF)
        (16) 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, n^1).


The TRS R consists of the following rules:

   flatten(nil) -> nil
   flatten(unit(x)) -> flatten(x)
   flatten(++(x, y)) -> ++(flatten(x), flatten(y))
   flatten(++(unit(x), y)) -> ++(flatten(x), flatten(y))
   flatten(flatten(x)) -> flatten(x)
   rev(nil) -> nil
   rev(unit(x)) -> unit(x)
   rev(++(x, y)) -> ++(rev(y), rev(x))
   rev(rev(x)) -> x
   ++(x, nil) -> x
   ++(nil, y) -> y
   ++(++(x, y), z) -> ++(x, ++(y, z))

S is empty.
Rewrite Strategy: FULL
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(1) DependencyGraphProof (UPPER BOUND(ID))
The following rules are not reachable from basic terms in the dependency graph and can be removed:

rev(++(x, y)) -> ++(rev(y), rev(x))
rev(rev(x)) -> x

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(2)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   flatten(nil) -> nil
   flatten(unit(x)) -> flatten(x)
   flatten(++(x, y)) -> ++(flatten(x), flatten(y))
   flatten(++(unit(x), y)) -> ++(flatten(x), flatten(y))
   flatten(flatten(x)) -> flatten(x)
   rev(nil) -> nil
   rev(unit(x)) -> unit(x)
   ++(x, nil) -> x
   ++(nil, y) -> y
   ++(++(x, y), z) -> ++(x, ++(y, z))

S is empty.
Rewrite Strategy: FULL
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(3) NestedDefinedSymbolProof (UPPER BOUND(ID))
The TRS does not nest defined symbols.
Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
flatten(++(x, y)) -> ++(flatten(x), flatten(y))
flatten(++(unit(x), y)) -> ++(flatten(x), flatten(y))
flatten(flatten(x)) -> flatten(x)
++(++(x, y), z) -> ++(x, ++(y, z))

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(4)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   flatten(nil) -> nil
   flatten(unit(x)) -> flatten(x)
   rev(nil) -> nil
   rev(unit(x)) -> unit(x)
   ++(x, nil) -> x
   ++(nil, y) -> y

S is empty.
Rewrite Strategy: FULL
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(5) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
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(6)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   flatten(nil) -> nil
   flatten(unit(x)) -> flatten(x)
   rev(nil) -> nil
   rev(unit(x)) -> unit(x)
   ++(x, nil) -> x
   ++(nil, y) -> y

S is empty.
Rewrite Strategy: FULL
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(7) CpxTrsMatchBoundsTAProof (FINISHED)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 1. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2, 3]
transitions: 
nil0() -> 0
unit0(0) -> 0
flatten0(0) -> 1
rev0(0) -> 2
++0(0, 0) -> 3
nil1() -> 1
flatten1(0) -> 1
nil1() -> 2
unit1(0) -> 2
0 -> 3

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(8)
BOUNDS(1, n^1)

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(9) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(10)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   flatten(nil) -> nil
   flatten(unit(x)) -> flatten(x)
   flatten(++(x, y)) -> ++(flatten(x), flatten(y))
   flatten(++(unit(x), y)) -> ++(flatten(x), flatten(y))
   flatten(flatten(x)) -> flatten(x)
   rev(nil) -> nil
   rev(unit(x)) -> unit(x)
   rev(++(x, y)) -> ++(rev(y), rev(x))
   rev(rev(x)) -> x
   ++(x, nil) -> x
   ++(nil, y) -> y
   ++(++(x, y), z) -> ++(x, ++(y, z))

S is empty.
Rewrite Strategy: FULL
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(11) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

flatten(unit(x)) ->^+ flatten(x)

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

The pumping substitution is [x / unit(x)].

The result substitution is [ ].




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(12)
Complex Obligation (BEST)

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(13)
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, n^1).


The TRS R consists of the following rules:

   flatten(nil) -> nil
   flatten(unit(x)) -> flatten(x)
   flatten(++(x, y)) -> ++(flatten(x), flatten(y))
   flatten(++(unit(x), y)) -> ++(flatten(x), flatten(y))
   flatten(flatten(x)) -> flatten(x)
   rev(nil) -> nil
   rev(unit(x)) -> unit(x)
   rev(++(x, y)) -> ++(rev(y), rev(x))
   rev(rev(x)) -> x
   ++(x, nil) -> x
   ++(nil, y) -> y
   ++(++(x, y), z) -> ++(x, ++(y, z))

S is empty.
Rewrite Strategy: FULL
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(14) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(15)
BOUNDS(n^1, INF)

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(16)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   flatten(nil) -> nil
   flatten(unit(x)) -> flatten(x)
   flatten(++(x, y)) -> ++(flatten(x), flatten(y))
   flatten(++(unit(x), y)) -> ++(flatten(x), flatten(y))
   flatten(flatten(x)) -> flatten(x)
   rev(nil) -> nil
   rev(unit(x)) -> unit(x)
   rev(++(x, y)) -> ++(rev(y), rev(x))
   rev(rev(x)) -> x
   ++(x, nil) -> x
   ++(nil, y) -> y
   ++(++(x, y), z) -> ++(x, ++(y, z))

S is empty.
Rewrite Strategy: FULL