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) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
Cons/0

(4) Obligation:

Runtime Complexity Relative TRS:
The TRS R consists of the following rules:

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

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(6) Obligation:

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

Types:
foldl#3 :: Nil:Cons → Nil:Cons → Nil:Cons
Nil :: Nil:Cons
Cons :: Nil:Cons → Nil:Cons
main :: Nil:Cons → Nil:Cons
hole_Nil:Cons1_0 :: Nil:Cons
gen_Nil:Cons2_0 :: Nat → Nil:Cons

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

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

(8) Obligation:

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

Types:
foldl#3 :: Nil:Cons → Nil:Cons → Nil:Cons
Nil :: Nil:Cons
Cons :: Nil:Cons → Nil:Cons
main :: Nil:Cons → Nil:Cons
hole_Nil:Cons1_0 :: Nil:Cons
gen_Nil:Cons2_0 :: Nat → Nil:Cons

Generator Equations:
gen_Nil:Cons2_0(0) ⇔ Nil
gen_Nil:Cons2_0(+(x, 1)) ⇔ Cons(gen_Nil:Cons2_0(x))

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

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

Proved the following rewrite lemma:
foldl#3(gen_Nil:Cons2_0(a), gen_Nil:Cons2_0(n4_0)) → gen_Nil:Cons2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)

Induction Base:
foldl#3(gen_Nil:Cons2_0(a), gen_Nil:Cons2_0(0)) →_R^Ω(1)
gen_Nil:Cons2_0(a)

Induction Step:
foldl#3(gen_Nil:Cons2_0(a), gen_Nil:Cons2_0(+(n4_0, 1))) →_R^Ω(1)
foldl#3(Cons(gen_Nil:Cons2_0(a)), gen_Nil:Cons2_0(n4_0)) →_IH
gen_Nil:Cons2_0(+(+(a, 1), c5_0))

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

Types:
foldl#3 :: Nil:Cons → Nil:Cons → Nil:Cons
Nil :: Nil:Cons
Cons :: Nil:Cons → Nil:Cons
main :: Nil:Cons → Nil:Cons
hole_Nil:Cons1_0 :: Nil:Cons
gen_Nil:Cons2_0 :: Nat → Nil:Cons

Lemmas:
foldl#3(gen_Nil:Cons2_0(a), gen_Nil:Cons2_0(n4_0)) → gen_Nil:Cons2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_Nil:Cons2_0(0) ⇔ Nil
gen_Nil:Cons2_0(+(x, 1)) ⇔ Cons(gen_Nil:Cons2_0(x))

No more defined symbols left to analyse.

(12) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
foldl#3(gen_Nil:Cons2_0(a), gen_Nil:Cons2_0(n4_0)) → gen_Nil:Cons2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)

(13) BOUNDS(n^1, INF)

(14) Obligation:

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

Types:
foldl#3 :: Nil:Cons → Nil:Cons → Nil:Cons
Nil :: Nil:Cons
Cons :: Nil:Cons → Nil:Cons
main :: Nil:Cons → Nil:Cons
hole_Nil:Cons1_0 :: Nil:Cons
gen_Nil:Cons2_0 :: Nat → Nil:Cons

Lemmas:
foldl#3(gen_Nil:Cons2_0(a), gen_Nil:Cons2_0(n4_0)) → gen_Nil:Cons2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_Nil:Cons2_0(0) ⇔ Nil
gen_Nil:Cons2_0(+(x, 1)) ⇔ Cons(gen_Nil:Cons2_0(x))

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
foldl#3(gen_Nil:Cons2_0(a), gen_Nil:Cons2_0(n4_0)) → gen_Nil:Cons2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)

(16) BOUNDS(n^1, INF)