/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), O(n^1))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxTRS
(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) CpxTrsMatchBoundsTAProof [FINISHED, 84 ms]
    (4) BOUNDS(1, n^1)
(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(6) TRS for Loop Detection
    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (8) BEST
        (9) proven lower bound
            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
            (11) BOUNDS(n^1, INF)
        (12) TRS for Loop Detection


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(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   @(Cons(x, xs), ys) -> Cons(x, @(xs, ys))
   @(Nil, ys) -> ys
   game(p1, Cons(x', xs'), Cons(Capture, xs)) -> game(Cons(x', p1), xs', xs)
   game(p1, p2, Cons(Swap, xs)) -> game(p2, p1, xs)
   equal(Capture, Capture) -> True
   equal(Capture, Swap) -> False
   equal(Swap, Capture) -> False
   equal(Swap, Swap) -> True
   game(p1, p2, Nil) -> @(p1, p2)
   goal(p1, p2, moves) -> game(p1, p2, moves)

S is empty.
Rewrite Strategy: INNERMOST
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(1) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
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(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   @(Cons(x, xs), ys) -> Cons(x, @(xs, ys))
   @(Nil, ys) -> ys
   game(p1, Cons(x', xs'), Cons(Capture, xs)) -> game(Cons(x', p1), xs', xs)
   game(p1, p2, Cons(Swap, xs)) -> game(p2, p1, xs)
   equal(Capture, Capture) -> True
   equal(Capture, Swap) -> False
   equal(Swap, Capture) -> False
   equal(Swap, Swap) -> True
   game(p1, p2, Nil) -> @(p1, p2)
   goal(p1, p2, moves) -> game(p1, p2, moves)

S is empty.
Rewrite Strategy: INNERMOST
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(3) 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 2. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2, 3, 4]
transitions: 
Cons0(0, 0) -> 0
Nil0() -> 0
Capture0() -> 0
Swap0() -> 0
True0() -> 0
False0() -> 0
@0(0, 0) -> 1
game0(0, 0, 0) -> 2
equal0(0, 0) -> 3
goal0(0, 0, 0) -> 4
@1(0, 0) -> 5
Cons1(0, 5) -> 1
Cons1(0, 0) -> 6
game1(6, 0, 0) -> 2
game1(0, 0, 0) -> 2
True1() -> 3
False1() -> 3
@1(0, 0) -> 2
game1(0, 0, 0) -> 4
Cons1(0, 5) -> 2
Cons1(0, 5) -> 5
Cons1(0, 6) -> 6
game1(6, 0, 0) -> 4
game1(0, 6, 0) -> 2
@1(6, 0) -> 2
@1(0, 0) -> 4
Cons1(0, 5) -> 4
@2(0, 0) -> 7
Cons2(0, 7) -> 2
@2(6, 0) -> 7
game1(6, 6, 0) -> 2
game1(0, 6, 0) -> 4
@1(0, 6) -> 2
@1(6, 0) -> 4
@1(0, 6) -> 5
Cons2(0, 7) -> 4
game1(6, 6, 0) -> 4
@1(6, 6) -> 2
@1(0, 6) -> 4
Cons1(0, 5) -> 7
Cons2(0, 7) -> 7
@2(0, 6) -> 7
@2(6, 6) -> 7
@1(6, 6) -> 4
0 -> 1
0 -> 2
0 -> 5
0 -> 4
0 -> 7
6 -> 2
6 -> 4
6 -> 5
6 -> 7

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(4)
BOUNDS(1, n^1)

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(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(6)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   @(Cons(x, xs), ys) -> Cons(x, @(xs, ys))
   @(Nil, ys) -> ys
   game(p1, Cons(x', xs'), Cons(Capture, xs)) -> game(Cons(x', p1), xs', xs)
   game(p1, p2, Cons(Swap, xs)) -> game(p2, p1, xs)
   equal(Capture, Capture) -> True
   equal(Capture, Swap) -> False
   equal(Swap, Capture) -> False
   equal(Swap, Swap) -> True
   game(p1, p2, Nil) -> @(p1, p2)
   goal(p1, p2, moves) -> game(p1, p2, moves)

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

The rewrite sequence

game(p1, Cons(x', xs'), Cons(Capture, xs)) ->^+ game(Cons(x', p1), xs', xs)

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

The pumping substitution is [xs' / Cons(x', xs'), xs / Cons(Capture, xs)].

The result substitution is [p1 / Cons(x', p1)].




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(8)
Complex Obligation (BEST)

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(9)
Obligation:
Proved the lower bound n^1 for the following obligation:

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


The TRS R consists of the following rules:

   @(Cons(x, xs), ys) -> Cons(x, @(xs, ys))
   @(Nil, ys) -> ys
   game(p1, Cons(x', xs'), Cons(Capture, xs)) -> game(Cons(x', p1), xs', xs)
   game(p1, p2, Cons(Swap, xs)) -> game(p2, p1, xs)
   equal(Capture, Capture) -> True
   equal(Capture, Swap) -> False
   equal(Swap, Capture) -> False
   equal(Swap, Swap) -> True
   game(p1, p2, Nil) -> @(p1, p2)
   goal(p1, p2, moves) -> game(p1, p2, moves)

S is empty.
Rewrite Strategy: INNERMOST
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(10) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(11)
BOUNDS(n^1, INF)

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(12)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   @(Cons(x, xs), ys) -> Cons(x, @(xs, ys))
   @(Nil, ys) -> ys
   game(p1, Cons(x', xs'), Cons(Capture, xs)) -> game(Cons(x', p1), xs', xs)
   game(p1, p2, Cons(Swap, xs)) -> game(p2, p1, xs)
   equal(Capture, Capture) -> True
   equal(Capture, Swap) -> False
   equal(Swap, Capture) -> False
   equal(Swap, Swap) -> True
   game(p1, p2, Nil) -> @(p1, p2)
   goal(p1, p2, moves) -> game(p1, p2, moves)

S is empty.
Rewrite Strategy: INNERMOST