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:

foldl#3(x2, Nil) → x2
foldl#3(x16, Cons(x24, x6)) → foldl#3(Cons(x24, x16), x6)
main(x1) → foldl#3(Nil, x1)

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:

foldl#3(x2, Nil) → x2
foldl#3(x16, Cons(x24, x6)) → foldl#3(Cons(x24, x16), x6)
main(x1) → foldl#3(Nil, x1)

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
foldl#3(x2, Nil) → x2
foldl#3(x16, Cons(x24, x6)) → foldl#3(Cons(x24, x16), x6)
main(x1) → foldl#3(Nil, x1)

Types:
foldl#3 :: Nil:Cons → Nil:Cons → Nil:Cons
Nil :: Nil:Cons
Cons :: a → Nil:Cons → Nil:Cons
main :: Nil:Cons → Nil:Cons
hole_Nil:Cons1_0 :: Nil:Cons
hole_a2_0 :: a
gen_Nil:Cons3_0 :: Nat → Nil:Cons

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

Heuristically decided to analyse the following defined symbols:
foldl#3

(6) Obligation:

Innermost TRS:
Rules:
foldl#3(x2, Nil) → x2
foldl#3(x16, Cons(x24, x6)) → foldl#3(Cons(x24, x16), x6)
main(x1) → foldl#3(Nil, x1)

Types:
foldl#3 :: Nil:Cons → Nil:Cons → Nil:Cons
Nil :: Nil:Cons
Cons :: a → Nil:Cons → Nil:Cons
main :: Nil:Cons → Nil:Cons
hole_Nil:Cons1_0 :: Nil:Cons
hole_a2_0 :: a
gen_Nil:Cons3_0 :: Nat → Nil:Cons

Generator Equations:
gen_Nil:Cons3_0(0) ⇔ Nil
gen_Nil:Cons3_0(+(x, 1)) ⇔ Cons(hole_a2_0, gen_Nil:Cons3_0(x))

The following defined symbols remain to be analysed:
foldl#3

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

Proved the following rewrite lemma:
foldl#3(gen_Nil:Cons3_0(a), gen_Nil:Cons3_0(n5_0)) → gen_Nil:Cons3_0(+(n5_0, a)), rt ∈ Ω(1 + n5₀)

Induction Base:
foldl#3(gen_Nil:Cons3_0(a), gen_Nil:Cons3_0(0)) →_R^Ω(1)
gen_Nil:Cons3_0(a)

Induction Step:
foldl#3(gen_Nil:Cons3_0(a), gen_Nil:Cons3_0(+(n5_0, 1))) →_R^Ω(1)
foldl#3(Cons(hole_a2_0, gen_Nil:Cons3_0(a)), gen_Nil:Cons3_0(n5_0)) →_IH
gen_Nil:Cons3_0(+(+(a, 1), c6_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:
foldl#3(x2, Nil) → x2
foldl#3(x16, Cons(x24, x6)) → foldl#3(Cons(x24, x16), x6)
main(x1) → foldl#3(Nil, x1)

Types:
foldl#3 :: Nil:Cons → Nil:Cons → Nil:Cons
Nil :: Nil:Cons
Cons :: a → Nil:Cons → Nil:Cons
main :: Nil:Cons → Nil:Cons
hole_Nil:Cons1_0 :: Nil:Cons
hole_a2_0 :: a
gen_Nil:Cons3_0 :: Nat → Nil:Cons

Lemmas:
foldl#3(gen_Nil:Cons3_0(a), gen_Nil:Cons3_0(n5_0)) → gen_Nil:Cons3_0(+(n5_0, a)), rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_Nil:Cons3_0(0) ⇔ Nil
gen_Nil:Cons3_0(+(x, 1)) ⇔ Cons(hole_a2_0, gen_Nil:Cons3_0(x))

No more defined symbols left to analyse.

(10) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
foldl#3(gen_Nil:Cons3_0(a), gen_Nil:Cons3_0(n5_0)) → gen_Nil:Cons3_0(+(n5_0, a)), rt ∈ Ω(1 + n5₀)

(11) BOUNDS(n^1, INF)

(12) Obligation:

Innermost TRS:
Rules:
foldl#3(x2, Nil) → x2
foldl#3(x16, Cons(x24, x6)) → foldl#3(Cons(x24, x16), x6)
main(x1) → foldl#3(Nil, x1)

Types:
foldl#3 :: Nil:Cons → Nil:Cons → Nil:Cons
Nil :: Nil:Cons
Cons :: a → Nil:Cons → Nil:Cons
main :: Nil:Cons → Nil:Cons
hole_Nil:Cons1_0 :: Nil:Cons
hole_a2_0 :: a
gen_Nil:Cons3_0 :: Nat → Nil:Cons

Lemmas:
foldl#3(gen_Nil:Cons3_0(a), gen_Nil:Cons3_0(n5_0)) → gen_Nil:Cons3_0(+(n5_0, a)), rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_Nil:Cons3_0(0) ⇔ Nil
gen_Nil:Cons3_0(+(x, 1)) ⇔ Cons(hole_a2_0, gen_Nil:Cons3_0(x))

No more defined symbols left to analyse.

(13) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
foldl#3(gen_Nil:Cons3_0(a), gen_Nil:Cons3_0(n5_0)) → gen_Nil:Cons3_0(+(n5_0, a)), rt ∈ Ω(1 + n5₀)

(14) BOUNDS(n^1, INF)