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

(0) Obligation:

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

@(Cons(x, xs), ys) → Cons(x, @(xs, ys))
@(Nil, ys) → ys
game(p1, p2, Cons(Capture, xs)) → game[Ite][False][Ite][False][Ite](True, p1, p2, Cons(Capture, 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)

Rewrite Strategy: INNERMOST

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted Cpx (relative) TRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

@'(Cons(z0, z1), z2) → c(@'(z1, z2))
@'(Nil, z0) → c1
GAME(z0, z1, Cons(Capture, z2)) → c2
GAME(z0, z1, Cons(Swap, z2)) → c3(GAME(z1, z0, z2))
GAME(z0, z1, Nil) → c4(@'(z0, z1))
EQUAL(Capture, Capture) → c5
EQUAL(Capture, Swap) → c6
EQUAL(Swap, Capture) → c7
EQUAL(Swap, Swap) → c8
GOAL(z0, z1, z2) → c9(GAME(z0, z1, z2))

S tuples:

@'(Cons(z0, z1), z2) → c(@'(z1, z2))
@'(Nil, z0) → c1
GAME(z0, z1, Cons(Capture, z2)) → c2
GAME(z0, z1, Cons(Swap, z2)) → c3(GAME(z1, z0, z2))
GAME(z0, z1, Nil) → c4(@'(z0, z1))
EQUAL(Capture, Capture) → c5
EQUAL(Capture, Swap) → c6
EQUAL(Swap, Capture) → c7
EQUAL(Swap, Swap) → c8
GOAL(z0, z1, z2) → c9(GAME(z0, z1, z2))

K tuples:none
Defined Rule Symbols:

@, game, equal, goal

Defined Pair Symbols:

@', GAME, EQUAL, GOAL

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9

(3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

GOAL(z0, z1, z2) → c9(GAME(z0, z1, z2))

Removed 6 trailing nodes:

EQUAL(Capture, Swap) → c6
GAME(z0, z1, Cons(Capture, z2)) → c2
@'(Nil, z0) → c1
EQUAL(Swap, Capture) → c7
EQUAL(Swap, Swap) → c8
EQUAL(Capture, Capture) → c5

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

@'(Cons(z0, z1), z2) → c(@'(z1, z2))
GAME(z0, z1, Cons(Swap, z2)) → c3(GAME(z1, z0, z2))
GAME(z0, z1, Nil) → c4(@'(z0, z1))

S tuples:

@'(Cons(z0, z1), z2) → c(@'(z1, z2))
GAME(z0, z1, Cons(Swap, z2)) → c3(GAME(z1, z0, z2))
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

(5) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

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

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

@'(Cons(z0, z1), z2) → c(@'(z1, z2))
GAME(z0, z1, Cons(Swap, z2)) → c3(GAME(z1, z0, z2))
GAME(z0, z1, Nil) → c4(@'(z0, z1))

S tuples:

@'(Cons(z0, z1), z2) → c(@'(z1, z2))
GAME(z0, z1, Cons(Swap, z2)) → c3(GAME(z1, z0, z2))
GAME(z0, z1, Nil) → c4(@'(z0, z1))

K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

@', GAME

Compound Symbols:

c, c3, c4

(7) CdtRuleRemovalProof (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))
GAME(z0, z1, Cons(Swap, z2)) → c3(GAME(z1, z0, z2))
GAME(z0, z1, Nil) → c4(@'(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

@'(Cons(z0, z1), z2) → c(@'(z1, z2))
GAME(z0, z1, Cons(Swap, z2)) → c3(GAME(z1, z0, z2))
GAME(z0, z1, Nil) → c4(@'(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(@'(x₁, x₂)) = [3] + [2]x₁ + [2]x₂   
POL(Cons(x₁, x₂)) = [5] + x₁ + x₂   
POL(GAME(x₁, x₂, x₃)) = [4] + [5]x₁ + [5]x₂ + [5]x₃   
POL(Nil) = [4]   
POL(Swap) = [5]   
POL(c(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c4(x₁)) = x₁   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

@'(Cons(z0, z1), z2) → c(@'(z1, z2))
GAME(z0, z1, Cons(Swap, z2)) → c3(GAME(z1, z0, z2))
GAME(z0, z1, Nil) → c4(@'(z0, z1))

S tuples:none
K tuples:

@'(Cons(z0, z1), z2) → c(@'(z1, z2))
GAME(z0, z1, Cons(Swap, z2)) → c3(GAME(z1, z0, z2))
GAME(z0, z1, Nil) → c4(@'(z0, z1))

Defined Rule Symbols:none

Defined Pair Symbols:

@', GAME

Compound Symbols:

c, c3, c4

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

The set S is empty

(10) BOUNDS(O(1), O(1))