The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(2^n, 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
binom(Cons(x, xs), Cons(x', xs')) → @(binom(xs, xs'), binom(xs, Cons(x', xs')))
binom(Cons(x, xs), Nil) → Cons(Nil, Nil)
binom(Nil, k) → Cons(Nil, Nil)
goal(x, y) → binom(x, y)

Rewrite Strategy: INNERMOST

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(2ⁿ):
The rewrite sequence
binom(Cons(x, xs), Cons(x', xs')) →⁺ @(binom(xs, xs'), binom(xs, Cons(x', xs')))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [xs / Cons(x, xs), xs' / Cons(x', xs')].
The result substitution is [ ].

The rewrite sequence
binom(Cons(x, xs), Cons(x', xs')) →⁺ @(binom(xs, xs'), binom(xs, Cons(x', xs')))
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(2^n, 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
binom(Cons(x, xs), Cons(x', xs')) → @(binom(xs, xs'), binom(xs, Cons(x', xs')))
binom(Cons(x, xs), Nil) → Cons(Nil, Nil)
binom(Nil, k) → Cons(Nil, Nil)
goal(x, y) → binom(x, y)

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
binom(Cons(x, xs), Cons(x', xs')) → @(binom(xs, xs'), binom(xs, Cons(x', xs')))
binom(Cons(x, xs), Nil) → Cons(Nil, Nil)
binom(Nil, k) → Cons(Nil, Nil)
goal(x, y) → binom(x, y)

Types:
@ :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
binom :: Cons:Nil → Cons:Nil → Cons:Nil
goal :: Cons:Nil → Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
gen_Cons:Nil2_0 :: Nat → Cons:Nil

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

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

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

(8) Obligation:

Innermost TRS:
Rules:
@(Cons(x, xs), ys) → Cons(x, @(xs, ys))
@(Nil, ys) → ys
binom(Cons(x, xs), Cons(x', xs')) → @(binom(xs, xs'), binom(xs, Cons(x', xs')))
binom(Cons(x, xs), Nil) → Cons(Nil, Nil)
binom(Nil, k) → Cons(Nil, Nil)
goal(x, y) → binom(x, y)

Types:
@ :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
binom :: Cons:Nil → Cons:Nil → Cons:Nil
goal :: Cons:Nil → Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
gen_Cons:Nil2_0 :: Nat → Cons:Nil

Generator Equations:
gen_Cons:Nil2_0(0) ⇔ Nil
gen_Cons:Nil2_0(+(x, 1)) ⇔ Cons(Nil, gen_Cons:Nil2_0(x))

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

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

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

Proved the following rewrite lemma:
@(gen_Cons:Nil2_0(n4_0), gen_Cons:Nil2_0(b)) → gen_Cons:Nil2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)

Induction Base:
@(gen_Cons:Nil2_0(0), gen_Cons:Nil2_0(b)) →_R^Ω(1)
gen_Cons:Nil2_0(b)

Induction Step:
@(gen_Cons:Nil2_0(+(n4_0, 1)), gen_Cons:Nil2_0(b)) →_R^Ω(1)
Cons(Nil, @(gen_Cons:Nil2_0(n4_0), gen_Cons:Nil2_0(b))) →_IH
Cons(Nil, gen_Cons:Nil2_0(+(b, c5_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
binom(Cons(x, xs), Cons(x', xs')) → @(binom(xs, xs'), binom(xs, Cons(x', xs')))
binom(Cons(x, xs), Nil) → Cons(Nil, Nil)
binom(Nil, k) → Cons(Nil, Nil)
goal(x, y) → binom(x, y)

Types:
@ :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
binom :: Cons:Nil → Cons:Nil → Cons:Nil
goal :: Cons:Nil → Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
gen_Cons:Nil2_0 :: Nat → Cons:Nil

Lemmas:
@(gen_Cons:Nil2_0(n4_0), gen_Cons:Nil2_0(b)) → gen_Cons:Nil2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_Cons:Nil2_0(0) ⇔ Nil
gen_Cons:Nil2_0(+(x, 1)) ⇔ Cons(Nil, gen_Cons:Nil2_0(x))

The following defined symbols remain to be analysed:
binom

(12) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol binom.

(13) Obligation:

Innermost TRS:
Rules:
@(Cons(x, xs), ys) → Cons(x, @(xs, ys))
@(Nil, ys) → ys
binom(Cons(x, xs), Cons(x', xs')) → @(binom(xs, xs'), binom(xs, Cons(x', xs')))
binom(Cons(x, xs), Nil) → Cons(Nil, Nil)
binom(Nil, k) → Cons(Nil, Nil)
goal(x, y) → binom(x, y)

Types:
@ :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
binom :: Cons:Nil → Cons:Nil → Cons:Nil
goal :: Cons:Nil → Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
gen_Cons:Nil2_0 :: Nat → Cons:Nil

Lemmas:
@(gen_Cons:Nil2_0(n4_0), gen_Cons:Nil2_0(b)) → gen_Cons:Nil2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_Cons:Nil2_0(0) ⇔ Nil
gen_Cons:Nil2_0(+(x, 1)) ⇔ Cons(Nil, gen_Cons:Nil2_0(x))

No more defined symbols left to analyse.

(14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
@(gen_Cons:Nil2_0(n4_0), gen_Cons:Nil2_0(b)) → gen_Cons:Nil2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)

(15) BOUNDS(n^1, INF)

(16) Obligation:

Innermost TRS:
Rules:
@(Cons(x, xs), ys) → Cons(x, @(xs, ys))
@(Nil, ys) → ys
binom(Cons(x, xs), Cons(x', xs')) → @(binom(xs, xs'), binom(xs, Cons(x', xs')))
binom(Cons(x, xs), Nil) → Cons(Nil, Nil)
binom(Nil, k) → Cons(Nil, Nil)
goal(x, y) → binom(x, y)

Types:
@ :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
binom :: Cons:Nil → Cons:Nil → Cons:Nil
goal :: Cons:Nil → Cons:Nil → Cons:Nil
hole_Cons:Nil1_0 :: Cons:Nil
gen_Cons:Nil2_0 :: Nat → Cons:Nil

Lemmas:
@(gen_Cons:Nil2_0(n4_0), gen_Cons:Nil2_0(b)) → gen_Cons:Nil2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_Cons:Nil2_0(0) ⇔ Nil
gen_Cons:Nil2_0(+(x, 1)) ⇔ Cons(Nil, gen_Cons:Nil2_0(x))

No more defined symbols left to analyse.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
@(gen_Cons:Nil2_0(n4_0), gen_Cons:Nil2_0(b)) → gen_Cons:Nil2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)

(18) BOUNDS(n^1, INF)