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

(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, 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)

Rewrite Strategy: INNERMOST

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
@(Cons(x, xs), ys) →⁺ Cons(x, @(xs, ys))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
The pumping substitution is [xs / Cons(x, xs)].
The result substitution is [ ].

(2) BOUNDS(n^1, INF)

(3) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(4) Obligation:

Runtime Complexity Relative TRS:
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

(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(6) Obligation:

Innermost TRS:
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)

Types:
@ :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Capture:Swap → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
game :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil
Capture :: Capture:Swap
Swap :: Capture:Swap
equal :: Capture:Swap → Capture:Swap → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
hole_Capture:Swap2_0 :: Capture:Swap
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat → Cons:Nil

(7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
@, game

They will be analysed ascendingly in the following order:
@ < game

(8) Obligation:

Innermost TRS:
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)

Types:
@ :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Capture:Swap → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
game :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil
Capture :: Capture:Swap
Swap :: Capture:Swap
equal :: Capture:Swap → Capture:Swap → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
hole_Capture:Swap2_0 :: Capture:Swap
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat → Cons:Nil

Generator Equations:
gen_Cons:Nil4_0(0) ⇔ Nil
gen_Cons:Nil4_0(+(x, 1)) ⇔ Cons(Capture, gen_Cons:Nil4_0(x))

The following defined symbols remain to be analysed:
@, game

They will be analysed ascendingly in the following order:
@ < game

(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
@(gen_Cons:Nil4_0(n6_0), gen_Cons:Nil4_0(b)) → gen_Cons:Nil4_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)

Induction Base:
@(gen_Cons:Nil4_0(0), gen_Cons:Nil4_0(b)) →_R^Ω(1)
gen_Cons:Nil4_0(b)

Induction Step:
@(gen_Cons:Nil4_0(+(n6_0, 1)), gen_Cons:Nil4_0(b)) →_R^Ω(1)
Cons(Capture, @(gen_Cons:Nil4_0(n6_0), gen_Cons:Nil4_0(b))) →_IH
Cons(Capture, gen_Cons:Nil4_0(+(b, c7_0)))

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(10) Complex Obligation (BEST)

(11) Obligation:

Innermost TRS:
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)

Types:
@ :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Capture:Swap → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
game :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil
Capture :: Capture:Swap
Swap :: Capture:Swap
equal :: Capture:Swap → Capture:Swap → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
hole_Capture:Swap2_0 :: Capture:Swap
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat → Cons:Nil

Lemmas:
@(gen_Cons:Nil4_0(n6_0), gen_Cons:Nil4_0(b)) → gen_Cons:Nil4_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_Cons:Nil4_0(0) ⇔ Nil
gen_Cons:Nil4_0(+(x, 1)) ⇔ Cons(Capture, gen_Cons:Nil4_0(x))

The following defined symbols remain to be analysed:
game

(12) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
game(gen_Cons:Nil4_0(a), gen_Cons:Nil4_0(n573_0), gen_Cons:Nil4_0(n573_0)) → *5_0, rt ∈ Ω(n573₀)

Induction Base:
game(gen_Cons:Nil4_0(a), gen_Cons:Nil4_0(0), gen_Cons:Nil4_0(0))

Induction Step:
game(gen_Cons:Nil4_0(a), gen_Cons:Nil4_0(+(n573_0, 1)), gen_Cons:Nil4_0(+(n573_0, 1))) →_R^Ω(1)
game(Cons(Capture, gen_Cons:Nil4_0(a)), gen_Cons:Nil4_0(n573_0), gen_Cons:Nil4_0(n573_0)) →_IH
*5_0

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(13) Complex Obligation (BEST)

(14) Obligation:

Innermost TRS:
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)

Types:
@ :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Capture:Swap → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
game :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil
Capture :: Capture:Swap
Swap :: Capture:Swap
equal :: Capture:Swap → Capture:Swap → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
hole_Capture:Swap2_0 :: Capture:Swap
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat → Cons:Nil

Lemmas:
@(gen_Cons:Nil4_0(n6_0), gen_Cons:Nil4_0(b)) → gen_Cons:Nil4_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)
game(gen_Cons:Nil4_0(a), gen_Cons:Nil4_0(n573_0), gen_Cons:Nil4_0(n573_0)) → *5_0, rt ∈ Ω(n573₀)

Generator Equations:
gen_Cons:Nil4_0(0) ⇔ Nil
gen_Cons:Nil4_0(+(x, 1)) ⇔ Cons(Capture, gen_Cons:Nil4_0(x))

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
@(gen_Cons:Nil4_0(n6_0), gen_Cons:Nil4_0(b)) → gen_Cons:Nil4_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)

(16) BOUNDS(n^1, INF)

(17) Obligation:

Innermost TRS:
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)

Types:
@ :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Capture:Swap → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
game :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil
Capture :: Capture:Swap
Swap :: Capture:Swap
equal :: Capture:Swap → Capture:Swap → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
hole_Capture:Swap2_0 :: Capture:Swap
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat → Cons:Nil

Lemmas:
@(gen_Cons:Nil4_0(n6_0), gen_Cons:Nil4_0(b)) → gen_Cons:Nil4_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)
game(gen_Cons:Nil4_0(a), gen_Cons:Nil4_0(n573_0), gen_Cons:Nil4_0(n573_0)) → *5_0, rt ∈ Ω(n573₀)

Generator Equations:
gen_Cons:Nil4_0(0) ⇔ Nil
gen_Cons:Nil4_0(+(x, 1)) ⇔ Cons(Capture, gen_Cons:Nil4_0(x))

No more defined symbols left to analyse.

(18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
@(gen_Cons:Nil4_0(n6_0), gen_Cons:Nil4_0(b)) → gen_Cons:Nil4_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)

(19) BOUNDS(n^1, INF)

(20) Obligation:

Innermost TRS:
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)

Types:
@ :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Capture:Swap → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
game :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil
Capture :: Capture:Swap
Swap :: Capture:Swap
equal :: Capture:Swap → Capture:Swap → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
hole_Capture:Swap2_0 :: Capture:Swap
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat → Cons:Nil

Lemmas:
@(gen_Cons:Nil4_0(n6_0), gen_Cons:Nil4_0(b)) → gen_Cons:Nil4_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_Cons:Nil4_0(0) ⇔ Nil
gen_Cons:Nil4_0(+(x, 1)) ⇔ Cons(Capture, gen_Cons:Nil4_0(x))

No more defined symbols left to analyse.

(21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
@(gen_Cons:Nil4_0(n6_0), gen_Cons:Nil4_0(b)) → gen_Cons:Nil4_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)

(22) BOUNDS(n^1, INF)