l3lab/ntp-mathlib huggingface dataset:
lemma eventually_zero_of_frequently_zero (hf : GrowsPolynomially f) (hf' : ∃ᶠ x in atTop, f x = 0) :
    ∀ᶠ x in atTop, f x = 0 := by
  obtain ⟨c₁, hc₁_mem, c₂, hc₂_mem, hf⟩ := hf (1/2) (by norm_num)
  rw [frequently_atTop] at hf'
  filter_upwards [eventually_forall_ge_atTop.mpr hf, eventually_gt_atTop 0] with x hx hx_pos
  obtain ⟨x₀, hx₀_ge, hx₀⟩ := hf' (max x 1)
  have x₀_pos := calc
    0 < 1 := by norm_num
    _ ≤ x₀ := le_of_max_le_right hx₀_ge
  have hmain : ∀ (m : ℕ) (z : ℝ), x ≤ z →
      z ∈ Set.Icc ((2:ℝ)^(-(m:ℤ) -1) * x₀) ((2:ℝ)^(-(m:ℤ)) * x₀) → f z = 0 := by
    intro m
    induction m
    case zero =>
      simp only [Nat.zero_eq, CharP.cast_eq_zero, neg_zero, zero_sub, zpow_zero, one_mul] at *
      specialize hx x₀ (le_of_max_le_left hx₀_ge)
      simp only [hx₀, mul_zero, Set.Icc_self, Set.mem_singleton_iff] at hx
      refine fun z _ hz => hx _ ?_
      simp only [zpow_neg, zpow_one] at hz
      simp only [one_div, hz]
    case succ k ih =>
      intro z hxz hz
      simp only [Nat.succ_eq_add_one, Nat.cast_add, Nat.cast_one] at *
      have hx' : x ≤ (2:ℝ)^(-(k:ℤ) - 1) * x₀ := by
        calc x ≤ z := hxz
          _ ≤ _ := by simp only [neg_add, ← sub_eq_add_neg] at hz; exact hz.2
      specialize hx ((2:ℝ)^(-(k:ℤ) - 1) * x₀) hx' z
      specialize ih ((2:ℝ)^(-(k:ℤ) - 1) * x₀) hx' ?ineq
      case ineq =>
        rw [Set.left_mem_Icc]
        gcongr
        · norm_num
        · linarith
      simp only [ih, mul_zero, Set.Icc_self, Set.mem_singleton_iff] at hx
      refine hx ⟨?lb₁, ?ub₁⟩
      case lb₁ =>
        rw [one_div, ← zpow_neg_one, ← mul_assoc, ← zpow_add₀ (by norm_num)]
        have h₁ : (-1 : ℤ)  + (-k - 1) = -k - 2 := by ring
        have h₂ : -(k + (1:ℤ)) - 1 = -k - 2 := by 

Proof state:
f : ℝ → ℝ
hf✝ : GrowsPolynomially f
hf' : ∀ (a : ℝ), ∃ b ≥ a, f b = 0
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
x : ℝ
hx_pos : 0 < x
x₀ : ℝ
hx₀_ge : x₀ ≥ max x 1
hx₀ : f x₀ = 0
x₀_pos : 0 < x₀
k : ℕ
z : ℝ
hxz : x ≤ z
hz : z ∈ Set.Icc (2 ^ (-(↑k + 1) - 1) * x₀) (2 ^ (-(↑k + 1)) * x₀)
hx' : x ≤ 2 ^ (-↑k - 1) * x₀
ih : f (2 ^ (-↑k - 1) * x₀) = 0
hx : z ∈ Set.Icc (1 / 2 * (2 ^ (-↑k - 1) * x₀)) (2 ^ (-↑k - 1) * x₀) → f z = 0
h₁ : -1 + (-↑k - 1) = -↑k - 2
⊢ -(↑k + 1) - 1 = -↑k - 2

Next tactic:
ring
lemma eventually_zero_of_frequently_zero (hf : GrowsPolynomially f) (hf' : ∃ᶠ x in atTop, f x = 0) :
    ∀ᶠ x in atTop, f x = 0 := by
  obtain ⟨c₁, hc₁_mem, c₂, hc₂_mem, hf⟩ := hf (1/2) (by norm_num)
  rw [frequently_atTop] at hf'
  filter_upwards [eventually_forall_ge_atTop.mpr hf, eventually_gt_atTop 0] with x hx hx_pos
  obtain ⟨x₀, hx₀_ge, hx₀⟩ := hf' (max x 1)
  have x₀_pos := calc
    0 < 1 := by norm_num
    _ ≤ x₀ := le_of_max_le_right hx₀_ge
  have hmain : ∀ (m : ℕ) (z : ℝ), x ≤ z →
      z ∈ Set.Icc ((2:ℝ)^(-(m:ℤ) -1) * x₀) ((2:ℝ)^(-(m:ℤ)) * x₀) → f z = 0 := by
    intro m
    induction m
    case zero =>
      simp only [Nat.zero_eq, CharP.cast_eq_zero, neg_zero, zero_sub, zpow_zero, one_mul] at *
      specialize hx x₀ (le_of_max_le_left hx₀_ge)
      simp only [hx₀, mul_zero, Set.Icc_self, Set.mem_singleton_iff] at hx
      refine fun z _ hz => hx _ ?_
      simp only [zpow_neg, zpow_one] at hz
      simp only [one_div, hz]
    case succ k ih =>
      intro z hxz hz
      simp only [Nat.succ_eq_add_one, Nat.cast_add, Nat.cast_one] at *
      have hx' : x ≤ (2:ℝ)^(-(k:ℤ) - 1) * x₀ := by
        calc x ≤ z := hxz
          _ ≤ _ := by simp only [neg_add, ← sub_eq_add_neg] at hz; exact hz.2
      specialize hx ((2:ℝ)^(-(k:ℤ) - 1) * x₀) hx' z
      specialize ih ((2:ℝ)^(-(k:ℤ) - 1) * x₀) hx' ?ineq
      case ineq =>
        rw [Set.left_mem_Icc]
        gcongr
        · norm_num
        · linarith
      simp only [ih, mul_zero, Set.Icc_self, Set.mem_singleton_iff] at hx
      refine hx ⟨?lb₁, ?ub₁⟩
      case lb₁ =>
        rw [one_div, ← zpow_neg_one, ← mul_assoc, ← zpow_add₀ (by norm_num)]
        have h₁ : (-1 : ℤ)  + (-k - 1) = -k - 2 := by ring
        have h₂ : -(k + (1:ℤ)) - 1 = -k - 2 := by ring
        

Proof state:
f : ℝ → ℝ
hf✝ : GrowsPolynomially f
hf' : ∀ (a : ℝ), ∃ b ≥ a, f b = 0
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
x : ℝ
hx_pos : 0 < x
x₀ : ℝ
hx₀_ge : x₀ ≥ max x 1
hx₀ : f x₀ = 0
x₀_pos : 0 < x₀
k : ℕ
z : ℝ
hxz : x ≤ z
hz : z ∈ Set.Icc (2 ^ (-(↑k + 1) - 1) * x₀) (2 ^ (-(↑k + 1)) * x₀)
hx' : x ≤ 2 ^ (-↑k - 1) * x₀
ih : f (2 ^ (-↑k - 1) * x₀) = 0
hx : z ∈ Set.Icc (1 / 2 * (2 ^ (-↑k - 1) * x₀)) (2 ^ (-↑k - 1) * x₀) → f z = 0
h₁ : -1 + (-↑k - 1) = -↑k - 2
h₂ : -(↑k + 1) - 1 = -↑k - 2
⊢ 2 ^ (-1 + (-↑k - 1)) * x₀ ≤ z

Next tactic:
rw [h₁]
lemma eventually_zero_of_frequently_zero (hf : GrowsPolynomially f) (hf' : ∃ᶠ x in atTop, f x = 0) :
    ∀ᶠ x in atTop, f x = 0 := by
  obtain ⟨c₁, hc₁_mem, c₂, hc₂_mem, hf⟩ := hf (1/2) (by norm_num)
  rw [frequently_atTop] at hf'
  filter_upwards [eventually_forall_ge_atTop.mpr hf, eventually_gt_atTop 0] with x hx hx_pos
  obtain ⟨x₀, hx₀_ge, hx₀⟩ := hf' (max x 1)
  have x₀_pos := calc
    0 < 1 := by norm_num
    _ ≤ x₀ := le_of_max_le_right hx₀_ge
  have hmain : ∀ (m : ℕ) (z : ℝ), x ≤ z →
      z ∈ Set.Icc ((2:ℝ)^(-(m:ℤ) -1) * x₀) ((2:ℝ)^(-(m:ℤ)) * x₀) → f z = 0 := by
    intro m
    induction m
    case zero =>
      simp only [Nat.zero_eq, CharP.cast_eq_zero, neg_zero, zero_sub, zpow_zero, one_mul] at *
      specialize hx x₀ (le_of_max_le_left hx₀_ge)
      simp only [hx₀, mul_zero, Set.Icc_self, Set.mem_singleton_iff] at hx
      refine fun z _ hz => hx _ ?_
      simp only [zpow_neg, zpow_one] at hz
      simp only [one_div, hz]
    case succ k ih =>
      intro z hxz hz
      simp only [Nat.succ_eq_add_one, Nat.cast_add, Nat.cast_one] at *
      have hx' : x ≤ (2:ℝ)^(-(k:ℤ) - 1) * x₀ := by
        calc x ≤ z := hxz
          _ ≤ _ := by simp only [neg_add, ← sub_eq_add_neg] at hz; exact hz.2
      specialize hx ((2:ℝ)^(-(k:ℤ) - 1) * x₀) hx' z
      specialize ih ((2:ℝ)^(-(k:ℤ) - 1) * x₀) hx' ?ineq
      case ineq =>
        rw [Set.left_mem_Icc]
        gcongr
        · norm_num
        · linarith
      simp only [ih, mul_zero, Set.Icc_self, Set.mem_singleton_iff] at hx
      refine hx ⟨?lb₁, ?ub₁⟩
      case lb₁ =>
        rw [one_div, ← zpow_neg_one, ← mul_assoc, ← zpow_add₀ (by norm_num)]
        have h₁ : (-1 : ℤ)  + (-k - 1) = -k - 2 := by ring
        have h₂ : -(k + (1:ℤ)) - 1 = -k - 2 := by ring
        rw [h₁]
        

Proof state:
f : ℝ → ℝ
hf✝ : GrowsPolynomially f
hf' : ∀ (a : ℝ), ∃ b ≥ a, f b = 0
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
x : ℝ
hx_pos : 0 < x
x₀ : ℝ
hx₀_ge : x₀ ≥ max x 1
hx₀ : f x₀ = 0
x₀_pos : 0 < x₀
k : ℕ
z : ℝ
hxz : x ≤ z
hz : z ∈ Set.Icc (2 ^ (-(↑k + 1) - 1) * x₀) (2 ^ (-(↑k + 1)) * x₀)
hx' : x ≤ 2 ^ (-↑k - 1) * x₀
ih : f (2 ^ (-↑k - 1) * x₀) = 0
hx : z ∈ Set.Icc (1 / 2 * (2 ^ (-↑k - 1) * x₀)) (2 ^ (-↑k - 1) * x₀) → f z = 0
h₁ : -1 + (-↑k - 1) = -↑k - 2
h₂ : -(↑k + 1) - 1 = -↑k - 2
⊢ 2 ^ (-↑k - 2) * x₀ ≤ z

Next tactic:
rw [h₂] at hz
lemma eventually_zero_of_frequently_zero (hf : GrowsPolynomially f) (hf' : ∃ᶠ x in atTop, f x = 0) :
    ∀ᶠ x in atTop, f x = 0 := by
  obtain ⟨c₁, hc₁_mem, c₂, hc₂_mem, hf⟩ := hf (1/2) (by norm_num)
  rw [frequently_atTop] at hf'
  filter_upwards [eventually_forall_ge_atTop.mpr hf, eventually_gt_atTop 0] with x hx hx_pos
  obtain ⟨x₀, hx₀_ge, hx₀⟩ := hf' (max x 1)
  have x₀_pos := calc
    0 < 1 := by norm_num
    _ ≤ x₀ := le_of_max_le_right hx₀_ge
  have hmain : ∀ (m : ℕ) (z : ℝ), x ≤ z →
      z ∈ Set.Icc ((2:ℝ)^(-(m:ℤ) -1) * x₀) ((2:ℝ)^(-(m:ℤ)) * x₀) → f z = 0 := by
    intro m
    induction m
    case zero =>
      simp only [Nat.zero_eq, CharP.cast_eq_zero, neg_zero, zero_sub, zpow_zero, one_mul] at *
      specialize hx x₀ (le_of_max_le_left hx₀_ge)
      simp only [hx₀, mul_zero, Set.Icc_self, Set.mem_singleton_iff] at hx
      refine fun z _ hz => hx _ ?_
      simp only [zpow_neg, zpow_one] at hz
      simp only [one_div, hz]
    case succ k ih =>
      intro z hxz hz
      simp only [Nat.succ_eq_add_one, Nat.cast_add, Nat.cast_one] at *
      have hx' : x ≤ (2:ℝ)^(-(k:ℤ) - 1) * x₀ := by
        calc x ≤ z := hxz
          _ ≤ _ := by simp only [neg_add, ← sub_eq_add_neg] at hz; exact hz.2
      specialize hx ((2:ℝ)^(-(k:ℤ) - 1) * x₀) hx' z
      specialize ih ((2:ℝ)^(-(k:ℤ) - 1) * x₀) hx' ?ineq
      case ineq =>
        rw [Set.left_mem_Icc]
        gcongr
        · norm_num
        · linarith
      simp only [ih, mul_zero, Set.Icc_self, Set.mem_singleton_iff] at hx
      refine hx ⟨?lb₁, ?ub₁⟩
      case lb₁ =>
        rw [one_div, ← zpow_neg_one, ← mul_assoc, ← zpow_add₀ (by norm_num)]
        have h₁ : (-1 : ℤ)  + (-k - 1) = -k - 2 := by ring
        have h₂ : -(k + (1:ℤ)) - 1 = -k - 2 := by ring
        rw [h₁]
        rw [h₂] at hz
        

Proof state:
f : ℝ → ℝ
hf✝ : GrowsPolynomially f
hf' : ∀ (a : ℝ), ∃ b ≥ a, f b = 0
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
x : ℝ
hx_pos : 0 < x
x₀ : ℝ
hx₀_ge : x₀ ≥ max x 1
hx₀ : f x₀ = 0
x₀_pos : 0 < x₀
k : ℕ
z : ℝ
hxz : x ≤ z
hz : z ∈ Set.Icc (2 ^ (-↑k - 2) * x₀) (2 ^ (-(↑k + 1)) * x₀)
hx' : x ≤ 2 ^ (-↑k - 1) * x₀
ih : f (2 ^ (-↑k - 1) * x₀) = 0
hx : z ∈ Set.Icc (1 / 2 * (2 ^ (-↑k - 1) * x₀)) (2 ^ (-↑k - 1) * x₀) → f z = 0
h₁ : -1 + (-↑k - 1) = -↑k - 2
h₂ : -(↑k + 1) - 1 = -↑k - 2
⊢ 2 ^ (-↑k - 2) * x₀ ≤ z

Next tactic:
exact hz.1
lemma eventually_zero_of_frequently_zero (hf : GrowsPolynomially f) (hf' : ∃ᶠ x in atTop, f x = 0) :
    ∀ᶠ x in atTop, f x = 0 := by
  obtain ⟨c₁, hc₁_mem, c₂, hc₂_mem, hf⟩ := hf (1/2) (by norm_num)
  rw [frequently_atTop] at hf'
  filter_upwards [eventually_forall_ge_atTop.mpr hf, eventually_gt_atTop 0] with x hx hx_pos
  obtain ⟨x₀, hx₀_ge, hx₀⟩ := hf' (max x 1)
  have x₀_pos := calc
    0 < 1 := by norm_num
    _ ≤ x₀ := le_of_max_le_right hx₀_ge
  have hmain : ∀ (m : ℕ) (z : ℝ), x ≤ z →
      z ∈ Set.Icc ((2:ℝ)^(-(m:ℤ) -1) * x₀) ((2:ℝ)^(-(m:ℤ)) * x₀) → f z = 0 := by
    intro m
    induction m
    case zero =>
      simp only [Nat.zero_eq, CharP.cast_eq_zero, neg_zero, zero_sub, zpow_zero, one_mul] at *
      specialize hx x₀ (le_of_max_le_left hx₀_ge)
      simp only [hx₀, mul_zero, Set.Icc_self, Set.mem_singleton_iff] at hx
      refine fun z _ hz => hx _ ?_
      simp only [zpow_neg, zpow_one] at hz
      simp only [one_div, hz]
    case succ k ih =>
      intro z hxz hz
      simp only [Nat.succ_eq_add_one, Nat.cast_add, Nat.cast_one] at *
      have hx' : x ≤ (2:ℝ)^(-(k:ℤ) - 1) * x₀ := by
        calc x ≤ z := hxz
          _ ≤ _ := by simp only [neg_add, ← sub_eq_add_neg] at hz; exact hz.2
      specialize hx ((2:ℝ)^(-(k:ℤ) - 1) * x₀) hx' z
      specialize ih ((2:ℝ)^(-(k:ℤ) - 1) * x₀) hx' ?ineq
      case ineq =>
        rw [Set.left_mem_Icc]
        gcongr
        · norm_num
        · linarith
      simp only [ih, mul_zero, Set.Icc_self, Set.mem_singleton_iff] at hx
      refine hx ⟨?lb₁, ?ub₁⟩
      case lb₁ =>
        rw [one_div, ← zpow_neg_one, ← mul_assoc, ← zpow_add₀ (by norm_num)]
        have h₁ : (-1 : ℤ)  + (-k - 1) = -k - 2 := by ring
        have h₂ : -(k + (1:ℤ)) - 1 = -k - 2 := by ring
        rw [h₁]
        rw [h₂] at hz
        exact hz.1
      

Proof state:
case ub₁
f : ℝ → ℝ
hf✝ : GrowsPolynomially f
hf' : ∀ (a : ℝ), ∃ b ≥ a, f b = 0
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
x : ℝ
hx_pos : 0 < x
x₀ : ℝ
hx₀_ge : x₀ ≥ max x 1
hx₀ : f x₀ = 0
x₀_pos : 0 < x₀
k : ℕ
z : ℝ
hxz : x ≤ z
hz : z ∈ Set.Icc (2 ^ (-(↑k + 1) - 1) * x₀) (2 ^ (-(↑k + 1)) * x₀)
hx' : x ≤ 2 ^ (-↑k - 1) * x₀
ih : f (2 ^ (-↑k - 1) * x₀) = 0
hx : z ∈ Set.Icc (1 / 2 * (2 ^ (-↑k - 1) * x₀)) (2 ^ (-↑k - 1) * x₀) → f z = 0
⊢ z ≤ 2 ^ (-↑k - 1) * x₀

Next tactic:
case ub₁ =>
        have := hz.2
        simp only [neg_add, ← sub_eq_add_neg] at this
        exact this
lemma eventually_zero_of_frequently_zero (hf : GrowsPolynomially f) (hf' : ∃ᶠ x in atTop, f x = 0) :
    ∀ᶠ x in atTop, f x = 0 := by
  obtain ⟨c₁, hc₁_mem, c₂, hc₂_mem, hf⟩ := hf (1/2) (by norm_num)
  rw [frequently_atTop] at hf'
  filter_upwards [eventually_forall_ge_atTop.mpr hf, eventually_gt_atTop 0] with x hx hx_pos
  obtain ⟨x₀, hx₀_ge, hx₀⟩ := hf' (max x 1)
  have x₀_pos := calc
    0 < 1 := by norm_num
    _ ≤ x₀ := le_of_max_le_right hx₀_ge
  have hmain : ∀ (m : ℕ) (z : ℝ), x ≤ z →
      z ∈ Set.Icc ((2:ℝ)^(-(m:ℤ) -1) * x₀) ((2:ℝ)^(-(m:ℤ)) * x₀) → f z = 0 := by
    intro m
    induction m
    case zero =>
      simp only [Nat.zero_eq, CharP.cast_eq_zero, neg_zero, zero_sub, zpow_zero, one_mul] at *
      specialize hx x₀ (le_of_max_le_left hx₀_ge)
      simp only [hx₀, mul_zero, Set.Icc_self, Set.mem_singleton_iff] at hx
      refine fun z _ hz => hx _ ?_
      simp only [zpow_neg, zpow_one] at hz
      simp only [one_div, hz]
    case succ k ih =>
      intro z hxz hz
      simp only [Nat.succ_eq_add_one, Nat.cast_add, Nat.cast_one] at *
      have hx' : x ≤ (2:ℝ)^(-(k:ℤ) - 1) * x₀ := by
        calc x ≤ z := hxz
          _ ≤ _ := by simp only [neg_add, ← sub_eq_add_neg] at hz; exact hz.2
      specialize hx ((2:ℝ)^(-(k:ℤ) - 1) * x₀) hx' z
      specialize ih ((2:ℝ)^(-(k:ℤ) - 1) * x₀) hx' ?ineq
      case ineq =>
        rw [Set.left_mem_Icc]
        gcongr
        · norm_num
        · linarith
      simp only [ih, mul_zero, Set.Icc_self, Set.mem_singleton_iff] at hx
      refine hx ⟨?lb₁, ?ub₁⟩
      case lb₁ =>
        rw [one_div, ← zpow_neg_one, ← mul_assoc, ← zpow_add₀ (by norm_num)]
        have h₁ : (-1 : ℤ)  + (-k - 1) = -k - 2 := by ring
        have h₂ : -(k + (1:ℤ)) - 1 = -k - 2 := by ring
        rw [h₁]
        rw [h₂] at hz
        exact hz.1
      

Proof state:
f : ℝ → ℝ
hf✝ : GrowsPolynomially f
hf' : ∀ (a : ℝ), ∃ b ≥ a, f b = 0
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
x : ℝ
hx_pos : 0 < x
x₀ : ℝ
hx₀_ge : x₀ ≥ max x 1
hx₀ : f x₀ = 0
x₀_pos : 0 < x₀
k : ℕ
z : ℝ
hxz : x ≤ z
hz : z ∈ Set.Icc (2 ^ (-(↑k + 1) - 1) * x₀) (2 ^ (-(↑k + 1)) * x₀)
hx' : x ≤ 2 ^ (-↑k - 1) * x₀
ih : f (2 ^ (-↑k - 1) * x₀) = 0
hx : z ∈ Set.Icc (1 / 2 * (2 ^ (-↑k - 1) * x₀)) (2 ^ (-↑k - 1) * x₀) → f z = 0
⊢ z ≤ 2 ^ (-↑k - 1) * x₀

Next tactic:
case ub₁ =>
        have := hz.2
        simp only [neg_add, ← sub_eq_add_neg] at this
        exact this
lemma eventually_zero_of_frequently_zero (hf : GrowsPolynomially f) (hf' : ∃ᶠ x in atTop, f x = 0) :
    ∀ᶠ x in atTop, f x = 0 := by
  obtain ⟨c₁, hc₁_mem, c₂, hc₂_mem, hf⟩ := hf (1/2) (by norm_num)
  rw [frequently_atTop] at hf'
  filter_upwards [eventually_forall_ge_atTop.mpr hf, eventually_gt_atTop 0] with x hx hx_pos
  obtain ⟨x₀, hx₀_ge, hx₀⟩ := hf' (max x 1)
  have x₀_pos := calc
    0 < 1 := by norm_num
    _ ≤ x₀ := le_of_max_le_right hx₀_ge
  have hmain : ∀ (m : ℕ) (z : ℝ), x ≤ z →
      z ∈ Set.Icc ((2:ℝ)^(-(m:ℤ) -1) * x₀) ((2:ℝ)^(-(m:ℤ)) * x₀) → f z = 0 := by
    intro m
    induction m
    case zero =>
      simp only [Nat.zero_eq, CharP.cast_eq_zero, neg_zero, zero_sub, zpow_zero, one_mul] at *
      specialize hx x₀ (le_of_max_le_left hx₀_ge)
      simp only [hx₀, mul_zero, Set.Icc_self, Set.mem_singleton_iff] at hx
      refine fun z _ hz => hx _ ?_
      simp only [zpow_neg, zpow_one] at hz
      simp only [one_div, hz]
    case succ k ih =>
      intro z hxz hz
      simp only [Nat.succ_eq_add_one, Nat.cast_add, Nat.cast_one] at *
      have hx' : x ≤ (2:ℝ)^(-(k:ℤ) - 1) * x₀ := by
        calc x ≤ z := hxz
          _ ≤ _ := by simp only [neg_add, ← sub_eq_add_neg] at hz; exact hz.2
      specialize hx ((2:ℝ)^(-(k:ℤ) - 1) * x₀) hx' z
      specialize ih ((2:ℝ)^(-(k:ℤ) - 1) * x₀) hx' ?ineq
      case ineq =>
        rw [Set.left_mem_Icc]
        gcongr
        · norm_num
        · linarith
      simp only [ih, mul_zero, Set.Icc_self, Set.mem_singleton_iff] at hx
      refine hx ⟨?lb₁, ?ub₁⟩
      case lb₁ =>
        rw [one_div, ← zpow_neg_one, ← mul_assoc, ← zpow_add₀ (by norm_num)]
        have h₁ : (-1 : ℤ)  + (-k - 1) = -k - 2 := by ring
        have h₂ : -(k + (1:ℤ)) - 1 = -k - 2 := by ring
        rw [h₁]
        rw [h₂] at hz
        exact hz.1
      case ub₁ =>
        

Proof state:
f : ℝ → ℝ
hf✝ : GrowsPolynomially f
hf' : ∀ (a : ℝ), ∃ b ≥ a, f b = 0
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
x : ℝ
hx_pos : 0 < x
x₀ : ℝ
hx₀_ge : x₀ ≥ max x 1
hx₀ : f x₀ = 0
x₀_pos : 0 < x₀
k : ℕ
z : ℝ
hxz : x ≤ z
hz : z ∈ Set.Icc (2 ^ (-(↑k + 1) - 1) * x₀) (2 ^ (-(↑k + 1)) * x₀)
hx' : x ≤ 2 ^ (-↑k - 1) * x₀
ih : f (2 ^ (-↑k - 1) * x₀) = 0
hx : z ∈ Set.Icc (1 / 2 * (2 ^ (-↑k - 1) * x₀)) (2 ^ (-↑k - 1) * x₀) → f z = 0
⊢ z ≤ 2 ^ (-↑k - 1) * x₀

Next tactic:
have := hz.2
lemma eventually_zero_of_frequently_zero (hf : GrowsPolynomially f) (hf' : ∃ᶠ x in atTop, f x = 0) :
    ∀ᶠ x in atTop, f x = 0 := by
  obtain ⟨c₁, hc₁_mem, c₂, hc₂_mem, hf⟩ := hf (1/2) (by norm_num)
  rw [frequently_atTop] at hf'
  filter_upwards [eventually_forall_ge_atTop.mpr hf, eventually_gt_atTop 0] with x hx hx_pos
  obtain ⟨x₀, hx₀_ge, hx₀⟩ := hf' (max x 1)
  have x₀_pos := calc
    0 < 1 := by norm_num
    _ ≤ x₀ := le_of_max_le_right hx₀_ge
  have hmain : ∀ (m : ℕ) (z : ℝ), x ≤ z →
      z ∈ Set.Icc ((2:ℝ)^(-(m:ℤ) -1) * x₀) ((2:ℝ)^(-(m:ℤ)) * x₀) → f z = 0 := by
    intro m
    induction m
    case zero =>
      simp only [Nat.zero_eq, CharP.cast_eq_zero, neg_zero, zero_sub, zpow_zero, one_mul] at *
      specialize hx x₀ (le_of_max_le_left hx₀_ge)
      simp only [hx₀, mul_zero, Set.Icc_self, Set.mem_singleton_iff] at hx
      refine fun z _ hz => hx _ ?_
      simp only [zpow_neg, zpow_one] at hz
      simp only [one_div, hz]
    case succ k ih =>
      intro z hxz hz
      simp only [Nat.succ_eq_add_one, Nat.cast_add, Nat.cast_one] at *
      have hx' : x ≤ (2:ℝ)^(-(k:ℤ) - 1) * x₀ := by
        calc x ≤ z := hxz
          _ ≤ _ := by simp only [neg_add, ← sub_eq_add_neg] at hz; exact hz.2
      specialize hx ((2:ℝ)^(-(k:ℤ) - 1) * x₀) hx' z
      specialize ih ((2:ℝ)^(-(k:ℤ) - 1) * x₀) hx' ?ineq
      case ineq =>
        rw [Set.left_mem_Icc]
        gcongr
        · norm_num
        · linarith
      simp only [ih, mul_zero, Set.Icc_self, Set.mem_singleton_iff] at hx
      refine hx ⟨?lb₁, ?ub₁⟩
      case lb₁ =>
        rw [one_div, ← zpow_neg_one, ← mul_assoc, ← zpow_add₀ (by norm_num)]
        have h₁ : (-1 : ℤ)  + (-k - 1) = -k - 2 := by ring
        have h₂ : -(k + (1:ℤ)) - 1 = -k - 2 := by ring
        rw [h₁]
        rw [h₂] at hz
        exact hz.1
      case ub₁ =>
        have := hz.2
        

Proof state:
f : ℝ → ℝ
hf✝ : GrowsPolynomially f
hf' : ∀ (a : ℝ), ∃ b ≥ a, f b = 0
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
x : ℝ
hx_pos : 0 < x
x₀ : ℝ
hx₀_ge : x₀ ≥ max x 1
hx₀ : f x₀ = 0
x₀_pos : 0 < x₀
k : ℕ
z : ℝ
hxz : x ≤ z
hz : z ∈ Set.Icc (2 ^ (-(↑k + 1) - 1) * x₀) (2 ^ (-(↑k + 1)) * x₀)
hx' : x ≤ 2 ^ (-↑k - 1) * x₀
ih : f (2 ^ (-↑k - 1) * x₀) = 0
hx : z ∈ Set.Icc (1 / 2 * (2 ^ (-↑k - 1) * x₀)) (2 ^ (-↑k - 1) * x₀) → f z = 0
this : z ≤ 2 ^ (-(↑k + 1)) * x₀
⊢ z ≤ 2 ^ (-↑k - 1) * x₀

Next tactic:
simp only [neg_add, ← sub_eq_add_neg] at this
lemma eventually_zero_of_frequently_zero (hf : GrowsPolynomially f) (hf' : ∃ᶠ x in atTop, f x = 0) :
    ∀ᶠ x in atTop, f x = 0 := by
  obtain ⟨c₁, hc₁_mem, c₂, hc₂_mem, hf⟩ := hf (1/2) (by norm_num)
  rw [frequently_atTop] at hf'
  filter_upwards [eventually_forall_ge_atTop.mpr hf, eventually_gt_atTop 0] with x hx hx_pos
  obtain ⟨x₀, hx₀_ge, hx₀⟩ := hf' (max x 1)
  have x₀_pos := calc
    0 < 1 := by norm_num
    _ ≤ x₀ := le_of_max_le_right hx₀_ge
  have hmain : ∀ (m : ℕ) (z : ℝ), x ≤ z →
      z ∈ Set.Icc ((2:ℝ)^(-(m:ℤ) -1) * x₀) ((2:ℝ)^(-(m:ℤ)) * x₀) → f z = 0 := by
    intro m
    induction m
    case zero =>
      simp only [Nat.zero_eq, CharP.cast_eq_zero, neg_zero, zero_sub, zpow_zero, one_mul] at *
      specialize hx x₀ (le_of_max_le_left hx₀_ge)
      simp only [hx₀, mul_zero, Set.Icc_self, Set.mem_singleton_iff] at hx
      refine fun z _ hz => hx _ ?_
      simp only [zpow_neg, zpow_one] at hz
      simp only [one_div, hz]
    case succ k ih =>
      intro z hxz hz
      simp only [Nat.succ_eq_add_one, Nat.cast_add, Nat.cast_one] at *
      have hx' : x ≤ (2:ℝ)^(-(k:ℤ) - 1) * x₀ := by
        calc x ≤ z := hxz
          _ ≤ _ := by simp only [neg_add, ← sub_eq_add_neg] at hz; exact hz.2
      specialize hx ((2:ℝ)^(-(k:ℤ) - 1) * x₀) hx' z
      specialize ih ((2:ℝ)^(-(k:ℤ) - 1) * x₀) hx' ?ineq
      case ineq =>
        rw [Set.left_mem_Icc]
        gcongr
        · norm_num
        · linarith
      simp only [ih, mul_zero, Set.Icc_self, Set.mem_singleton_iff] at hx
      refine hx ⟨?lb₁, ?ub₁⟩
      case lb₁ =>
        rw [one_div, ← zpow_neg_one, ← mul_assoc, ← zpow_add₀ (by norm_num)]
        have h₁ : (-1 : ℤ)  + (-k - 1) = -k - 2 := by ring
        have h₂ : -(k + (1:ℤ)) - 1 = -k - 2 := by ring
        rw [h₁]
        rw [h₂] at hz
        exact hz.1
      case ub₁ =>
        have := hz.2
        simp only [neg_add, ← sub_eq_add_neg] at this
        

Proof state:
f : ℝ → ℝ
hf✝ : GrowsPolynomially f
hf' : ∀ (a : ℝ), ∃ b ≥ a, f b = 0
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
x : ℝ
hx_pos : 0 < x
x₀ : ℝ
hx₀_ge : x₀ ≥ max x 1
hx₀ : f x₀ = 0
x₀_pos : 0 < x₀
k : ℕ
z : ℝ
hxz : x ≤ z
hz : z ∈ Set.Icc (2 ^ (-(↑k + 1) - 1) * x₀) (2 ^ (-(↑k + 1)) * x₀)
hx' : x ≤ 2 ^ (-↑k - 1) * x₀
ih : f (2 ^ (-↑k - 1) * x₀) = 0
hx : z ∈ Set.Icc (1 / 2 * (2 ^ (-↑k - 1) * x₀)) (2 ^ (-↑k - 1) * x₀) → f z = 0
this : z ≤ 2 ^ (-↑k - 1) * x₀
⊢ z ≤ 2 ^ (-↑k - 1) * x₀

Next tactic:
exact this