awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (goal)
Goal:
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
hb₀ : 0 < b
h_tendsto : Tendsto (β := ℝ) (fun (x : ℝ) ↦ 1 / 2 * log x) atTop atTop
x : ℝ
hx_pos : 1 < x
hx : ∀ (y : ℝ), b * id x ≤ y → -log b < 1 / 2 * log y
u : ℝ
hu : u ∈ Set.Icc (b * x) x
rw : log (x * b) = log (b * x)
⊢ log x + log b = log (b * x)
have rw₁ : log (b * x) = log (b * x)
Tactic: rw [← Real.log_mul (by positivity) (by positivity), mul_comm]
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (goal)
Goal:
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
hb₀ : 0 < b
h_tendsto : Tendsto (β := ℝ) (fun (x : ℝ) ↦ 1 / 2 * log x) atTop atTop
x : ℝ
hx_pos : 1 < x
hx : ∀ (y : ℝ), b * id x ≤ y → -log b < 1 / 2 * log y
u : ℝ
hu : u ∈ Set.Icc (b * x) x
⊢ log (b * x) ≤ log u
have hxy : b * x ≤ u
Tactic: gcongr
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (goal)
Goal:
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
hb₀ : 0 < b
h_tendsto : Tendsto (β := ℝ) (fun (x : ℝ) ↦ 1 / 2 * log x) atTop atTop
x : ℝ
hx_pos : 1 < x
hx : ∀ (y : ℝ), b * id x ≤ y → -log b < 1 / 2 * log y
u : ℝ
hu : u ∈ Set.Icc (b * x) x
⊢ 1 / 2 * log x ≤ log u
have x₁ : 1 / 2 * log x = log x + -1 / 2 * log x
Tactic: calc
    1 / 2 * Real.log x = Real.log x + (-1 / 2) * Real.log x := by ring
      _ ≤ Real.log x + Real.log b := by grind
      _ = Real.log (b * x) := by rw [← Real.log_mul (by positivity) (by positivity), mul_comm]
      _ ≤ Real.log u := by gcongr; exact hu.1
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (goal)
Goal:
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
hb₀ : 0 < b
h_tendsto : Tendsto (β := ℝ) (fun (x : ℝ) ↦ 1 / 2 * log x) atTop atTop
x : ℝ
hx_pos : 1 < x
hx : ∀ (y : ℝ), b * id x ≤ y → -log b < 1 / 2 * log y
u : ℝ
hu : u ∈ Set.Icc (b * x) x
x₁ : 1 / 2 * log x = log x + -1 / 2 * log x
⊢ 1 / 2 * log x ≤ log u
have x₂ : log x + -1 / 2 * log x ≤ log x + log b
Tactic: calc
    1 / 2 * Real.log x = Real.log x + (-1 / 2) * Real.log x := by ring
      _ ≤ Real.log x + Real.log b := by grind
      _ = Real.log (b * x) := by rw [← Real.log_mul (by positivity) (by positivity), mul_comm]
      _ ≤ Real.log u := by gcongr; exact hu.1
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (goal)
Goal:
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
hb₀ : 0 < b
h_tendsto : Tendsto (β := ℝ) (fun (x : ℝ) ↦ 1 / 2 * log x) atTop atTop
x : ℝ
hx_pos : 1 < x
hx : ∀ (y : ℝ), b * id x ≤ y → -log b < 1 / 2 * log y
u : ℝ
hu : u ∈ Set.Icc (b * x) x
x₁ : 1 / 2 * log x = log x + -1 / 2 * log x
x₂ : log x + -1 / 2 * log x ≤ log x + log b
⊢ 1 / 2 * log x ≤ log u
have x₃ : log x + log b = log (b * x)
Tactic: calc
    1 / 2 * Real.log x = Real.log x + (-1 / 2) * Real.log x := by ring
      _ ≤ Real.log x + Real.log b := by grind
      _ = Real.log (b * x) := by rw [← Real.log_mul (by positivity) (by positivity), mul_comm]
      _ ≤ Real.log u := by gcongr; exact hu.1
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (goal)
Goal:
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
hb₀ : 0 < b
h_tendsto : Tendsto (β := ℝ) (fun (x : ℝ) ↦ 1 / 2 * log x) atTop atTop
x : ℝ
hx_pos : 1 < x
hx : ∀ (y : ℝ), b * id x ≤ y → -log b < 1 / 2 * log y
u : ℝ
hu : u ∈ Set.Icc (b * x) x
x₁ : 1 / 2 * log x = log x + -1 / 2 * log x
x₂ : log x + -1 / 2 * log x ≤ log x + log b
x₃ : log x + log b = log (b * x)
⊢ 1 / 2 * log x ≤ log u
have x₄ : log (b * x) ≤ log u
Tactic: calc
    1 / 2 * Real.log x = Real.log x + (-1 / 2) * Real.log x := by ring
      _ ≤ Real.log x + Real.log b := by grind
      _ = Real.log (b * x) := by rw [← Real.log_mul (by positivity) (by positivity), mul_comm]
      _ ≤ Real.log u := by gcongr; exact hu.1
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (goal)
Goal:
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
hb₀ : 0 < b
h_tendsto : Tendsto (β := ℝ) (fun (x : ℝ) ↦ 1 / 2 * log x) atTop atTop
x : ℝ
hx_pos : 1 < x
hx : ∀ (y : ℝ), b * id x ≤ y → -log b < 1 / 2 * log y
u : ℝ
hu : u ∈ Set.Icc (b * x) x
⊢ log u ≤ 1 * log x
have rw : log u ≤ log x
Tactic: rw [one_mul]
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (goal)
Goal:
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
hb₀ : 0 < b
h_tendsto : Tendsto (β := ℝ) (fun (x : ℝ) ↦ 1 / 2 * log x) atTop atTop
x : ℝ
hx_pos : 1 < x
hx : ∀ (y : ℝ), b * id x ≤ y → -log b < 1 / 2 * log y
u : ℝ
hu : u ∈ Set.Icc (b * x) x
⊢ log u ≤ log x
have hx₁ : 0 < u
Tactic: gcongr
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (goal)
Goal:
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
hb₀ : 0 < b
h_tendsto : Tendsto (β := ℝ) (fun (x : ℝ) ↦ 1 / 2 * log x) atTop atTop
x : ℝ
hx_pos : 1 < x
hx : ∀ (y : ℝ), b * id x ≤ y → -log b < 1 / 2 * log y
u : ℝ
hu : u ∈ Set.Icc (b * x) x
hx₁ : 0 < u
⊢ log u ≤ log x
have hxy : u ≤ x
Tactic: gcongr
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (goal)
Goal:
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
hb₀ : 0 < b
h_tendsto : Tendsto (β := ℝ) (fun (x : ℝ) ↦ 1 / 2 * log x) atTop atTop
x : ℝ
hx_pos : 1 < x
hx : ∀ (y : ℝ), b * id x ≤ y → -log b < 1 / 2 * log y
u : ℝ
hu : u ∈ Set.Icc (b * x) x
⊢ 0 < u
have calc : 0 < b * x
Tactic: calc 0 < b * x := by positivity
         _ ≤ u := by exact hu.1
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (goal)
Goal:
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
hb₀ : 0 < b
h_tendsto : Tendsto (β := ℝ) (fun (x : ℝ) ↦ 1 / 2 * log x) atTop atTop
x : ℝ
hx_pos : 1 < x
hx : ∀ (y : ℝ), b * id x ≤ y → -log b < 1 / 2 * log y
u : ℝ
hu : u ∈ Set.Icc (b * x) x
calc : 0 < b * x
⊢ 0 < u
have calc₁ : b * x ≤ u
Tactic: calc 0 < b * x := by positivity
         _ ≤ u := by exact hu.1
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (goal)
Goal:
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
hb₀ : 0 < b
h_tendsto : Tendsto (β := ℝ) (fun (x : ℝ) ↦ 1 / 2 * log x) atTop atTop
x : ℝ
hx_pos : 1 < x
hx : ∀ (y : ℝ), b * id x ≤ y → -log b < 1 / 2 * log y
u : ℝ
hu : u ∈ Set.Icc (b * x) x
⊢ log u ≤ 1 * log x
have case : log u ≤ log x
Tactic: case ub =>
    rw [one_mul]
    gcongr
    · calc 0 < b * x := by positivity
         _ ≤ u := by exact hu.1
    · exact hu.2
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (goal)
Goal:
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
hb₀ : 0 < b
h_tendsto : Tendsto (β := ℝ) (fun (x : ℝ) ↦ 1 / 2 * log x) atTop atTop
x : ℝ
hx_pos : 1 < x
hx : ∀ (y : ℝ), b * id x ≤ y → -log b < 1 / 2 * log y
u : ℝ
hu : u ∈ Set.Icc (b * x) x
case case₁ case₂ : log u ≤ log x
⊢ log u ≤ 1 * log x
have hx₁ : 0 < u
Tactic: case ub =>
    rw [one_mul]
    gcongr
    · calc 0 < b * x := by positivity
         _ ≤ u := by exact hu.1
    · exact hu.2
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (goal)
Goal:
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
hb₀ : 0 < b
h_tendsto : Tendsto (β := ℝ) (fun (x : ℝ) ↦ 1 / 2 * log x) atTop atTop
x : ℝ
hx_pos : 1 < x
hx : ∀ (y : ℝ), b * id x ≤ y → -log b < 1 / 2 * log y
u : ℝ
hu : u ∈ Set.Icc (b * x) x
case case₁ case₂ : log u ≤ log x
hx₁ : 0 < u
⊢ log u ≤ 1 * log x
have hxy : u ≤ x
Tactic: case ub =>
    rw [one_mul]
    gcongr
    · calc 0 < b * x := by positivity
         _ ≤ u := by exact hu.1
    · exact hu.2
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (hypothesis)
Goal:
f g : ℝ → ℝ
hg : GrowsPolynomially g
hf : f =Θ[atTop] g
hf' : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
⊢ ∃ c₁ > 0, ∃ c₂ > 0, ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
have hb_pos : 0 < b
Tactic: have hb_pos := hb.1
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (hypothesis)
Goal:
f g : ℝ → ℝ
hg : GrowsPolynomially g
hf : f =Θ[atTop] g
hf' : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
hb_pos : 0 < b
⊢ ∃ c₁ > 0, ∃ c₂ > 0, ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
have hf_lb : ∃ c > 0, ∀ᶠ (x : ℝ) in atTop, c * ‖g x‖ ≤ ‖f x‖
Tactic: have hf_lb := isBigO_iff''.mp hf.isBigO_symm
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (hypothesis)
Goal:
f g : ℝ → ℝ
hg : GrowsPolynomially g
hf : f =Θ[atTop] g
hf' : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
hb_pos : 0 < b
hf_lb : ∃ c > 0, ∀ᶠ (x : ℝ) in atTop, c * ‖g x‖ ≤ ‖f x‖
⊢ ∃ c₁ > 0, ∃ c₂ > 0, ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
have hf_ub : ∃ c > 0, ∀ᶠ (x : ℝ) in atTop, ‖f x‖ ≤ c * ‖g x‖
Tactic: have hf_ub := isBigO_iff'.mp hf.isBigO
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (goal)
Goal:
f g : ℝ → ℝ
hg : GrowsPolynomially g
hf : f =Θ[atTop] g
hf' : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
hb_pos : 0 < b
hf_lb : ∃ c > 0, ∀ᶠ (x : ℝ) in atTop, c * ‖g x‖ ≤ ‖f x‖
hf_ub : ∃ c > 0, ∀ᶠ (x : ℝ) in atTop, ‖f x‖ ≤ c * ‖g x‖
⊢ ∃ c₁ > 0, ∃ c₂ > 0, ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
have obtain : ∀ (c₁ : ℝ) (hc₁_pos : 0 < c₁) (hf_lb : ∀ᶠ (x : ℝ) in atTop, c₁ * ‖g x‖ ≤ ‖f x‖),
  ∃ c₁ > 0, ∃ c₂ > 0, ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
Tactic: obtain ⟨c₁, hc₁_pos : 0 < c₁, hf_lb⟩ := hf_lb
Cleared hypotheses: hf_lb
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (goal)
Goal:
f g : ℝ → ℝ
hg : GrowsPolynomially g
hf : f =Θ[atTop] g
hf' : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
hb_pos : 0 < b
hf_ub : ∃ c > 0, ∀ᶠ (x : ℝ) in atTop, ‖f x‖ ≤ c * ‖g x‖
c₁ : ℝ
hc₁_pos : 0 < c₁
hf_lb : ∀ᶠ (x : ℝ) in atTop, c₁ * ‖g x‖ ≤ ‖f x‖
⊢ ∃ c₁ > 0, ∃ c₂ > 0, ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
have obtain : ∀ (c₂ : ℝ) (hc₂_pos : 0 < c₂) (hf_ub : ∀ᶠ (x : ℝ) in atTop, ‖f x‖ ≤ c₂ * ‖g x‖),
  ∃ c₁ > 0, ∃ c₂ > 0, ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
Tactic: obtain ⟨c₂, hc₂_pos : 0 < c₂, hf_ub⟩ := hf_ub
Cleared hypotheses: hf_ub
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (hypothesis)
Goal:
f g : ℝ → ℝ
hg : GrowsPolynomially g
hf : f =Θ[atTop] g
hf' : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
hb_pos : 0 < b
c₁ : ℝ
hc₁_pos : 0 < c₁
hf_lb : ∀ᶠ (x : ℝ) in atTop, c₁ * ‖g x‖ ≤ ‖f x‖
c₂ : ℝ
hc₂_pos : 0 < c₂
hf_ub : ∀ᶠ (x : ℝ) in atTop, ‖f x‖ ≤ c₂ * ‖g x‖
⊢ ∃ c₁ > 0, ∃ c₂ > 0, ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
have hg₁ : ∃ c₁ > 0,
  ∃ c₂ > 0,
    ∀ᶠ (x : ℝ) in atTop,
      ∀ u ∈ Set.Icc (b * x) x,
        (fun (x : ℝ) ↦ ‖g x‖) u ∈ Set.Icc (c₁ * (fun (x : ℝ) ↦ ‖g x‖) x) (c₂ * (fun (x : ℝ) ↦ ‖g x‖) x)
Tactic: have hg := hg.norm b hb
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (goal)
Goal:
f g : ℝ → ℝ
hg_1 : GrowsPolynomially g
hf : f =Θ[atTop] g
hf' : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
hb_pos : 0 < b
c₁ : ℝ
hc₁_pos : 0 < c₁
hf_lb : ∀ᶠ (x : ℝ) in atTop, c₁ * ‖g x‖ ≤ ‖f x‖
c₂ : ℝ
hc₂_pos : 0 < c₂
hf_ub : ∀ᶠ (x : ℝ) in atTop, ‖f x‖ ≤ c₂ * ‖g x‖
hg :
  ∃ c₁ > 0,
    ∃ c₂ > 0,
      ∀ᶠ (x : ℝ) in atTop,
        ∀ u ∈ Set.Icc (b * x) x,
          (fun (x : ℝ) ↦ ‖g x‖) u ∈ Set.Icc (c₁ * (fun (x : ℝ) ↦ ‖g x‖) x) (c₂ * (fun (x : ℝ) ↦ ‖g x‖) x)
⊢ ∃ c₁ > 0, ∃ c₂ > 0, ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
have obtain : ∀ (c₃ : ℝ) (hc₃_pos : 0 < c₃)
  (hg :
    ∃ c₂ > 0,
      ∀ᶠ (x : ℝ) in atTop,
        ∀ u ∈ Set.Icc (b * x) x,
          (fun (x : ℝ) ↦ ‖g x‖) u ∈ Set.Icc (c₃ * (fun (x : ℝ) ↦ ‖g x‖) x) (c₂ * (fun (x : ℝ) ↦ ‖g x‖) x)),
  ∃ c₁ > 0, ∃ c₂ > 0, ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
Tactic: obtain ⟨c₃, hc₃_pos : 0 < c₃, hg⟩ := hg
Cleared hypotheses: hg
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (goal)
Goal:
f g : ℝ → ℝ
hg_1 : GrowsPolynomially g
hf : f =Θ[atTop] g
hf' : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
hb_pos : 0 < b
c₁ : ℝ
hc₁_pos : 0 < c₁
hf_lb : ∀ᶠ (x : ℝ) in atTop, c₁ * ‖g x‖ ≤ ‖f x‖
c₂ : ℝ
hc₂_pos : 0 < c₂
hf_ub : ∀ᶠ (x : ℝ) in atTop, ‖f x‖ ≤ c₂ * ‖g x‖
c₃ : ℝ
hc₃_pos : 0 < c₃
hg :
  ∃ c₂ > 0,
    ∀ᶠ (x : ℝ) in atTop,
      ∀ u ∈ Set.Icc (b * x) x,
        (fun (x : ℝ) ↦ ‖g x‖) u ∈ Set.Icc (c₃ * (fun (x : ℝ) ↦ ‖g x‖) x) (c₂ * (fun (x : ℝ) ↦ ‖g x‖) x)
⊢ ∃ c₁ > 0, ∃ c₂ > 0, ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
have obtain : ∀ (c₄ : ℝ) (hc₄_pos : 0 < c₄)
  (hg :
    ∀ᶠ (x : ℝ) in atTop,
      ∀ u ∈ Set.Icc (b * x) x,
        (fun (x : ℝ) ↦ ‖g x‖) u ∈ Set.Icc (c₃ * (fun (x : ℝ) ↦ ‖g x‖) x) (c₄ * (fun (x : ℝ) ↦ ‖g x‖) x)),
  ∃ c₁ > 0, ∃ c₂ > 0, ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
Tactic: obtain ⟨c₄, hc₄_pos : 0 < c₄, hg⟩ := hg
Cleared hypotheses: hg
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (hypothesis)
Goal:
f g : ℝ → ℝ
hg_1 : GrowsPolynomially g
hf : f =Θ[atTop] g
hf' : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
hb_pos : 0 < b
c₁ : ℝ
hc₁_pos : 0 < c₁
hf_lb : ∀ᶠ (x : ℝ) in atTop, c₁ * ‖g x‖ ≤ ‖f x‖
c₂ : ℝ
hc₂_pos : 0 < c₂
hf_ub : ∀ᶠ (x : ℝ) in atTop, ‖f x‖ ≤ c₂ * ‖g x‖
c₃ : ℝ
hc₃_pos : 0 < c₃
c₄ : ℝ
hc₄_pos : 0 < c₄
hg :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x,
      (fun (x : ℝ) ↦ ‖g x‖) u ∈ Set.Icc (c₃ * (fun (x : ℝ) ↦ ‖g x‖) x) (c₄ * (fun (x : ℝ) ↦ ‖g x‖) x)
⊢ ∃ c₁ > 0, ∃ c₂ > 0, ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
have h_lb_pos : 0 < c₁ * c₂⁻¹ * c₃
Tactic: have h_lb_pos : 0 < c₁ * c₂⁻¹ * c₃ := by positivity
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (hypothesis)
Goal:
f g : ℝ → ℝ
hg_1 : GrowsPolynomially g
hf : f =Θ[atTop] g
hf' : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
hb_pos : 0 < b
c₁ : ℝ
hc₁_pos : 0 < c₁
hf_lb : ∀ᶠ (x : ℝ) in atTop, c₁ * ‖g x‖ ≤ ‖f x‖
c₂ : ℝ
hc₂_pos : 0 < c₂
hf_ub : ∀ᶠ (x : ℝ) in atTop, ‖f x‖ ≤ c₂ * ‖g x‖
c₃ : ℝ
hc₃_pos : 0 < c₃
c₄ : ℝ
hc₄_pos : 0 < c₄
hg :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x,
      (fun (x : ℝ) ↦ ‖g x‖) u ∈ Set.Icc (c₃ * (fun (x : ℝ) ↦ ‖g x‖) x) (c₄ * (fun (x : ℝ) ↦ ‖g x‖) x)
h_lb_pos : 0 < c₁ * c₂⁻¹ * c₃
⊢ ∃ c₁ > 0, ∃ c₂ > 0, ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
have h_ub_pos : 0 < c₂ * c₄ * c₁⁻¹
Tactic: have h_ub_pos : 0 < c₂ * c₄ * c₁⁻¹ := by positivity
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (goal)
Goal:
f g : ℝ → ℝ
hg_1 : GrowsPolynomially g
hf : f =Θ[atTop] g
hf' : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
hb_pos : 0 < b
c₁ : ℝ
hc₁_pos : 0 < c₁
hf_lb : ∀ᶠ (x : ℝ) in atTop, c₁ * ‖g x‖ ≤ ‖f x‖
c₂ : ℝ
hc₂_pos : 0 < c₂
hf_ub : ∀ᶠ (x : ℝ) in atTop, ‖f x‖ ≤ c₂ * ‖g x‖
c₃ : ℝ
hc₃_pos : 0 < c₃
c₄ : ℝ
hc₄_pos : 0 < c₄
hg :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x,
      (fun (x : ℝ) ↦ ‖g x‖) u ∈ Set.Icc (c₃ * (fun (x : ℝ) ↦ ‖g x‖) x) (c₄ * (fun (x : ℝ) ↦ ‖g x‖) x)
h_lb_pos : 0 < c₁ * c₂⁻¹ * c₃
h_ub_pos : 0 < c₂ * c₄ * c₁⁻¹
⊢ ∃ c₁ > 0, ∃ c₂ > 0, ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
have refine : ∃ c₂_1 > 0, ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * c₂⁻¹ * c₃ * f x) (c₂_1 * f x)
Tactic: refine ⟨c₁ * c₂⁻¹ * c₃, h_lb_pos, ?_⟩
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (goal)
Goal:
f g : ℝ → ℝ
hg_1 : GrowsPolynomially g
hf : f =Θ[atTop] g
hf' : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
hb_pos : 0 < b
c₁ : ℝ
hc₁_pos : 0 < c₁
hf_lb : ∀ᶠ (x : ℝ) in atTop, c₁ * ‖g x‖ ≤ ‖f x‖
c₂ : ℝ
hc₂_pos : 0 < c₂
hf_ub : ∀ᶠ (x : ℝ) in atTop, ‖f x‖ ≤ c₂ * ‖g x‖
c₃ : ℝ
hc₃_pos : 0 < c₃
c₄ : ℝ
hc₄_pos : 0 < c₄
hg :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x,
      (fun (x : ℝ) ↦ ‖g x‖) u ∈ Set.Icc (c₃ * (fun (x : ℝ) ↦ ‖g x‖) x) (c₄ * (fun (x : ℝ) ↦ ‖g x‖) x)
h_lb_pos : 0 < c₁ * c₂⁻¹ * c₃
h_ub_pos : 0 < c₂ * c₄ * c₁⁻¹
⊢ ∃ c₂_1 > 0, ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * c₂⁻¹ * c₃ * f x) (c₂_1 * f x)
have refine : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * c₂⁻¹ * c₃ * f x) (c₂ * c₄ * c₁⁻¹ * f x)
Tactic: refine ⟨c₂ * c₄ * c₁⁻¹, h_ub_pos, ?_⟩
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (hypothesis)
Goal:
f g : ℝ → ℝ
hg_1 : GrowsPolynomially g
hf : f =Θ[atTop] g
hf' : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
hb_pos : 0 < b
c₁ : ℝ
hc₁_pos : 0 < c₁
hf_lb : ∀ᶠ (x : ℝ) in atTop, c₁ * ‖g x‖ ≤ ‖f x‖
c₂ : ℝ
hc₂_pos : 0 < c₂
hf_ub : ∀ᶠ (x : ℝ) in atTop, ‖f x‖ ≤ c₂ * ‖g x‖
c₃ : ℝ
hc₃_pos : 0 < c₃
c₄ : ℝ
hc₄_pos : 0 < c₄
hg :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x,
      (fun (x : ℝ) ↦ ‖g x‖) u ∈ Set.Icc (c₃ * (fun (x : ℝ) ↦ ‖g x‖) x) (c₄ * (fun (x : ℝ) ↦ ‖g x‖) x)
h_lb_pos : 0 < c₁ * c₂⁻¹ * c₃
h_ub_pos : 0 < c₂ * c₄ * c₁⁻¹
⊢ ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * c₂⁻¹ * c₃ * f x) (c₂ * c₄ * c₁⁻¹ * f x)
have c₂_cancel : c₂⁻¹ * c₂ = 1
Tactic: have c₂_cancel : c₂⁻¹ * c₂ = 1 := inv_mul_cancel₀ (by positivity)
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (hypothesis)
Goal:
f g : ℝ → ℝ
hg_1 : GrowsPolynomially g
hf : f =Θ[atTop] g
hf' : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
hb_pos : 0 < b
c₁ : ℝ
hc₁_pos : 0 < c₁
hf_lb : ∀ᶠ (x : ℝ) in atTop, c₁ * ‖g x‖ ≤ ‖f x‖
c₂ : ℝ
hc₂_pos : 0 < c₂
hf_ub : ∀ᶠ (x : ℝ) in atTop, ‖f x‖ ≤ c₂ * ‖g x‖
c₃ : ℝ
hc₃_pos : 0 < c₃
c₄ : ℝ
hc₄_pos : 0 < c₄
hg :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x,
      (fun (x : ℝ) ↦ ‖g x‖) u ∈ Set.Icc (c₃ * (fun (x : ℝ) ↦ ‖g x‖) x) (c₄ * (fun (x : ℝ) ↦ ‖g x‖) x)
h_lb_pos : 0 < c₁ * c₂⁻¹ * c₃
h_ub_pos : 0 < c₂ * c₄ * c₁⁻¹
c₂_cancel : c₂⁻¹ * c₂ = 1
⊢ ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * c₂⁻¹ * c₃ * f x) (c₂ * c₄ * c₁⁻¹ * f x)
have c₁_cancel : c₁⁻¹ * c₁ = 1
Tactic: have c₁_cancel : c₁⁻¹ * c₁ = 1 := inv_mul_cancel₀ (by positivity)
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (goal)
Goal:
f g : ℝ → ℝ
hg_1 : GrowsPolynomially g
hf : f =Θ[atTop] g
hf' : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
hb_pos : 0 < b
c₁ : ℝ
hc₁_pos : 0 < c₁
hf_lb : ∀ᶠ (x : ℝ) in atTop, c₁ * ‖g x‖ ≤ ‖f x‖
c₂ : ℝ
hc₂_pos : 0 < c₂
hf_ub : ∀ᶠ (x : ℝ) in atTop, ‖f x‖ ≤ c₂ * ‖g x‖
c₃ : ℝ
hc₃_pos : 0 < c₃
c₄ : ℝ
hc₄_pos : 0 < c₄
hg :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x,
      (fun (x : ℝ) ↦ ‖g x‖) u ∈ Set.Icc (c₃ * (fun (x : ℝ) ↦ ‖g x‖) x) (c₄ * (fun (x : ℝ) ↦ ‖g x‖) x)
h_lb_pos : 0 < c₁ * c₂⁻¹ * c₃
h_ub_pos : 0 < c₂ * c₄ * c₁⁻¹
c₂_cancel : c₂⁻¹ * c₂ = 1
c₁_cancel : c₁⁻¹ * c₁ = 1
⊢ ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * c₂⁻¹ * c₃ * f x) (c₂ * c₄ * c₁⁻¹ * f x)
have h : ∀ (x : ℝ) (hf_pos : ∀ (y : ℝ), b * id x ≤ y → 0 ≤ f y) (h_lb : ∀ (y : ℝ), b * id x ≤ y → c₁ * ‖g y‖ ≤ ‖f y‖)
  (h_ub : ∀ (y : ℝ), b * id x ≤ y → ‖f y‖ ≤ c₂ * ‖g y‖)
  (hg_bound : ∀ (y : ℝ), b * id x ≤ y → ∀ u ∈ Set.Icc (b * y) y, ‖g u‖ ∈ Set.Icc (c₃ * ‖g y‖) (c₄ * ‖g y‖))
  (hx_pos : 0 ≤ x), ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * c₂⁻¹ * c₃ * f x) (c₂ * c₄ * c₁⁻¹ * f x)
Tactic: filter_upwards [(tendsto_id.const_mul_atTop hb_pos).eventually_forall_ge_atTop hf',
                  (tendsto_id.const_mul_atTop hb_pos).eventually_forall_ge_atTop hf_lb,
                  (tendsto_id.const_mul_atTop hb_pos).eventually_forall_ge_atTop hf_ub,
                  (tendsto_id.const_mul_atTop hb_pos).eventually_forall_ge_atTop hg,
                  eventually_ge_atTop 0]
    with x hf_pos h_lb h_ub hg_bound hx_pos
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (goal)
Goal:
f g : ℝ → ℝ
hg_1 : GrowsPolynomially g
hf : f =Θ[atTop] g
hf' : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
hb_pos : 0 < b
c₁ : ℝ
hc₁_pos : 0 < c₁
hf_lb : ∀ᶠ (x : ℝ) in atTop, c₁ * ‖g x‖ ≤ ‖f x‖
c₂ : ℝ
hc₂_pos : 0 < c₂
hf_ub : ∀ᶠ (x : ℝ) in atTop, ‖f x‖ ≤ c₂ * ‖g x‖
c₃ : ℝ
hc₃_pos : 0 < c₃
c₄ : ℝ
hc₄_pos : 0 < c₄
hg :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x,
      (fun (x : ℝ) ↦ ‖g x‖) u ∈ Set.Icc (c₃ * (fun (x : ℝ) ↦ ‖g x‖) x) (c₄ * (fun (x : ℝ) ↦ ‖g x‖) x)
h_lb_pos : 0 < c₁ * c₂⁻¹ * c₃
h_ub_pos : 0 < c₂ * c₄ * c₁⁻¹
c₂_cancel : c₂⁻¹ * c₂ = 1
c₁_cancel : c₁⁻¹ * c₁ = 1
x : ℝ
hf_pos : ∀ (y : ℝ), b * id x ≤ y → 0 ≤ f y
h_lb : ∀ (y : ℝ), b * id x ≤ y → c₁ * ‖g y‖ ≤ ‖f y‖
h_ub : ∀ (y : ℝ), b * id x ≤ y → ‖f y‖ ≤ c₂ * ‖g y‖
hg_bound : ∀ (y : ℝ), b * id x ≤ y → ∀ u ∈ Set.Icc (b * y) y, ‖g u‖ ∈ Set.Icc (c₃ * ‖g y‖) (c₄ * ‖g y‖)
hx_pos : 0 ≤ x
u : ℝ
hu : u ∈ Set.Icc (b * x) x
⊢ b * x ≤ 1 * x
have hbc : b ≤ 1
Tactic: gcongr
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (goal)
Goal:
f g : ℝ → ℝ
hg_1 : GrowsPolynomially g
hf : f =Θ[atTop] g
hf' : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
hb_pos : 0 < b
c₁ : ℝ
hc₁_pos : 0 < c₁
hf_lb : ∀ᶠ (x : ℝ) in atTop, c₁ * ‖g x‖ ≤ ‖f x‖
c₂ : ℝ
hc₂_pos : 0 < c₂
hf_ub : ∀ᶠ (x : ℝ) in atTop, ‖f x‖ ≤ c₂ * ‖g x‖
c₃ : ℝ
hc₃_pos : 0 < c₃
c₄ : ℝ
hc₄_pos : 0 < c₄
hg :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x,
      (fun (x : ℝ) ↦ ‖g x‖) u ∈ Set.Icc (c₃ * (fun (x : ℝ) ↦ ‖g x‖) x) (c₄ * (fun (x : ℝ) ↦ ‖g x‖) x)
h_lb_pos : 0 < c₁ * c₂⁻¹ * c₃
h_ub_pos : 0 < c₂ * c₄ * c₁⁻¹
c₂_cancel : c₂⁻¹ * c₂ = 1
c₁_cancel : c₁⁻¹ * c₁ = 1
x : ℝ
hf_pos : ∀ (y : ℝ), b * id x ≤ y → 0 ≤ f y
h_lb : ∀ (y : ℝ), b * id x ≤ y → c₁ * ‖g y‖ ≤ ‖f y‖
h_ub : ∀ (y : ℝ), b * id x ≤ y → ‖f y‖ ≤ c₂ * ‖g y‖
hg_bound : ∀ (y : ℝ), b * id x ≤ y → ∀ u ∈ Set.Icc (b * y) y, ‖g u‖ ∈ Set.Icc (c₃ * ‖g y‖) (c₄ * ‖g y‖)
hx_pos : 0 ≤ x
u : ℝ
hu : u ∈ Set.Icc (b * x) x
⊢ 1 * x = x
have rw : x = x
Tactic: rw [one_mul]
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (hypothesis)
Goal:
f g : ℝ → ℝ
hg_1 : GrowsPolynomially g
hf : f =Θ[atTop] g
hf' : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
hb_pos : 0 < b
c₁ : ℝ
hc₁_pos : 0 < c₁
hf_lb : ∀ᶠ (x : ℝ) in atTop, c₁ * ‖g x‖ ≤ ‖f x‖
c₂ : ℝ
hc₂_pos : 0 < c₂
hf_ub : ∀ᶠ (x : ℝ) in atTop, ‖f x‖ ≤ c₂ * ‖g x‖
c₃ : ℝ
hc₃_pos : 0 < c₃
c₄ : ℝ
hc₄_pos : 0 < c₄
hg :
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x,
      (fun (x : ℝ) ↦ ‖g x‖) u ∈ Set.Icc (c₃ * (fun (x : ℝ) ↦ ‖g x‖) x) (c₄ * (fun (x : ℝ) ↦ ‖g x‖) x)
h_lb_pos : 0 < c₁ * c₂⁻¹ * c₃
h_ub_pos : 0 < c₂ * c₄ * c₁⁻¹
c₂_cancel : c₂⁻¹ * c₂ = 1
c₁_cancel : c₁⁻¹ * c₁ = 1
x : ℝ
hf_pos : ∀ (y : ℝ), b * id x ≤ y → 0 ≤ f y
h_lb : ∀ (y : ℝ), b * id x ≤ y → c₁ * ‖g y‖ ≤ ‖f y‖
h_ub : ∀ (y : ℝ), b * id x ≤ y → ‖f y‖ ≤ c₂ * ‖g y‖
hg_bound : ∀ (y : ℝ), b * id x ≤ y → ∀ u ∈ Set.Icc (b * y) y, ‖g u‖ ∈ Set.Icc (c₃ * ‖g y‖) (c₄ * ‖g y‖)
hx_pos : 0 ≤ x
u : ℝ
hu : u ∈ Set.Icc (b * x) x
⊢ f u ∈ Set.Icc (c₁ * c₂⁻¹ * c₃ * f x) (c₂ * c₄ * c₁⁻¹ * f x)
have hbx : b * x ≤ x
Tactic: have hbx : b * x ≤ x :=
    calc b * x ≤ 1 * x := by gcongr; exact le_of_lt hb.2
             _ = x := by rw [one_mul]