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

Infered types.

(6) Obligation:

Innermost TRS:
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)

Types:
walk#1 :: Nil:Cons → walk_xs:comp_f_g
Nil :: Nil:Cons
walk_xs :: walk_xs:comp_f_g
Cons :: Nil:Cons → Nil:Cons
comp_f_g :: walk_xs:comp_f_g → walk_xs_3 → walk_xs:comp_f_g
walk_xs_3 :: 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_walk_xs_33_0 :: walk_xs_3
gen_walk_xs:comp_f_g4_0 :: Nat → walk_xs:comp_f_g
gen_Nil:Cons5_0 :: Nat → Nil:Cons

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

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

(8) Obligation:

Innermost TRS:
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)

Types:
walk#1 :: Nil:Cons → walk_xs:comp_f_g
Nil :: Nil:Cons
walk_xs :: walk_xs:comp_f_g
Cons :: Nil:Cons → Nil:Cons
comp_f_g :: walk_xs:comp_f_g → walk_xs_3 → walk_xs:comp_f_g
walk_xs_3 :: 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_walk_xs_33_0 :: walk_xs_3
gen_walk_xs:comp_f_g4_0 :: Nat → walk_xs:comp_f_g
gen_Nil:Cons5_0 :: Nat → Nil:Cons

Generator Equations:
gen_walk_xs:comp_f_g4_0(0) ⇔ walk_xs
gen_walk_xs:comp_f_g4_0(+(x, 1)) ⇔ comp_f_g(gen_walk_xs:comp_f_g4_0(x), walk_xs_3)
gen_Nil:Cons5_0(0) ⇔ Nil
gen_Nil:Cons5_0(+(x, 1)) ⇔ Cons(gen_Nil:Cons5_0(x))

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

(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
walk#1(gen_Nil:Cons5_0(n7_0)) → gen_walk_xs:comp_f_g4_0(n7_0), rt ∈ Ω(1 + n7₀)

Induction Base:
walk#1(gen_Nil:Cons5_0(0)) →_R^Ω(1)
walk_xs

Induction Step:
walk#1(gen_Nil:Cons5_0(+(n7_0, 1))) →_R^Ω(1)
comp_f_g(walk#1(gen_Nil:Cons5_0(n7_0)), walk_xs_3) →_IH
comp_f_g(gen_walk_xs:comp_f_g4_0(c8_0), walk_xs_3)

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

(10) Complex Obligation (BEST)

(11) Obligation:

Innermost TRS:
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)

Types:
walk#1 :: Nil:Cons → walk_xs:comp_f_g
Nil :: Nil:Cons
walk_xs :: walk_xs:comp_f_g
Cons :: Nil:Cons → Nil:Cons
comp_f_g :: walk_xs:comp_f_g → walk_xs_3 → walk_xs:comp_f_g
walk_xs_3 :: 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_walk_xs_33_0 :: walk_xs_3
gen_walk_xs:comp_f_g4_0 :: Nat → walk_xs:comp_f_g
gen_Nil:Cons5_0 :: Nat → Nil:Cons

Lemmas:
walk#1(gen_Nil:Cons5_0(n7_0)) → gen_walk_xs:comp_f_g4_0(n7_0), rt ∈ Ω(1 + n7₀)

Generator Equations:
gen_walk_xs:comp_f_g4_0(0) ⇔ walk_xs
gen_walk_xs:comp_f_g4_0(+(x, 1)) ⇔ comp_f_g(gen_walk_xs:comp_f_g4_0(x), walk_xs_3)
gen_Nil:Cons5_0(0) ⇔ Nil
gen_Nil:Cons5_0(+(x, 1)) ⇔ Cons(gen_Nil:Cons5_0(x))

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

(12) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
comp_f_g#1(gen_walk_xs:comp_f_g4_0(n199_0), walk_xs_3, gen_Nil:Cons5_0(b)) → gen_Nil:Cons5_0(+(+(1, n199_0), b)), rt ∈ Ω(1 + n199₀)

Induction Base:
comp_f_g#1(gen_walk_xs:comp_f_g4_0(0), walk_xs_3, gen_Nil:Cons5_0(b)) →_R^Ω(1)
Cons(gen_Nil:Cons5_0(b))

Induction Step:
comp_f_g#1(gen_walk_xs:comp_f_g4_0(+(n199_0, 1)), walk_xs_3, gen_Nil:Cons5_0(b)) →_R^Ω(1)
comp_f_g#1(gen_walk_xs:comp_f_g4_0(n199_0), walk_xs_3, Cons(gen_Nil:Cons5_0(b))) →_IH
gen_Nil:Cons5_0(+(+(1, +(b, 1)), c200_0))

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

(13) Complex Obligation (BEST)

(14) Obligation:

Innermost TRS:
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)

Types:
walk#1 :: Nil:Cons → walk_xs:comp_f_g
Nil :: Nil:Cons
walk_xs :: walk_xs:comp_f_g
Cons :: Nil:Cons → Nil:Cons
comp_f_g :: walk_xs:comp_f_g → walk_xs_3 → walk_xs:comp_f_g
walk_xs_3 :: 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_walk_xs_33_0 :: walk_xs_3
gen_walk_xs:comp_f_g4_0 :: Nat → walk_xs:comp_f_g
gen_Nil:Cons5_0 :: Nat → Nil:Cons

Lemmas:
walk#1(gen_Nil:Cons5_0(n7_0)) → gen_walk_xs:comp_f_g4_0(n7_0), rt ∈ Ω(1 + n7₀)
comp_f_g#1(gen_walk_xs:comp_f_g4_0(n199_0), walk_xs_3, gen_Nil:Cons5_0(b)) → gen_Nil:Cons5_0(+(+(1, n199_0), b)), rt ∈ Ω(1 + n199₀)

Generator Equations:
gen_walk_xs:comp_f_g4_0(0) ⇔ walk_xs
gen_walk_xs:comp_f_g4_0(+(x, 1)) ⇔ comp_f_g(gen_walk_xs:comp_f_g4_0(x), walk_xs_3)
gen_Nil:Cons5_0(0) ⇔ Nil
gen_Nil:Cons5_0(+(x, 1)) ⇔ Cons(gen_Nil:Cons5_0(x))

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
walk#1(gen_Nil:Cons5_0(n7_0)) → gen_walk_xs:comp_f_g4_0(n7_0), rt ∈ Ω(1 + n7₀)

(16) BOUNDS(n^1, INF)

(17) Obligation:

Innermost TRS:
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)

Types:
walk#1 :: Nil:Cons → walk_xs:comp_f_g
Nil :: Nil:Cons
walk_xs :: walk_xs:comp_f_g
Cons :: Nil:Cons → Nil:Cons
comp_f_g :: walk_xs:comp_f_g → walk_xs_3 → walk_xs:comp_f_g
walk_xs_3 :: 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_walk_xs_33_0 :: walk_xs_3
gen_walk_xs:comp_f_g4_0 :: Nat → walk_xs:comp_f_g
gen_Nil:Cons5_0 :: Nat → Nil:Cons

Lemmas:
walk#1(gen_Nil:Cons5_0(n7_0)) → gen_walk_xs:comp_f_g4_0(n7_0), rt ∈ Ω(1 + n7₀)
comp_f_g#1(gen_walk_xs:comp_f_g4_0(n199_0), walk_xs_3, gen_Nil:Cons5_0(b)) → gen_Nil:Cons5_0(+(+(1, n199_0), b)), rt ∈ Ω(1 + n199₀)

Generator Equations:
gen_walk_xs:comp_f_g4_0(0) ⇔ walk_xs
gen_walk_xs:comp_f_g4_0(+(x, 1)) ⇔ comp_f_g(gen_walk_xs:comp_f_g4_0(x), walk_xs_3)
gen_Nil:Cons5_0(0) ⇔ Nil
gen_Nil:Cons5_0(+(x, 1)) ⇔ Cons(gen_Nil:Cons5_0(x))

No more defined symbols left to analyse.

(18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
walk#1(gen_Nil:Cons5_0(n7_0)) → gen_walk_xs:comp_f_g4_0(n7_0), rt ∈ Ω(1 + n7₀)

(19) BOUNDS(n^1, INF)

(20) Obligation:

Innermost TRS:
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)

Types:
walk#1 :: Nil:Cons → walk_xs:comp_f_g
Nil :: Nil:Cons
walk_xs :: walk_xs:comp_f_g
Cons :: Nil:Cons → Nil:Cons
comp_f_g :: walk_xs:comp_f_g → walk_xs_3 → walk_xs:comp_f_g
walk_xs_3 :: 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_walk_xs_33_0 :: walk_xs_3
gen_walk_xs:comp_f_g4_0 :: Nat → walk_xs:comp_f_g
gen_Nil:Cons5_0 :: Nat → Nil:Cons

Lemmas:
walk#1(gen_Nil:Cons5_0(n7_0)) → gen_walk_xs:comp_f_g4_0(n7_0), rt ∈ Ω(1 + n7₀)

Generator Equations:
gen_walk_xs:comp_f_g4_0(0) ⇔ walk_xs
gen_walk_xs:comp_f_g4_0(+(x, 1)) ⇔ comp_f_g(gen_walk_xs:comp_f_g4_0(x), walk_xs_3)
gen_Nil:Cons5_0(0) ⇔ Nil
gen_Nil:Cons5_0(+(x, 1)) ⇔ Cons(gen_Nil:Cons5_0(x))

No more defined symbols left to analyse.

(21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
walk#1(gen_Nil:Cons5_0(n7_0)) → gen_walk_xs:comp_f_g4_0(n7_0), rt ∈ Ω(1 + n7₀)

(22) BOUNDS(n^1, INF)