-- Mathlib/Computability/AkraBazzi/AkraBazzi.lean
lemma isBigO_apply_r_sub_b (q : ℝ → ℝ) (hq_diff : DifferentiableOn ℝ q (Set.Ioi 1))
    (hq_poly : GrowsPolynomially fun x => ‖deriv q x‖) (i : α) :
    (fun n => q (r i n) - q (b i * n)) =O[atTop] fun n => (deriv q n) * (r i n - b i * n)

Proof state:
case diff
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
q : ℝ → ℝ
hq_diff : DifferentiableOn ℝ q (Set.Ioi 1)
hq_poly✝ : GrowsPolynomially fun x => ‖deriv q x‖
i : α
b' : ℝ := b (min_bi b) / 2
hb_pos : 0 < b'
hb_lt_one : b' < 1
hb : b' ∈ Set.Ioo 0 1
hb' : ∀ (i : α), b' ≤ b i
c₁ : ℝ
left✝¹ : c₁ > 0
c₂ : ℝ
left✝ : c₂ > 0
hq_poly :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b' * x) x,
      (fun x => ‖deriv q x‖) u ∈ Set.Icc (c₁ * (fun x => ‖deriv q x‖) x) (c₂ * (fun x => ‖deriv q x‖) x)
h_tendsto : Tendsto (fun x => b' * x) atTop atTop
n : ℕ
hn : ∀ u ∈ Set.Icc (b' * ↑n) ↑n, ‖deriv q u‖ ∈ Set.Icc (c₁ * ‖deriv q ↑n‖) (c₂ * ‖deriv q ↑n‖)
h_bi_le_r : ∀ (i : α), b (min_bi b) / 2 * ↑n ≤ ↑(r i n)
h_ge_n₀ : R.n₀ ≤ n
h_n_pos : 0 < n
h_bn : 1 < b' * ↑n
z : ℝ
hz : z ∈ Set.Icc (b' * ↑n) ↑n
⊢ DifferentiableAt ℝ q z

case mem_Icc
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
q : ℝ → ℝ
hq_diff : DifferentiableOn ℝ q (Set.Ioi 1)
hq_poly✝ : GrowsPolynomially fun x => ‖deriv q x‖
i : α
b' : ℝ := b (min_bi b) / 2
hb_pos : 0 < b'
hb_lt_one : b' < 1
hb : b' ∈ Set.Ioo 0 1
hb' : ∀ (i : α), b' ≤ b i
c₁ : ℝ
left✝¹ : c₁ > 0
c₂ : ℝ
left✝ : c₂ > 0
hq_poly :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b' * x) x,
      (fun x => ‖deriv q x‖) u ∈ Set.Icc (c₁ * (fun x => ‖deriv q x‖) x) (c₂ * (fun x => ‖deriv q x‖) x)
h_tendsto : Tendsto (fun x => b' * x) atTop atTop
n : ℕ
hn : ∀ u ∈ Set.Icc (b' * ↑n) ↑n, ‖deriv q u‖ ∈ Set.Icc (c₁ * ‖deriv q ↑n‖) (c₂ * ‖deriv q ↑n‖)
h_bi_le_r : ∀ (i : α), b (min_bi b) / 2 * ↑n ≤ ↑(r i n)
h_ge_n₀ : R.n₀ ≤ n
h_n_pos : 0 < n
h_bn : 1 < b' * ↑n
⊢ b i * ↑n ∈ Set.Icc (b' * ↑n) ↑n

Tactic:
case diff =>
  refine hq_diff.differentiableAt (Ioi_mem_nhds ?_)
  calc 1 < b' * n := by exact h_bn
       _ ≤ z := hz.1

Result:
case mem_Icc
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
q : ℝ → ℝ
hq_diff : DifferentiableOn ℝ q (Set.Ioi 1)
hq_poly✝ : GrowsPolynomially fun x => ‖deriv q x‖
i : α
b' : ℝ := b (min_bi b) / 2
hb_pos : 0 < b'
hb_lt_one : b' < 1
hb : b' ∈ Set.Ioo 0 1
hb' : ∀ (i : α), b' ≤ b i
c₁ : ℝ
left✝¹ : c₁ > 0
c₂ : ℝ
left✝ : c₂ > 0
hq_poly :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b' * x) x,
      (fun x => ‖deriv q x‖) u ∈ Set.Icc (c₁ * (fun x => ‖deriv q x‖) x) (c₂ * (fun x => ‖deriv q x‖) x)
h_tendsto : Tendsto (fun x => b' * x) atTop atTop
n : ℕ
hn : ∀ u ∈ Set.Icc (b' * ↑n) ↑n, ‖deriv q u‖ ∈ Set.Icc (c₁ * ‖deriv q ↑n‖) (c₂ * ‖deriv q ↑n‖)
h_bi_le_r : ∀ (i : α), b (min_bi b) / 2 * ↑n ≤ ↑(r i n)
h_ge_n₀ : R.n₀ ≤ n
h_n_pos : 0 < n
h_bn : 1 < b' * ↑n
⊢ b i * ↑n ∈ Set.Icc (b' * ↑n) ↑n

Proof state:
case intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
q : ℝ → ℝ
hq_diff : DifferentiableOn ℝ q (Set.Ioi 1)
hq_poly✝ : GrowsPolynomially fun x => ‖deriv q x‖
i : α
b' : ℝ := b (min_bi b) / 2
hb_pos : 0 < b'
hb_lt_one : b' < 1
hb : b' ∈ Set.Ioo 0 1
hb' : ∀ (i : α), b' ≤ b i
c₁ : ℝ
left✝¹ : c₁ > 0
c₂ : ℝ
left✝ : c₂ > 0
hq_poly :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b' * x) x,
      (fun x => ‖deriv q x‖) u ∈ Set.Icc (c₁ * (fun x => ‖deriv q x‖) x) (c₂ * (fun x => ‖deriv q x‖) x)
⊢ (fun n => q ↑(r i n) - q (b i * ↑n)) =O[atTop] fun n => deriv q ↑n * (↑(r i n) - b i * ↑n)

Tactic:
rw [isBigO_iff]

Result:
case intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
q : ℝ → ℝ
hq_diff : DifferentiableOn ℝ q (Set.Ioi 1)
hq_poly✝ : GrowsPolynomially fun x => ‖deriv q x‖
i : α
b' : ℝ := b (min_bi b) / 2
hb_pos : 0 < b'
hb_lt_one : b' < 1
hb : b' ∈ Set.Ioo 0 1
hb' : ∀ (i : α), b' ≤ b i
c₁ : ℝ
left✝¹ : c₁ > 0
c₂ : ℝ
left✝ : c₂ > 0
hq_poly :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b' * x) x,
      (fun x => ‖deriv q x‖) u ∈ Set.Icc (c₁ * (fun x => ‖deriv q x‖) x) (c₂ * (fun x => ‖deriv q x‖) x)
⊢ ∃ c, ∀ᶠ (x : ℕ) in atTop, ‖q ↑(r i x) - q (b i * ↑x)‖ ≤ c * ‖deriv q ↑x * (↑(r i x) - b i * ↑x)‖

Proof state:
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
q : ℝ → ℝ
hq_diff : DifferentiableOn ℝ q (Set.Ioi 1)
hq_poly : GrowsPolynomially fun x => ‖deriv q x‖
i : α
b' : ℝ := b (min_bi b) / 2
⊢ (fun n => q ↑(r i n) - q (b i * ↑n)) =O[atTop] fun n => deriv q ↑n * (↑(r i n) - b i * ↑n)

Tactic:
have hb_pos : 0 < b' := by have := R.b_pos (min_bi b); positivity

Result:
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
q : ℝ → ℝ
hq_diff : DifferentiableOn ℝ q (Set.Ioi 1)
hq_poly : GrowsPolynomially fun x => ‖deriv q x‖
i : α
b' : ℝ := b (min_bi b) / 2
hb_pos : 0 < b'
⊢ (fun n => q ↑(r i n) - q (b i * ↑n)) =O[atTop] fun n => deriv q ↑n * (↑(r i n) - b i * ↑n)

Proof state:
case mem_Icc
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
q : ℝ → ℝ
hq_diff : DifferentiableOn ℝ q (Set.Ioi 1)
hq_poly✝ : GrowsPolynomially fun x => ‖deriv q x‖
i : α
b' : ℝ := b (min_bi b) / 2
hb_pos : 0 < b'
hb_lt_one : b' < 1
hb : b' ∈ Set.Ioo 0 1
hb' : ∀ (i : α), b' ≤ b i
c₁ : ℝ
left✝¹ : c₁ > 0
c₂ : ℝ
left✝ : c₂ > 0
hq_poly :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b' * x) x,
      (fun x => ‖deriv q x‖) u ∈ Set.Icc (c₁ * (fun x => ‖deriv q x‖) x) (c₂ * (fun x => ‖deriv q x‖) x)
h_tendsto : Tendsto (fun x => b' * x) atTop atTop
n : ℕ
hn : ∀ u ∈ Set.Icc (b' * ↑n) ↑n, ‖deriv q u‖ ∈ Set.Icc (c₁ * ‖deriv q ↑n‖) (c₂ * ‖deriv q ↑n‖)
h_bi_le_r : ∀ (i : α), b (min_bi b) / 2 * ↑n ≤ ↑(r i n)
h_ge_n₀ : R.n₀ ≤ n
h_n_pos : 0 < n
h_bn : 1 < b' * ↑n
⊢ b i * ↑n ∈ Set.Icc (b' * ↑n) ↑n

Tactic:
case mem_Icc =>
  refine ⟨by gcongr; exact hb' i, ?_⟩
  calc b i * n ≤ 1 * n := by gcongr; exact le_of_lt <| R.b_lt_one i
               _ = n := by simp

Result:
no goals

Proof state:
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
q : ℝ → ℝ
hq_diff : DifferentiableOn ℝ q (Set.Ioi 1)
hq_poly : GrowsPolynomially fun x => ‖deriv q x‖
i : α
b' : ℝ := b (min_bi b) / 2
hb_pos : 0 < b'
hb_lt_one : b' < 1
hb : b' ∈ Set.Ioo 0 1
hb' : ∀ (i : α), b' ≤ b i
⊢ (fun n => q ↑(r i n) - q (b i * ↑n)) =O[atTop] fun n => deriv q ↑n * (↑(r i n) - b i * ↑n)

Tactic:
obtain ⟨c₁, _, c₂, _, hq_poly⟩ := hq_poly b' hb

Result:
case intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
q : ℝ → ℝ
hq_diff : DifferentiableOn ℝ q (Set.Ioi 1)
hq_poly✝ : GrowsPolynomially fun x => ‖deriv q x‖
i : α
b' : ℝ := b (min_bi b) / 2
hb_pos : 0 < b'
hb_lt_one : b' < 1
hb : b' ∈ Set.Ioo 0 1
hb' : ∀ (i : α), b' ≤ b i
c₁ : ℝ
left✝¹ : c₁ > 0
c₂ : ℝ
left✝ : c₂ > 0
hq_poly :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b' * x) x,
      (fun x => ‖deriv q x‖) u ∈ Set.Icc (c₁ * (fun x => ‖deriv q x‖) x) (c₂ * (fun x => ‖deriv q x‖) x)
⊢ (fun n => q ↑(r i n) - q (b i * ↑n)) =O[atTop] fun n => deriv q ↑n * (↑(r i n) - b i * ↑n)