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)

(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:

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')

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(6) Obligation:

Innermost TRS:
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')

Types:
comp_f_g#1 :: plus_x:0':S → comp_f_g:id → plus_x:0':S → plus_x:0':S
plus_x :: plus_x:0':S → plus_x:0':S
comp_f_g :: plus_x:0':S → comp_f_g:id → comp_f_g:id
0' :: plus_x:0':S
plus_x#1 :: plus_x:0':S → plus_x:0':S → plus_x:0':S
id :: comp_f_g:id
map#2 :: Nil:Cons → Nil:Cons
Nil :: Nil:Cons
Cons :: plus_x:0':S → Nil:Cons → Nil:Cons
S :: plus_x:0':S → plus_x:0':S
foldr_f#3 :: Nil:Cons → plus_x:0':S → plus_x:0':S
foldr#3 :: Nil:Cons → comp_f_g:id
main :: Nil:Cons → plus_x:0':S
hole_plus_x:0':S1_7 :: plus_x:0':S
hole_comp_f_g:id2_7 :: comp_f_g:id
hole_Nil:Cons3_7 :: Nil:Cons
gen_plus_x:0':S4_7 :: Nat → plus_x:0':S
gen_comp_f_g:id5_7 :: Nat → comp_f_g:id
gen_Nil:Cons6_7 :: Nat → Nil:Cons

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

Heuristically decided to analyse the following defined symbols:
comp_f_g#1, plus_x#1, map#2, foldr#3

They will be analysed ascendingly in the following order:
plus_x#1 < comp_f_g#1

(8) Obligation:

Innermost TRS:
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')

Types:
comp_f_g#1 :: plus_x:0':S → comp_f_g:id → plus_x:0':S → plus_x:0':S
plus_x :: plus_x:0':S → plus_x:0':S
comp_f_g :: plus_x:0':S → comp_f_g:id → comp_f_g:id
0' :: plus_x:0':S
plus_x#1 :: plus_x:0':S → plus_x:0':S → plus_x:0':S
id :: comp_f_g:id
map#2 :: Nil:Cons → Nil:Cons
Nil :: Nil:Cons
Cons :: plus_x:0':S → Nil:Cons → Nil:Cons
S :: plus_x:0':S → plus_x:0':S
foldr_f#3 :: Nil:Cons → plus_x:0':S → plus_x:0':S
foldr#3 :: Nil:Cons → comp_f_g:id
main :: Nil:Cons → plus_x:0':S
hole_plus_x:0':S1_7 :: plus_x:0':S
hole_comp_f_g:id2_7 :: comp_f_g:id
hole_Nil:Cons3_7 :: Nil:Cons
gen_plus_x:0':S4_7 :: Nat → plus_x:0':S
gen_comp_f_g:id5_7 :: Nat → comp_f_g:id
gen_Nil:Cons6_7 :: Nat → Nil:Cons

Generator Equations:
gen_plus_x:0':S4_7(0) ⇔ 0'
gen_plus_x:0':S4_7(+(x, 1)) ⇔ plus_x(gen_plus_x:0':S4_7(x))
gen_comp_f_g:id5_7(0) ⇔ id
gen_comp_f_g:id5_7(+(x, 1)) ⇔ comp_f_g(0', gen_comp_f_g:id5_7(x))
gen_Nil:Cons6_7(0) ⇔ Nil
gen_Nil:Cons6_7(+(x, 1)) ⇔ Cons(0', gen_Nil:Cons6_7(x))

The following defined symbols remain to be analysed:
plus_x#1, comp_f_g#1, map#2, foldr#3

They will be analysed ascendingly in the following order:
plus_x#1 < comp_f_g#1

(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol plus_x#1.

(10) Obligation:

Innermost TRS:
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')

Types:
comp_f_g#1 :: plus_x:0':S → comp_f_g:id → plus_x:0':S → plus_x:0':S
plus_x :: plus_x:0':S → plus_x:0':S
comp_f_g :: plus_x:0':S → comp_f_g:id → comp_f_g:id
0' :: plus_x:0':S
plus_x#1 :: plus_x:0':S → plus_x:0':S → plus_x:0':S
id :: comp_f_g:id
map#2 :: Nil:Cons → Nil:Cons
Nil :: Nil:Cons
Cons :: plus_x:0':S → Nil:Cons → Nil:Cons
S :: plus_x:0':S → plus_x:0':S
foldr_f#3 :: Nil:Cons → plus_x:0':S → plus_x:0':S
foldr#3 :: Nil:Cons → comp_f_g:id
main :: Nil:Cons → plus_x:0':S
hole_plus_x:0':S1_7 :: plus_x:0':S
hole_comp_f_g:id2_7 :: comp_f_g:id
hole_Nil:Cons3_7 :: Nil:Cons
gen_plus_x:0':S4_7 :: Nat → plus_x:0':S
gen_comp_f_g:id5_7 :: Nat → comp_f_g:id
gen_Nil:Cons6_7 :: Nat → Nil:Cons

Generator Equations:
gen_plus_x:0':S4_7(0) ⇔ 0'
gen_plus_x:0':S4_7(+(x, 1)) ⇔ plus_x(gen_plus_x:0':S4_7(x))
gen_comp_f_g:id5_7(0) ⇔ id
gen_comp_f_g:id5_7(+(x, 1)) ⇔ comp_f_g(0', gen_comp_f_g:id5_7(x))
gen_Nil:Cons6_7(0) ⇔ Nil
gen_Nil:Cons6_7(+(x, 1)) ⇔ Cons(0', gen_Nil:Cons6_7(x))

The following defined symbols remain to be analysed:
comp_f_g#1, map#2, foldr#3

(11) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol comp_f_g#1.

(12) Obligation:

Innermost TRS:
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')

Types:
comp_f_g#1 :: plus_x:0':S → comp_f_g:id → plus_x:0':S → plus_x:0':S
plus_x :: plus_x:0':S → plus_x:0':S
comp_f_g :: plus_x:0':S → comp_f_g:id → comp_f_g:id
0' :: plus_x:0':S
plus_x#1 :: plus_x:0':S → plus_x:0':S → plus_x:0':S
id :: comp_f_g:id
map#2 :: Nil:Cons → Nil:Cons
Nil :: Nil:Cons
Cons :: plus_x:0':S → Nil:Cons → Nil:Cons
S :: plus_x:0':S → plus_x:0':S
foldr_f#3 :: Nil:Cons → plus_x:0':S → plus_x:0':S
foldr#3 :: Nil:Cons → comp_f_g:id
main :: Nil:Cons → plus_x:0':S
hole_plus_x:0':S1_7 :: plus_x:0':S
hole_comp_f_g:id2_7 :: comp_f_g:id
hole_Nil:Cons3_7 :: Nil:Cons
gen_plus_x:0':S4_7 :: Nat → plus_x:0':S
gen_comp_f_g:id5_7 :: Nat → comp_f_g:id
gen_Nil:Cons6_7 :: Nat → Nil:Cons

Generator Equations:
gen_plus_x:0':S4_7(0) ⇔ 0'
gen_plus_x:0':S4_7(+(x, 1)) ⇔ plus_x(gen_plus_x:0':S4_7(x))
gen_comp_f_g:id5_7(0) ⇔ id
gen_comp_f_g:id5_7(+(x, 1)) ⇔ comp_f_g(0', gen_comp_f_g:id5_7(x))
gen_Nil:Cons6_7(0) ⇔ Nil
gen_Nil:Cons6_7(+(x, 1)) ⇔ Cons(0', gen_Nil:Cons6_7(x))

The following defined symbols remain to be analysed:
map#2, foldr#3