awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (goal)
Goal:
f : ℝ → ℝ
p : ℝ
hf : GrowsPolynomially f
hf_nonneg : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : c₂ > 0
hfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hc₁p : 0 < c₁ ^ p
hc₂p : 0 < c₂ ^ p
hp : p < 0
hzero : ∀ᶠ (x : ℝ) in atTop, f x = 0
⊢ ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x,
      (fun (x : ℝ) ↦ f x ^ p) u ∈
        Set.Icc (HMul.hMul (α := ℝ) 1 ((fun (x : ℝ) ↦ f x ^ p) x)) (HMul.hMul (α := ℝ) 1 ((fun (x : ℝ) ↦ f x ^ p) x))
have h : ∀ (x : ℝ) (hx : f x = 0) (hx' : ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)),
  ∀ u ∈ Set.Icc (b * x) x, f u ^ p ∈ Set.Icc (1 * f x ^ p) (1 * f x ^ p)
Tactic: filter_upwards [hzero, hfnew] with x hx hx'
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (goal)
Goal:
f : ℝ → ℝ
p : ℝ
hf : GrowsPolynomially f
hf_nonneg : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : c₂ > 0
hfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hc₁p : 0 < c₁ ^ p
hc₂p : 0 < c₂ ^ p
hp : p < 0
hzero : ∀ᶠ (x : ℝ) in atTop, f x = 0
x : ℝ
hx : f x = 0
hx' : ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
u : ℝ
hu : u ∈ Set.Icc (b * x) x
⊢ f u ^ p ∈ Set.Icc (1 * f x ^ p) (1 * f x ^ p)
have h : f u ^ p = 0
Tactic: simp only [hx, zero_rpow (ne_of_lt hp), mul_zero,
        Set.Icc_self, Set.mem_singleton_iff]
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (hypothesis)
Goal:
f : ℝ → ℝ
p : ℝ
hf : GrowsPolynomially f
hf_nonneg : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : c₂ > 0
hfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hc₁p : 0 < c₁ ^ p
hc₂p : 0 < c₂ ^ p
hp : p < 0
hzero : ∀ᶠ (x : ℝ) in atTop, f x = 0
x : ℝ
hx : f x = 0
hx' : ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
u : ℝ
hu : u ∈ Set.Icc (b * x) x
⊢ f u ^ p = 0
have hx'₁ : ∀ u ∈ Set.Icc (b * x) x, f u = 0
Tactic: simp only [hx, mul_zero, Set.Icc_self, Set.mem_singleton_iff] at hx'
Cleared hypotheses: hx'
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (goal)
Goal:
f : ℝ → ℝ
p : ℝ
hf : GrowsPolynomially f
hf_nonneg : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : c₂ > 0
hfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hc₁p : 0 < c₁ ^ p
hc₂p : 0 < c₂ ^ p
hp : p < 0
hzero : ∀ᶠ (x : ℝ) in atTop, f x = 0
x : ℝ
hx : f x = 0
u : ℝ
hu : u ∈ Set.Icc (b * x) x
hx' : ∀ u ∈ Set.Icc (b * x) x, f u = 0
⊢ f u ^ p = 0
have h : HPow.hPow (α := ℝ) 0 p = 0
Tactic: rw [hx' u hu, zero_rpow (ne_of_lt hp)]
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (goal)
Goal:
f : ℝ → ℝ
p : ℝ
hf : GrowsPolynomially f
hf_nonneg : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : c₂ > 0
hfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hc₁p : 0 < c₁ ^ p
hc₂p : 0 < c₂ ^ p
hp : p < 0
hzero : ∀ᶠ (x : ℝ) in atTop, f x = 0
x : ℝ
hx : f x = 0
u : ℝ
hu : u ∈ Set.Icc (b * x) x
hx' : ∀ u ∈ Set.Icc (b * x) x, f u = 0
h : HPow.hPow (α := ℝ) 0 p = 0
⊢ f u ^ p = 0
have h₁ : Eq (α := ℝ) 0 0
Tactic: rw [hx' u hu, zero_rpow (ne_of_lt hp)]
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (goal)
Goal:
f : ℝ → ℝ
p : ℝ
hf : GrowsPolynomially f
hf_nonneg : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : c₂ > 0
hfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hc₁p : 0 < c₁ ^ p
hc₂p : 0 < c₂ ^ p
hp : p < 0
hpos : ∀ᶠ (x : ℝ) in atTop, 0 < f x
⊢ ∃ c₁ > 0,
    ∃ c₂ > 0,
      ∀ᶠ (x : ℝ) in atTop,
        ∀ u ∈ Set.Icc (b * x) x,
          (fun (x : ℝ) ↦ f x ^ p) u ∈ Set.Icc (c₁ * (fun (x : ℝ) ↦ f x ^ p) x) (c₂ * (fun (x : ℝ) ↦ f x ^ p) x)
have refine : ∃ c₂_1 > 0,
  ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x,
      (fun (x : ℝ) ↦ f x ^ p) u ∈ Set.Icc (c₂ ^ p * (fun (x : ℝ) ↦ f x ^ p) x) (c₂_1 * (fun (x : ℝ) ↦ f x ^ p) x)
Tactic: refine ⟨c₂^p, hc₂p, ?_⟩
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (goal)
Goal:
f : ℝ → ℝ
p : ℝ
hf : GrowsPolynomially f
hf_nonneg : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : c₂ > 0
hfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hc₁p : 0 < c₁ ^ p
hc₂p : 0 < c₂ ^ p
hp : p < 0
hpos : ∀ᶠ (x : ℝ) in atTop, 0 < f x
⊢ ∃ c₂_1 > 0,
    ∀ᶠ (x : ℝ) in atTop,
      ∀ u ∈ Set.Icc (b * x) x,
        (fun (x : ℝ) ↦ f x ^ p) u ∈ Set.Icc (c₂ ^ p * (fun (x : ℝ) ↦ f x ^ p) x) (c₂_1 * (fun (x : ℝ) ↦ f x ^ p) x)
have refine : ∀ᶠ (x : ℝ) in atTop,
  ∀ u ∈ Set.Icc (b * x) x,
    (fun (x : ℝ) ↦ f x ^ p) u ∈ Set.Icc (c₂ ^ p * (fun (x : ℝ) ↦ f x ^ p) x) (c₁ ^ p * (fun (x : ℝ) ↦ f x ^ p) x)
Tactic: refine ⟨c₁^p, hc₁p, ?_⟩
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (goal)
Goal:
f : ℝ → ℝ
p : ℝ
hf : GrowsPolynomially f
hf_nonneg : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : c₂ > 0
hfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hc₁p : 0 < c₁ ^ p
hc₂p : 0 < c₂ ^ p
hp : p < 0
hpos : ∀ᶠ (x : ℝ) in atTop, 0 < f x
⊢ ∀ᶠ (x : ℝ) in atTop,
    ∀ u ∈ Set.Icc (b * x) x,
      (fun (x : ℝ) ↦ f x ^ p) u ∈ Set.Icc (c₂ ^ p * (fun (x : ℝ) ↦ f x ^ p) x) (c₁ ^ p * (fun (x : ℝ) ↦ f x ^ p) x)
have h : ∀ (x : ℝ),
  0 < x →
    ∀ (hf₁ : ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)) (hf_pos : 0 < f x)
      (hf_pos₂ : ∀ (y : ℝ), b * id x ≤ y → 0 < f y),
      ∀ u ∈ Set.Icc (b * x) x, f u ^ p ∈ Set.Icc (c₂ ^ p * f x ^ p) (c₁ ^ p * f x ^ p)
Tactic: filter_upwards [eventually_gt_atTop 0, hfnew, hpos,
          (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hpos]
          with x _ hf₁ hf_pos hf_pos₂
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (goal)
Goal:
f : ℝ → ℝ
p : ℝ
hf : GrowsPolynomially f
hf_nonneg : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : c₂ > 0
hfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hc₁p : 0 < c₁ ^ p
hc₂p : 0 < c₂ ^ p
hp : p < 0
hpos : ∀ᶠ (x : ℝ) in atTop, 0 < f x
x : ℝ
h : 0 < x
hf₁ : ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hf_pos : 0 < f x
hf_pos₂ : ∀ (y : ℝ), b * id x ≤ y → 0 < f y
u : ℝ
hu : u ∈ Set.Icc (b * x) x
⊢ f u ^ p ∈ Set.Icc (c₂ ^ p * f x ^ p) (c₁ ^ p * f x ^ p)
have refine : c₂ ^ p * f x ^ p ≤ f u ^ p
Tactic: refine ⟨?lb, ?ub⟩
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (goal)
Goal:
f : ℝ → ℝ
p : ℝ
hf : GrowsPolynomially f
hf_nonneg : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : c₂ > 0
hfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hc₁p : 0 < c₁ ^ p
hc₂p : 0 < c₂ ^ p
hp : p < 0
hpos : ∀ᶠ (x : ℝ) in atTop, 0 < f x
x : ℝ
h : 0 < x
hf₁ : ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hf_pos : 0 < f x
hf_pos₂ : ∀ (y : ℝ), b * id x ≤ y → 0 < f y
u : ℝ
hu : u ∈ Set.Icc (b * x) x
refine : c₂ ^ p * f x ^ p ≤ f u ^ p
⊢ f u ^ p ∈ Set.Icc (c₂ ^ p * f x ^ p) (c₁ ^ p * f x ^ p)
have refine₁ : f u ^ p ≤ c₁ ^ p * f x ^ p
Tactic: refine ⟨?lb, ?ub⟩
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (goal)
Goal:
f : ℝ → ℝ
p : ℝ
hf : GrowsPolynomially f
hf_nonneg : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : c₂ > 0
hfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hc₁p : 0 < c₁ ^ p
hc₂p : 0 < c₂ ^ p
hp : p < 0
hpos : ∀ᶠ (x : ℝ) in atTop, 0 < f x
x : ℝ
h : 0 < x
hf₁ : ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hf_pos : 0 < f x
hf_pos₂ : ∀ (y : ℝ), b * id x ≤ y → 0 < f y
u : ℝ
hu : u ∈ Set.Icc (b * x) x
⊢ c₂ ^ p * f x ^ p = (c₂ * f x) ^ p
have rw : c₂ ^ p * f x ^ p = c₂ ^ p * f x ^ p
Tactic: rw [mul_rpow (le_of_lt hc₂_mem) (le_of_lt hf_pos)]
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (goal)
Goal:
f : ℝ → ℝ
p : ℝ
hf : GrowsPolynomially f
hf_nonneg : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : c₂ > 0
hfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hc₁p : 0 < c₁ ^ p
hc₂p : 0 < c₂ ^ p
hp : p < 0
hpos : ∀ᶠ (x : ℝ) in atTop, 0 < f x
x : ℝ
h : 0 < x
hf₁ : ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hf_pos : 0 < f x
hf_pos₂ : ∀ (y : ℝ), b * id x ≤ y → 0 < f y
u : ℝ
hu : u ∈ Set.Icc (b * x) x
⊢ c₂ ^ p * f x ^ p ≤ f u ^ p
have c₃ : c₂ ^ p * f x ^ p = (c₂ * f x) ^ p
Tactic: calc
        c₂ ^ p * (f x) ^ p = (c₂ * f x) ^ p := by rw [mul_rpow (le_of_lt hc₂_mem) (le_of_lt hf_pos)]
          _ ≤ _ := rpow_le_rpow_of_nonpos (hf_pos₂ u hu.1) (hf₁ u hu).2 (le_of_lt hp)
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (goal)
Goal:
f : ℝ → ℝ
p : ℝ
hf : GrowsPolynomially f
hf_nonneg : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : c₂ > 0
hfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hc₁p : 0 < c₁ ^ p
hc₂p : 0 < c₂ ^ p
hp : p < 0
hpos : ∀ᶠ (x : ℝ) in atTop, 0 < f x
x : ℝ
h : 0 < x
hf₁ : ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hf_pos : 0 < f x
hf_pos₂ : ∀ (y : ℝ), b * id x ≤ y → 0 < f y
u : ℝ
hu : u ∈ Set.Icc (b * x) x
⊢ (c₁ * f x) ^ p = c₁ ^ p * f x ^ p
have rw : (c₁ * f x) ^ p = (c₁ * f x) ^ p
Tactic: rw [← mul_rpow (le_of_lt hc₁_mem) (le_of_lt hf_pos)]
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (goal)
Goal:
f : ℝ → ℝ
p : ℝ
hf : GrowsPolynomially f
hf_nonneg : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : c₂ > 0
hfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hc₁p : 0 < c₁ ^ p
hc₂p : 0 < c₂ ^ p
hp : p < 0
hpos : ∀ᶠ (x : ℝ) in atTop, 0 < f x
x : ℝ
h : 0 < x
hf₁ : ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hf_pos : 0 < f x
hf_pos₂ : ∀ (y : ℝ), b * id x ≤ y → 0 < f y
u : ℝ
hu : u ∈ Set.Icc (b * x) x
⊢ f u ^ p ≤ c₁ ^ p * f x ^ p
have p₁ : f u ^ p ≤ (c₁ * f x) ^ p
Tactic: calc
        (f u) ^ p ≤ (c₁ * f x) ^ p := by
              exact rpow_le_rpow_of_nonpos (by positivity) (hf₁ u hu).1 (le_of_lt hp)
          _ = _ := by rw [← mul_rpow (le_of_lt hc₁_mem) (le_of_lt hf_pos)]
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (goal)
Goal:
f : ℝ → ℝ
p : ℝ
hf : GrowsPolynomially f
hf_nonneg : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : c₂ > 0
hfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hc₁p : 0 < c₁ ^ p
hc₂p : 0 < c₂ ^ p
hp : p < 0
hpos : ∀ᶠ (x : ℝ) in atTop, 0 < f x
x : ℝ
h : 0 < x
hf₁ : ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hf_pos : 0 < f x
hf_pos₂ : ∀ (y : ℝ), b * id x ≤ y → 0 < f y
u : ℝ
hu : u ∈ Set.Icc (b * x) x
p₁ : f u ^ p ≤ (c₁ * f x) ^ p
⊢ f u ^ p ≤ c₁ ^ p * f x ^ p
have p₂ : (c₁ * f x) ^ p = c₁ ^ p * f x ^ p
Tactic: calc
        (f u) ^ p ≤ (c₁ * f x) ^ p := by
              exact rpow_le_rpow_of_nonpos (by positivity) (hf₁ u hu).1 (le_of_lt hp)
          _ = _ := by rw [← mul_rpow (le_of_lt hc₁_mem) (le_of_lt hf_pos)]
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (goal)
Goal:
f : ℝ → ℝ
p : ℝ
hf : GrowsPolynomially f
hf_nonneg : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : c₂ > 0
hfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hc₁p : 0 < c₁ ^ p
hc₂p : 0 < c₂ ^ p
hp : p < 0
hpos : ∀ᶠ (x : ℝ) in atTop, 0 < f x
x : ℝ
h : 0 < x
hf₁ : ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hf_pos : 0 < f x
hf_pos₂ : ∀ (y : ℝ), b * id x ≤ y → 0 < f y
u : ℝ
hu : u ∈ Set.Icc (b * x) x
⊢ f u ^ p ≤ c₁ ^ p * f x ^ p
have case : f u ^ p ≤ (c₁ * f x) ^ p
Tactic: case ub => calc
        (f u) ^ p ≤ (c₁ * f x) ^ p := by
              exact rpow_le_rpow_of_nonpos (by positivity) (hf₁ u hu).1 (le_of_lt hp)
          _ = _ := by rw [← mul_rpow (le_of_lt hc₁_mem) (le_of_lt hf_pos)]
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (goal)
Goal:
f : ℝ → ℝ
p : ℝ
hf : GrowsPolynomially f
hf_nonneg : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : c₂ > 0
hfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hc₁p : 0 < c₁ ^ p
hc₂p : 0 < c₂ ^ p
hp : p < 0
hpos : ∀ᶠ (x : ℝ) in atTop, 0 < f x
x : ℝ
h : 0 < x
hf₁ : ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hf_pos : 0 < f x
hf_pos₂ : ∀ (y : ℝ), b * id x ≤ y → 0 < f y
u : ℝ
hu : u ∈ Set.Icc (b * x) x
case : f u ^ p ≤ (c₁ * f x) ^ p
⊢ f u ^ p ≤ c₁ ^ p * f x ^ p
have case₁ : (c₁ * f x) ^ p = c₁ ^ p * f x ^ p
Tactic: case ub => calc
        (f u) ^ p ≤ (c₁ * f x) ^ p := by
              exact rpow_le_rpow_of_nonpos (by positivity) (hf₁ u hu).1 (le_of_lt hp)
          _ = _ := by rw [← mul_rpow (le_of_lt hc₁_mem) (le_of_lt hf_pos)]
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (goal)
Goal:
f : ℝ → ℝ
p : ℝ
hf : GrowsPolynomially f
hf_nonneg : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : c₂ > 0
hfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hc₁p : 0 < c₁ ^ p
hc₂p : 0 < c₂ ^ p
hp : p < 0
hneg : ∀ᶠ (x : ℝ) in atTop, f x < 0
⊢ ∀ᶠ (x : ℝ) in atTop, False
have h : ∀ (x : ℝ) (hx : 0 ≤ f x) (hx' : f x < 0), False
Tactic: filter_upwards [hf_nonneg, hneg] with x hx hx'
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (goal)
Goal:
f : ℝ → ℝ
p : ℝ
hf : GrowsPolynomially f
hf_nonneg : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : c₂ > 0
hfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hc₁p : 0 < c₁ ^ p
hc₂p : 0 < c₂ ^ p
hp : p < 0
hneg : ∀ᶠ (x : ℝ) in atTop, f x < 0
x : ℝ
hx : 0 ≤ f x
hx' : f x < 0
⊢ False
have linarith : 0 - f x + (f x - 0) = 0
Tactic: linarith
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (hypothesis)
Goal:
f : ℝ → ℝ
p : ℝ
hf : GrowsPolynomially f
hf_nonneg : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : c₂ > 0
hfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hc₁p : 0 < c₁ ^ p
hc₂p : 0 < c₂ ^ p
hp : p < 0
hneg : ∀ᶠ (x : ℝ) in atTop, f x < 0
⊢ ∃ c₁ > 0,
    ∃ c₂ > 0,
      ∀ᶠ (x : ℝ) in atTop,
        ∀ u ∈ Set.Icc (b * x) x,
          (fun (x : ℝ) ↦ f x ^ p) u ∈ Set.Icc (c₁ * (fun (x : ℝ) ↦ f x ^ p) x) (c₂ * (fun (x : ℝ) ↦ f x ^ p) x)
have this : ∀ᶠ (x : ℝ) in atTop, False
Tactic: have : ∀ᶠ (_ : ℝ) in atTop, False := by
        filter_upwards [hf_nonneg, hneg] with x hx hx'; linarith
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (hypothesis)
Goal:
f : ℝ → ℝ
p : ℝ
hf : GrowsPolynomially f
hf_nonneg : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : c₂ > 0
hfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hc₁p : 0 < c₁ ^ p
hc₂p : 0 < c₂ ^ p
hp : p < 0
hneg : ∀ᶠ (x : ℝ) in atTop, f x < 0
this : ∀ᶠ (x : ℝ) in atTop, False
⊢ ∃ c₁ > 0,
    ∃ c₂ > 0,
      ∀ᶠ (x : ℝ) in atTop,
        ∀ u ∈ Set.Icc (b * x) x,
          (fun (x : ℝ) ↦ f x ^ p) u ∈ Set.Icc (c₁ * (fun (x : ℝ) ↦ f x ^ p) x) (c₂ * (fun (x : ℝ) ↦ f x ^ p) x)
have this₁ : Eq (α := Filter ℝ) atTop ⊥
Tactic: rw [Filter.eventually_false_iff_eq_bot] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (hypothesis)
Goal:
f : ℝ → ℝ
p : ℝ
hf : GrowsPolynomially f
hf_nonneg : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : c₂ > 0
hfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hc₁p : 0 < c₁ ^ p
hc₂p : 0 < c₂ ^ p
⊢ ∃ c₁ > 0,
    ∃ c₂ > 0,
      ∀ᶠ (x : ℝ) in atTop,
        ∀ u ∈ Set.Icc (b * x) x,
          (fun (x : ℝ) ↦ f x ^ p) u ∈ Set.Icc (c₁ * (fun (x : ℝ) ↦ f x ^ p) x) (c₂ * (fun (x : ℝ) ↦ f x ^ p) x)
have x : 0 ≤ p ∨ p < 0
Tactic: cases le_or_gt 0 p with
  | inl => -- 0 ≤ p
    refine ⟨c₁^p, hc₁p, ?_⟩
    refine ⟨c₂^p, hc₂p, ?_⟩
    filter_upwards [eventually_gt_atTop 0, hfnew, hf_nonneg,
        (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hf_nonneg]
        with x _ hf₁ hf_nonneg hf_nonneg₂
    intro u hu
    have fu_nonneg : 0 ≤ f u := hf_nonneg₂ u hu.1
    refine ⟨?lb, ?ub⟩
    case lb => calc
      c₁ ^ p * (f x) ^ p = (c₁ * f x) ^ p := by rw [mul_rpow (le_of_lt hc₁_mem) hf_nonneg]
        _ ≤ _ := by gcongr; exact (hf₁ u hu).1
    case ub => calc
      (f u) ^ p ≤ (c₂ * f x) ^ p := by gcongr; exact (hf₁ u hu).2
        _ = _ := by rw [← mul_rpow (le_of_lt hc₂_mem) hf_nonneg]
  | inr hp => -- p < 0
    match hf.eventually_atTop_zero_or_pos_or_neg with
    | .inl hzero => -- eventually zero
      refine ⟨1, by norm_num, 1, by norm_num, ?_⟩
      filter_upwards [hzero, hfnew] with x hx hx'
      intro u hu
      simp only [hx, zero_rpow (ne_of_lt hp), mul_zero,
        Set.Icc_self, Set.mem_singleton_iff]
      simp only [hx, mul_zero, Set.Icc_self, Set.mem_singleton_iff] at hx'
      rw [hx' u hu, zero_rpow (ne_of_lt hp)]
    | .inr (.inl hpos) => -- eventually positive
      refine ⟨c₂^p, hc₂p, ?_⟩
      refine ⟨c₁^p, hc₁p, ?_⟩
      filter_upwards [eventually_gt_atTop 0, hfnew, hpos,
          (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop hpos]
          with x _ hf₁ hf_pos hf_pos₂
      intro u hu
      refine ⟨?lb, ?ub⟩
      case lb => calc
        c₂ ^ p * (f x) ^ p = (c₂ * f x) ^ p := by rw [mul_rpow (le_of_lt hc₂_mem) (le_of_lt hf_pos)]
          _ ≤ _ := rpow_le_rpow_of_nonpos (hf_pos₂ u hu.1) (hf₁ u hu).2 (le_of_lt hp)
      case ub => calc
        (f u) ^ p ≤ (c₁ * f x) ^ p := by
              exact rpow_le_rpow_of_nonpos (by positivity) (hf₁ u hu).1 (le_of_lt hp)
          _ = _ := by rw [← mul_rpow (le_of_lt hc₁_mem) (le_of_lt hf_pos)]
    | .inr (.inr hneg) => -- eventually negative (which is impossible)
      have : ∀ᶠ (_ : ℝ) in atTop, False := by
        filter_upwards [hf_nonneg, hneg] with x hx hx'; linarith
      rw [Filter.eventually_false_iff_eq_bot] at this
      exact False.elim <| atTop_neBot.ne this
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (goal)
Goal:
f : ℝ → ℝ
p : ℕ
hf : GrowsPolynomially f
hf_nonneg : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
⊢ GrowsPolynomially fun (x : ℝ) ↦ f x ^ p
have simp_rw : GrowsPolynomially fun (x : ℝ) ↦ HPow.hPow (β := ℝ) (f x) (↑p : ℝ)
Tactic: simp_rw [← rpow_natCast]
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (goal)
Goal:
f : ℝ → ℝ
p : ℤ
hf : GrowsPolynomially f
hf_nonneg : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
⊢ GrowsPolynomially fun (x : ℝ) ↦ f x ^ p
have simp_rw : GrowsPolynomially fun (x : ℝ) ↦ HPow.hPow (β := ℝ) (f x) (↑p : ℝ)
Tactic: simp_rw [← rpow_intCast]
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (hypothesis)
Goal:
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, log u ∈ Set.Icc (c₁ * log x) (c₂ * log x)
have hb₀ : 0 < b
Tactic: have hb₀ : 0 < b := hb.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
⊢ ∃ c₁ > 0, ∃ c₂ > 0, ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, log u ∈ Set.Icc (c₁ * log x) (c₂ * log x)
have refine : ∃ c₂ > 0, ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, log u ∈ Set.Icc (1 / 2 * log x) (c₂ * log x)
Tactic: refine ⟨1 / 2, by norm_num, ?_⟩
-- 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
⊢ ∃ c₂ > 0, ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, log u ∈ Set.Icc (1 / 2 * log x) (c₂ * log x)
have refine : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, log u ∈ Set.Icc (1 / 2 * log x) (1 * log x)
Tactic: refine ⟨1, by norm_num, ?_⟩
-- Lean 4 proof step from Mathlib.Computability.AkraBazzi.GrowsPolynomially (hypothesis)
Goal:
b : ℝ
hb : Membership.mem (γ := Set ℝ) (Set.Ioo 0 1) b
hb₀ : 0 < b
⊢ ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, log u ∈ Set.Icc (1 / 2 * log x) (1 * log x)
have h_tendsto : Tendsto (β := ℝ) (fun (x : ℝ) ↦ 1 / 2 * log x) atTop atTop
Tactic: have h_tendsto : Tendsto (fun x => 1 / 2 * Real.log x) atTop atTop :=
    Tendsto.const_mul_atTop (by norm_num) Real.tendsto_log_atTop
-- 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
⊢ ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, log u ∈ Set.Icc (1 / 2 * log x) (1 * log x)
have h : ∀ (x : ℝ) (hx_pos : 1 < x) (hx : ∀ (y : ℝ), b * id x ≤ y → -log b < 1 / 2 * log y),
  ∀ u ∈ Set.Icc (b * x) x, log u ∈ Set.Icc (1 / 2 * log x) (1 * log x)
Tactic: filter_upwards [eventually_gt_atTop 1,
                  (tendsto_id.const_mul_atTop hb.1).eventually_forall_ge_atTop
                    <| h_tendsto.eventually (eventually_gt_atTop (-Real.log b))] with x hx_pos hx
-- 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 ∈ Set.Icc (1 / 2 * log x) (1 * log x)
have refine : 1 / 2 * log x ≤ log u
Tactic: refine ⟨?lb, ?ub⟩
-- 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
refine : 1 / 2 * log x ≤ log u
⊢ log u ∈ Set.Icc (1 / 2 * log x) (1 * log x)
have refine₁ : log u ≤ 1 * log x
Tactic: refine ⟨?lb, ?ub⟩
-- 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 x + log b = log (b * x)
have rw : log (x * b) = log (b * x)
Tactic: rw [← Real.log_mul (by positivity) (by positivity), mul_comm]