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) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

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

Infered types.

(4) Obligation:

Innermost TRS:
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)

Types:
walk#1 :: Nil:Cons → walk_xs:comp_f_g
Nil :: Nil:Cons
walk_xs :: walk_xs:comp_f_g
Cons :: a → Nil:Cons → Nil:Cons
comp_f_g :: walk_xs:comp_f_g → walk_xs_3 → walk_xs:comp_f_g
walk_xs_3 :: a → walk_xs_3
comp_f_g#1 :: walk_xs:comp_f_g → walk_xs_3 → Nil:Cons → Nil:Cons
main :: Nil:Cons → Nil:Cons
hole_walk_xs:comp_f_g1_0 :: walk_xs:comp_f_g
hole_Nil:Cons2_0 :: Nil:Cons
hole_a3_0 :: a
hole_walk_xs_34_0 :: walk_xs_3
gen_walk_xs:comp_f_g5_0 :: Nat → walk_xs:comp_f_g
gen_Nil:Cons6_0 :: Nat → Nil:Cons

(5) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
walk#1, comp_f_g#1

(6) Obligation:

Innermost TRS:
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)

Types:
walk#1 :: Nil:Cons → walk_xs:comp_f_g
Nil :: Nil:Cons
walk_xs :: walk_xs:comp_f_g
Cons :: a → Nil:Cons → Nil:Cons
comp_f_g :: walk_xs:comp_f_g → walk_xs_3 → walk_xs:comp_f_g
walk_xs_3 :: a → walk_xs_3
comp_f_g#1 :: walk_xs:comp_f_g → walk_xs_3 → Nil:Cons → Nil:Cons
main :: Nil:Cons → Nil:Cons
hole_walk_xs:comp_f_g1_0 :: walk_xs:comp_f_g
hole_Nil:Cons2_0 :: Nil:Cons
hole_a3_0 :: a
hole_walk_xs_34_0 :: walk_xs_3
gen_walk_xs:comp_f_g5_0 :: Nat → walk_xs:comp_f_g
gen_Nil:Cons6_0 :: Nat → Nil:Cons

Generator Equations:
gen_walk_xs:comp_f_g5_0(0) ⇔ walk_xs
gen_walk_xs:comp_f_g5_0(+(x, 1)) ⇔ comp_f_g(gen_walk_xs:comp_f_g5_0(x), walk_xs_3(hole_a3_0))
gen_Nil:Cons6_0(0) ⇔ Nil
gen_Nil:Cons6_0(+(x, 1)) ⇔ Cons(hole_a3_0, gen_Nil:Cons6_0(x))

The following defined symbols remain to be analysed:
walk#1, comp_f_g#1

(7) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
walk#1(gen_Nil:Cons6_0(n8_0)) → gen_walk_xs:comp_f_g5_0(n8_0), rt ∈ Ω(1 + n8₀)

Induction Base:
walk#1(gen_Nil:Cons6_0(0)) →_R^Ω(1)
walk_xs

Induction Step:
walk#1(gen_Nil:Cons6_0(+(n8_0, 1))) →_R^Ω(1)
comp_f_g(walk#1(gen_Nil:Cons6_0(n8_0)), walk_xs_3(hole_a3_0)) →_IH
comp_f_g(gen_walk_xs:comp_f_g5_0(c9_0), walk_xs_3(hole_a3_0))

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(8) Complex Obligation (BEST)

(9) Obligation:

Innermost TRS:
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)

Types:
walk#1 :: Nil:Cons → walk_xs:comp_f_g
Nil :: Nil:Cons
walk_xs :: walk_xs:comp_f_g
Cons :: a → Nil:Cons → Nil:Cons
comp_f_g :: walk_xs:comp_f_g → walk_xs_3 → walk_xs:comp_f_g
walk_xs_3 :: a → walk_xs_3
comp_f_g#1 :: walk_xs:comp_f_g → walk_xs_3 → Nil:Cons → Nil:Cons
main :: Nil:Cons → Nil:Cons
hole_walk_xs:comp_f_g1_0 :: walk_xs:comp_f_g
hole_Nil:Cons2_0 :: Nil:Cons
hole_a3_0 :: a
hole_walk_xs_34_0 :: walk_xs_3
gen_walk_xs:comp_f_g5_0 :: Nat → walk_xs:comp_f_g
gen_Nil:Cons6_0 :: Nat → Nil:Cons

Lemmas:
walk#1(gen_Nil:Cons6_0(n8_0)) → gen_walk_xs:comp_f_g5_0(n8_0), rt ∈ Ω(1 + n8₀)

Generator Equations:
gen_walk_xs:comp_f_g5_0(0) ⇔ walk_xs
gen_walk_xs:comp_f_g5_0(+(x, 1)) ⇔ comp_f_g(gen_walk_xs:comp_f_g5_0(x), walk_xs_3(hole_a3_0))
gen_Nil:Cons6_0(0) ⇔ Nil
gen_Nil:Cons6_0(+(x, 1)) ⇔ Cons(hole_a3_0, gen_Nil:Cons6_0(x))

The following defined symbols remain to be analysed:
comp_f_g#1

(10) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
comp_f_g#1(gen_walk_xs:comp_f_g5_0(n218_0), walk_xs_3(hole_a3_0), gen_Nil:Cons6_0(b)) → gen_Nil:Cons6_0(+(+(1, n218_0), b)), rt ∈ Ω(1 + n218₀)

Induction Base:
comp_f_g#1(gen_walk_xs:comp_f_g5_0(0), walk_xs_3(hole_a3_0), gen_Nil:Cons6_0(b)) →_R^Ω(1)
Cons(hole_a3_0, gen_Nil:Cons6_0(b))

Induction Step:
comp_f_g#1(gen_walk_xs:comp_f_g5_0(+(n218_0, 1)), walk_xs_3(hole_a3_0), gen_Nil:Cons6_0(b)) →_R^Ω(1)
comp_f_g#1(gen_walk_xs:comp_f_g5_0(n218_0), walk_xs_3(hole_a3_0), Cons(hole_a3_0, gen_Nil:Cons6_0(b))) →_IH
gen_Nil:Cons6_0(+(+(1, +(b, 1)), c219_0))

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(11) Complex Obligation (BEST)

(12) Obligation:

Innermost TRS:
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)

Types:
walk#1 :: Nil:Cons → walk_xs:comp_f_g
Nil :: Nil:Cons
walk_xs :: walk_xs:comp_f_g
Cons :: a → Nil:Cons → Nil:Cons
comp_f_g :: walk_xs:comp_f_g → walk_xs_3 → walk_xs:comp_f_g
walk_xs_3 :: a → walk_xs_3
comp_f_g#1 :: walk_xs:comp_f_g → walk_xs_3 → Nil:Cons → Nil:Cons
main :: Nil:Cons → Nil:Cons
hole_walk_xs:comp_f_g1_0 :: walk_xs:comp_f_g
hole_Nil:Cons2_0 :: Nil:Cons
hole_a3_0 :: a
hole_walk_xs_34_0 :: walk_xs_3
gen_walk_xs:comp_f_g5_0 :: Nat → walk_xs:comp_f_g
gen_Nil:Cons6_0 :: Nat → Nil:Cons

Lemmas:
walk#1(gen_Nil:Cons6_0(n8_0)) → gen_walk_xs:comp_f_g5_0(n8_0), rt ∈ Ω(1 + n8₀)
comp_f_g#1(gen_walk_xs:comp_f_g5_0(n218_0), walk_xs_3(hole_a3_0), gen_Nil:Cons6_0(b)) → gen_Nil:Cons6_0(+(+(1, n218_0), b)), rt ∈ Ω(1 + n218₀)

Generator Equations:
gen_walk_xs:comp_f_g5_0(0) ⇔ walk_xs
gen_walk_xs:comp_f_g5_0(+(x, 1)) ⇔ comp_f_g(gen_walk_xs:comp_f_g5_0(x), walk_xs_3(hole_a3_0))
gen_Nil:Cons6_0(0) ⇔ Nil
gen_Nil:Cons6_0(+(x, 1)) ⇔ Cons(hole_a3_0, gen_Nil:Cons6_0(x))

No more defined symbols left to analyse.

(13) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
walk#1(gen_Nil:Cons6_0(n8_0)) → gen_walk_xs:comp_f_g5_0(n8_0), rt ∈ Ω(1 + n8₀)

(14) BOUNDS(n^1, INF)

(15) Obligation:

Innermost TRS:
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)

Types:
walk#1 :: Nil:Cons → walk_xs:comp_f_g
Nil :: Nil:Cons
walk_xs :: walk_xs:comp_f_g
Cons :: a → Nil:Cons → Nil:Cons
comp_f_g :: walk_xs:comp_f_g → walk_xs_3 → walk_xs:comp_f_g
walk_xs_3 :: a → walk_xs_3
comp_f_g#1 :: walk_xs:comp_f_g → walk_xs_3 → Nil:Cons → Nil:Cons
main :: Nil:Cons → Nil:Cons
hole_walk_xs:comp_f_g1_0 :: walk_xs:comp_f_g
hole_Nil:Cons2_0 :: Nil:Cons
hole_a3_0 :: a
hole_walk_xs_34_0 :: walk_xs_3
gen_walk_xs:comp_f_g5_0 :: Nat → walk_xs:comp_f_g
gen_Nil:Cons6_0 :: Nat → Nil:Cons

Lemmas:
walk#1(gen_Nil:Cons6_0(n8_0)) → gen_walk_xs:comp_f_g5_0(n8_0), rt ∈ Ω(1 + n8₀)
comp_f_g#1(gen_walk_xs:comp_f_g5_0(n218_0), walk_xs_3(hole_a3_0), gen_Nil:Cons6_0(b)) → gen_Nil:Cons6_0(+(+(1, n218_0), b)), rt ∈ Ω(1 + n218₀)

Generator Equations:
gen_walk_xs:comp_f_g5_0(0) ⇔ walk_xs
gen_walk_xs:comp_f_g5_0(+(x, 1)) ⇔ comp_f_g(gen_walk_xs:comp_f_g5_0(x), walk_xs_3(hole_a3_0))
gen_Nil:Cons6_0(0) ⇔ Nil
gen_Nil:Cons6_0(+(x, 1)) ⇔ Cons(hole_a3_0, gen_Nil:Cons6_0(x))

No more defined symbols left to analyse.

(16) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
walk#1(gen_Nil:Cons6_0(n8_0)) → gen_walk_xs:comp_f_g5_0(n8_0), rt ∈ Ω(1 + n8₀)

(17) BOUNDS(n^1, INF)

(18) Obligation:

Innermost TRS:
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)

Types:
walk#1 :: Nil:Cons → walk_xs:comp_f_g
Nil :: Nil:Cons
walk_xs :: walk_xs:comp_f_g
Cons :: a → Nil:Cons → Nil:Cons
comp_f_g :: walk_xs:comp_f_g → walk_xs_3 → walk_xs:comp_f_g
walk_xs_3 :: a → walk_xs_3
comp_f_g#1 :: walk_xs:comp_f_g → walk_xs_3 → Nil:Cons → Nil:Cons
main :: Nil:Cons → Nil:Cons
hole_walk_xs:comp_f_g1_0 :: walk_xs:comp_f_g
hole_Nil:Cons2_0 :: Nil:Cons
hole_a3_0 :: a
hole_walk_xs_34_0 :: walk_xs_3
gen_walk_xs:comp_f_g5_0 :: Nat → walk_xs:comp_f_g
gen_Nil:Cons6_0 :: Nat → Nil:Cons

Lemmas:
walk#1(gen_Nil:Cons6_0(n8_0)) → gen_walk_xs:comp_f_g5_0(n8_0), rt ∈ Ω(1 + n8₀)

Generator Equations:
gen_walk_xs:comp_f_g5_0(0) ⇔ walk_xs
gen_walk_xs:comp_f_g5_0(+(x, 1)) ⇔ comp_f_g(gen_walk_xs:comp_f_g5_0(x), walk_xs_3(hole_a3_0))
gen_Nil:Cons6_0(0) ⇔ Nil
gen_Nil:Cons6_0(+(x, 1)) ⇔ Cons(hole_a3_0, gen_Nil:Cons6_0(x))

No more defined symbols left to analyse.

(19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
walk#1(gen_Nil:Cons6_0(n8_0)) → gen_walk_xs:comp_f_g5_0(n8_0), rt ∈ Ω(1 + n8₀)

(20) BOUNDS(n^1, INF)