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:

comp_f_g#1(plus_x(x3), comp_f_g(x1, x2), 0) → plus_x#1(x3, comp_f_g#1(x1, x2, 0))
comp_f_g#1(plus_x(x3), id, 0) → plus_x#1(x3, 0)
map#2(Nil) → Nil
map#2(Cons(x16, x6)) → Cons(plus_x(x16), map#2(x6))
plus_x#1(0, x6) → x6
plus_x#1(S(x8), x10) → S(plus_x#1(x8, x10))
foldr_f#3(Nil, 0) → 0
foldr_f#3(Cons(x16, x5), x24) → comp_f_g#1(x16, foldr#3(x5), x24)
foldr#3(Nil) → id
foldr#3(Cons(x32, x6)) → comp_f_g(x32, foldr#3(x6))
main(x3) → foldr_f#3(map#2(x3), 0)

Rewrite Strategy: INNERMOST

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
map#2(Cons(x16, x6)) →⁺ Cons(plus_x(x16), map#2(x6))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
The pumping substitution is [x6 / Cons(x16, x6)].
The result substitution is [ ].

(2) BOUNDS(n^1, INF)