YES(O(1), O(n^1)) 0.00/0.89 YES(O(1), O(n^1)) 0.00/0.92 0.00/0.92 0.00/0.92 0.00/0.92 0.00/0.92 0.00/0.92 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 0.00/0.92 0.00/0.92 0.00/0.92
0.00/0.92 0.00/0.92 0.00/0.92
0.00/0.92
0.00/0.92

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

@(Cons(x, xs), ys) → Cons(x, @(xs, ys)) 0.00/0.92
@(Nil, ys) → ys 0.00/0.92
game(p1, p2, Cons(Capture, xs)) → game[Ite][False][Ite][False][Ite](True, p1, p2, Cons(Capture, xs)) 0.00/0.92
game(p1, p2, Cons(Swap, xs)) → game(p2, p1, xs) 0.00/0.92
equal(Capture, Capture) → True 0.00/0.92
equal(Capture, Swap) → False 0.00/0.92
equal(Swap, Capture) → False 0.00/0.92
equal(Swap, Swap) → True 0.00/0.92
game(p1, p2, Nil) → @(p1, p2) 0.00/0.92
goal(p1, p2, moves) → game(p1, p2, moves)

Rewrite Strategy: INNERMOST
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(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT
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0.00/0.92

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

@(Cons(z0, z1), z2) → Cons(z0, @(z1, z2)) 0.00/0.92
@(Nil, z0) → z0 0.00/0.92
game(z0, z1, Cons(Capture, z2)) → game[Ite][False][Ite][False][Ite](True, z0, z1, Cons(Capture, z2)) 0.00/0.92
game(z0, z1, Cons(Swap, z2)) → game(z1, z0, z2) 0.00/0.92
game(z0, z1, Nil) → @(z0, z1) 0.00/0.92
equal(Capture, Capture) → True 0.00/0.92
equal(Capture, Swap) → False 0.00/0.92
equal(Swap, Capture) → False 0.00/0.92
equal(Swap, Swap) → True 0.00/0.92
goal(z0, z1, z2) → game(z0, z1, z2)
Tuples:

@'(Cons(z0, z1), z2) → c(@'(z1, z2)) 0.00/0.92
GAME(z0, z1, Cons(Swap, z2)) → c3(GAME(z1, z0, z2)) 0.00/0.92
GAME(z0, z1, Nil) → c4(@'(z0, z1)) 0.00/0.92
GOAL(z0, z1, z2) → c9(GAME(z0, z1, z2))
S tuples:

@'(Cons(z0, z1), z2) → c(@'(z1, z2)) 0.00/0.92
GAME(z0, z1, Cons(Swap, z2)) → c3(GAME(z1, z0, z2)) 0.00/0.92
GAME(z0, z1, Nil) → c4(@'(z0, z1)) 0.00/0.92
GOAL(z0, z1, z2) → c9(GAME(z0, z1, z2))
K tuples:none
Defined Rule Symbols:

@, game, equal, goal

Defined Pair Symbols:

@', GAME, GOAL

Compound Symbols:

c, c3, c4, c9

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(3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

GOAL(z0, z1, z2) → c9(GAME(z0, z1, z2))
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(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

@(Cons(z0, z1), z2) → Cons(z0, @(z1, z2)) 0.00/0.92
@(Nil, z0) → z0 0.00/0.92
game(z0, z1, Cons(Capture, z2)) → game[Ite][False][Ite][False][Ite](True, z0, z1, Cons(Capture, z2)) 0.00/0.92
game(z0, z1, Cons(Swap, z2)) → game(z1, z0, z2) 0.00/0.92
game(z0, z1, Nil) → @(z0, z1) 0.00/0.92
equal(Capture, Capture) → True 0.00/0.92
equal(Capture, Swap) → False 0.00/0.92
equal(Swap, Capture) → False 0.00/0.92
equal(Swap, Swap) → True 0.00/0.92
goal(z0, z1, z2) → game(z0, z1, z2)
Tuples:

@'(Cons(z0, z1), z2) → c(@'(z1, z2)) 0.00/0.92
GAME(z0, z1, Cons(Swap, z2)) → c3(GAME(z1, z0, z2)) 0.00/0.92
GAME(z0, z1, Nil) → c4(@'(z0, z1))
S tuples:

@'(Cons(z0, z1), z2) → c(@'(z1, z2)) 0.00/0.92
GAME(z0, z1, Cons(Swap, z2)) → c3(GAME(z1, z0, z2)) 0.00/0.92
GAME(z0, z1, Nil) → c4(@'(z0, z1))
K tuples:none
Defined Rule Symbols:

@, game, equal, goal

Defined Pair Symbols:

@', GAME

Compound Symbols:

c, c3, c4

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(5) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

GAME(z0, z1, Nil) → c4(@'(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

@'(Cons(z0, z1), z2) → c(@'(z1, z2)) 0.00/0.92
GAME(z0, z1, Cons(Swap, z2)) → c3(GAME(z1, z0, z2)) 0.00/0.92
GAME(z0, z1, Nil) → c4(@'(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 0.00/0.92

POL(@'(x1, x2)) = [3] + x1 + [2]x2    0.00/0.92
POL(Cons(x1, x2)) = x2    0.00/0.92
POL(GAME(x1, x2, x3)) = [3] + [3]x1 + [3]x2 + [4]x3    0.00/0.92
POL(Nil) = [2]    0.00/0.92
POL(Swap) = 0    0.00/0.92
POL(c(x1)) = x1    0.00/0.92
POL(c3(x1)) = x1    0.00/0.92
POL(c4(x1)) = x1   
0.00/0.92
0.00/0.92

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

@(Cons(z0, z1), z2) → Cons(z0, @(z1, z2)) 0.00/0.92
@(Nil, z0) → z0 0.00/0.92
game(z0, z1, Cons(Capture, z2)) → game[Ite][False][Ite][False][Ite](True, z0, z1, Cons(Capture, z2)) 0.00/0.92
game(z0, z1, Cons(Swap, z2)) → game(z1, z0, z2) 0.00/0.92
game(z0, z1, Nil) → @(z0, z1) 0.00/0.92
equal(Capture, Capture) → True 0.00/0.92
equal(Capture, Swap) → False 0.00/0.92
equal(Swap, Capture) → False 0.00/0.92
equal(Swap, Swap) → True 0.00/0.92
goal(z0, z1, z2) → game(z0, z1, z2)
Tuples:

@'(Cons(z0, z1), z2) → c(@'(z1, z2)) 0.00/0.92
GAME(z0, z1, Cons(Swap, z2)) → c3(GAME(z1, z0, z2)) 0.00/0.92
GAME(z0, z1, Nil) → c4(@'(z0, z1))
S tuples:

@'(Cons(z0, z1), z2) → c(@'(z1, z2)) 0.00/0.92
GAME(z0, z1, Cons(Swap, z2)) → c3(GAME(z1, z0, z2))
K tuples:

GAME(z0, z1, Nil) → c4(@'(z0, z1))
Defined Rule Symbols:

@, game, equal, goal

Defined Pair Symbols:

@', GAME

Compound Symbols:

c, c3, c4

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0.00/0.92

(7) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

@'(Cons(z0, z1), z2) → c(@'(z1, z2))
We considered the (Usable) Rules:none
And the Tuples:

@'(Cons(z0, z1), z2) → c(@'(z1, z2)) 0.00/0.92
GAME(z0, z1, Cons(Swap, z2)) → c3(GAME(z1, z0, z2)) 0.00/0.92
GAME(z0, z1, Nil) → c4(@'(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 0.00/0.92

POL(@'(x1, x2)) = [3] + x1 + x2    0.00/0.92
POL(Cons(x1, x2)) = [2] + x2    0.00/0.92
POL(GAME(x1, x2, x3)) = [5] + [5]x1 + [5]x2    0.00/0.92
POL(Nil) = [3]    0.00/0.92
POL(Swap) = [1]    0.00/0.92
POL(c(x1)) = x1    0.00/0.92
POL(c3(x1)) = x1    0.00/0.92
POL(c4(x1)) = x1   
0.00/0.92
0.00/0.92

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

@(Cons(z0, z1), z2) → Cons(z0, @(z1, z2)) 0.00/0.92
@(Nil, z0) → z0 0.00/0.92
game(z0, z1, Cons(Capture, z2)) → game[Ite][False][Ite][False][Ite](True, z0, z1, Cons(Capture, z2)) 0.00/0.92
game(z0, z1, Cons(Swap, z2)) → game(z1, z0, z2) 0.00/0.92
game(z0, z1, Nil) → @(z0, z1) 0.00/0.92
equal(Capture, Capture) → True 0.00/0.92
equal(Capture, Swap) → False 0.00/0.92
equal(Swap, Capture) → False 0.00/0.92
equal(Swap, Swap) → True 0.00/0.92
goal(z0, z1, z2) → game(z0, z1, z2)
Tuples:

@'(Cons(z0, z1), z2) → c(@'(z1, z2)) 0.00/0.92
GAME(z0, z1, Cons(Swap, z2)) → c3(GAME(z1, z0, z2)) 0.00/0.92
GAME(z0, z1, Nil) → c4(@'(z0, z1))
S tuples:

GAME(z0, z1, Cons(Swap, z2)) → c3(GAME(z1, z0, z2))
K tuples:

GAME(z0, z1, Nil) → c4(@'(z0, z1)) 0.00/0.92
@'(Cons(z0, z1), z2) → c(@'(z1, z2))
Defined Rule Symbols:

@, game, equal, goal

Defined Pair Symbols:

@', GAME

Compound Symbols:

c, c3, c4

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0.00/0.92

(9) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

GAME(z0, z1, Cons(Swap, z2)) → c3(GAME(z1, z0, z2))
We considered the (Usable) Rules:none
And the Tuples:

@'(Cons(z0, z1), z2) → c(@'(z1, z2)) 0.00/0.92
GAME(z0, z1, Cons(Swap, z2)) → c3(GAME(z1, z0, z2)) 0.00/0.92
GAME(z0, z1, Nil) → c4(@'(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 0.00/0.92

POL(@'(x1, x2)) = [5] + [3]x1 + [3]x2    0.00/0.92
POL(Cons(x1, x2)) = x1 + x2    0.00/0.92
POL(GAME(x1, x2, x3)) = [3] + [3]x1 + [3]x2 + [2]x3    0.00/0.92
POL(Nil) = [3]    0.00/0.92
POL(Swap) = [1]    0.00/0.92
POL(c(x1)) = x1    0.00/0.92
POL(c3(x1)) = x1    0.00/0.92
POL(c4(x1)) = x1   
0.00/0.92
0.00/0.92

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

@(Cons(z0, z1), z2) → Cons(z0, @(z1, z2)) 0.00/0.92
@(Nil, z0) → z0 0.00/0.92
game(z0, z1, Cons(Capture, z2)) → game[Ite][False][Ite][False][Ite](True, z0, z1, Cons(Capture, z2)) 0.00/0.92
game(z0, z1, Cons(Swap, z2)) → game(z1, z0, z2) 0.00/0.92
game(z0, z1, Nil) → @(z0, z1) 0.00/0.92
equal(Capture, Capture) → True 0.00/0.92
equal(Capture, Swap) → False 0.00/0.92
equal(Swap, Capture) → False 0.00/0.92
equal(Swap, Swap) → True 0.00/0.92
goal(z0, z1, z2) → game(z0, z1, z2)
Tuples:

@'(Cons(z0, z1), z2) → c(@'(z1, z2)) 0.00/0.92
GAME(z0, z1, Cons(Swap, z2)) → c3(GAME(z1, z0, z2)) 0.00/0.92
GAME(z0, z1, Nil) → c4(@'(z0, z1))
S tuples:none
K tuples:

GAME(z0, z1, Nil) → c4(@'(z0, z1)) 0.00/0.92
@'(Cons(z0, z1), z2) → c(@'(z1, z2)) 0.00/0.92
GAME(z0, z1, Cons(Swap, z2)) → c3(GAME(z1, z0, z2))
Defined Rule Symbols:

@, game, equal, goal

Defined Pair Symbols:

@', GAME

Compound Symbols:

c, c3, c4

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(11) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty
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(12) BOUNDS(O(1), O(1))

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0.00/1.00 EOF