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:

walk#1(Nil) → walk_xs
walk#1(Cons(x4, x3)) → comp_f_g(walk#1(x3), walk_xs_3(x4))
comp_f_g#1(comp_f_g(x7, x9), walk_xs_3(x8), x12) → comp_f_g#1(x7, x9, Cons(x8, x12))
comp_f_g#1(walk_xs, walk_xs_3(x8), x12) → Cons(x8, x12)
main(Nil) → Nil
main(Cons(x4, x5)) → comp_f_g#1(walk#1(x5), walk_xs_3(x4), Nil)

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:

walk#1(Nil) → walk_xs
walk#1(Cons(x4, x3)) → comp_f_g(walk#1(x3), walk_xs_3(x4))
comp_f_g#1(comp_f_g(x7, x9), walk_xs_3(x8), x12) → comp_f_g#1(x7, x9, Cons(x8, x12))
comp_f_g#1(walk_xs, walk_xs_3(x8), x12) → Cons(x8, x12)
main(Nil) → Nil
main(Cons(x4, x5)) → comp_f_g#1(walk#1(x5), walk_xs_3(x4), Nil)

S is empty.
Rewrite Strategy: INNERMOST

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

Sliced the following arguments:
Cons/0
walk_xs_3/0

(4) Obligation:

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

walk#1(Nil) → walk_xs
walk#1(Cons(x3)) → comp_f_g(walk#1(x3), walk_xs_3)
comp_f_g#1(comp_f_g(x7, x9), walk_xs_3, x12) → comp_f_g#1(x7, x9, Cons(x12))
comp_f_g#1(walk_xs, walk_xs_3, x12) → Cons(x12)
main(Nil) → Nil
main(Cons(x5)) → comp_f_g#1(walk#1(x5), walk_xs_3, Nil)

S is empty.
Rewrite Strategy: INNERMOST

(5) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
walk#1(Cons(x3)) →⁺ comp_f_g(walk#1(x3), walk_xs_3)
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [x3 / Cons(x3)].
The result substitution is [ ].

(6) BOUNDS(n^1, INF)