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) CpxTrsToCpxRelTrsProof (BOTH BOUNDS(ID, ID) transformation)

Transformed TRS to relative TRS where S is empty.

(2) 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

(3) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
Cons/0

(4) Obligation:

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

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

S is empty.
Rewrite Strategy: INNERMOST

(5) DecreasingLoopProof (EQUIVALENT transformation)

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

The rewrite sequence
binom(Cons(xs), Cons(xs')) →⁺ @(binom(xs, xs'), binom(xs, Cons(xs')))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
The pumping substitution is [xs / Cons(xs)].
The result substitution is [ ].

(6) BOUNDS(2^n, INF)