lemma eventually_atTop_nonneg_or_nonpos (hf : GrowsPolynomially f) :
    (∀ᶠ x in atTop, 0 ≤ f x) ∨ (∀ᶠ x in atTop, f x ≤ 0) := by
  obtain ⟨c₁, _, c₂, _, h⟩ := hf (1/2) (by norm_num)
  rcases lt_trichotomy c₁ c₂ with hlt|heq|hgt
  case inl =>  -- c₁ < c₂
    left
    filter_upwards [h, eventually_ge_atTop 0] with x hx hx_nonneg
    have h' : 3 / 4 * x ∈ Set.Icc (1 / 2 * x) x := by
      rw [Set.mem_Icc]
      exact ⟨by gcongr ?_ * x; norm_num, by linarith⟩
    have hu := hx (3/4 * x) h'
    have hu := Set.nonempty_of_mem hu
    rw [Set.nonempty_Icc] at hu
    have hu' : 0 ≤ (c₂ - c₁) * f x := by linarith
    exact nonneg_of_mul_nonneg_right hu' (by linarith)
  case inr.inr =>   -- c₂ < c₁
    right
    filter_upwards [h, eventually_ge_atTop 0] with x hx hx_nonneg
    have h' : 3 / 4 * x ∈ Set.Icc (1 / 2 * x) x := by
      rw [Set.mem_Icc]
      exact ⟨by gcongr ?_ * x; norm_num, by linarith⟩
    have hu := hx (3/4 * x) h'
    have hu := Set.nonempty_of_mem hu
    rw [Set.nonempty_Icc] at hu
    have hu' : (c₁ - c₂) * f x ≤ 0 := by linarith
    exact nonpos_of_mul_nonpos_right hu' (by linarith)
  case inr.inl =>   -- c₁ = c₂
    have hmain : ∃ c, ∀ᶠ x in atTop, f x = c := by
      simp only [heq, Set.Icc_self, Set.mem_singleton_iff, one_mul] at h
      rw [eventually_atTop] at h
      obtain ⟨n₀, hn₀⟩ := h
      refine ⟨f (max n₀ 2), ?_⟩
      rw [eventually_atTop]
      refine ⟨max n₀ 2, ?_⟩
      refine Real.induction_Ico_mul _ 2 (by norm_num) (by positivity) ?base ?step
      case base =>
        intro x ⟨hxlb, hxub⟩
        have h₁ := calc n₀ ≤ 1 * max n₀ 2 := by simp
                        _ ≤ 2 * max n₀ 2 := by gcongr; norm_num
        have h₂ := hn₀ (2 * max n₀ 2) h₁ (max n₀ 2) ⟨by simp [-max_le_iff, hxlb], by linarith⟩
        rw [h₂]
        exact hn₀ (2 * max n₀ 2) h₁ x ⟨by simp [-max_le_iff, hxlb], le_of_lt hxub⟩
      case step =>
        intro n hn hyp_ind z hz
        have z_nonneg : 0 ≤ z := by
          calc (0:ℝ) ≤ (2:ℝ)^n * max n₀ 2 := by
                        exact mul_nonneg (pow_nonneg (by norm_num) _) (by norm_num)
                  _ ≤ z := by exact_mod_cast hz.1
        have le_2n : max n₀ 2 ≤ (2:ℝ)^n * max n₀ 2 := by
          nth_rewrite 1 [← one_mul (max n₀ 2)]
          gcongr
          exact one_le_pow_of_one_le (by norm_num : (1:ℝ) ≤ 2) _
        have n₀_le_z : n₀ ≤ z := by
          calc n₀ ≤ max n₀ 2 := by simp
                _ ≤ (2:ℝ)^n * max n₀ 2 := le_2n
                _ ≤ _ := by exact_mod_cast hz.1
        have fz_eq_c₂fz : f z = c₂ * f z := hn₀ z n₀_le_z z ⟨by linarith, le_rfl⟩
        have z_to_half_z' : f (1/2 * z) = c₂ * f z := hn₀ z n₀_le_z (1/2 * z) ⟨le_rfl, by linarith⟩
        have z_to_half_z : f (1/2 * z) = f z := by rwa [← fz_eq_c₂fz] at z_to_half_z'
        have half_z_to_base : f (1/2 * z) = f (max n₀ 2) := by
          

Proof state:
f : ℝ → ℝ
hf : GrowsPolynomially f
c₁ : ℝ
left✝¹ : c₁ > 0
c₂ : ℝ
left✝ : c₂ > 0
heq : c₁ = c₂
n₀ : ℝ
hn₀ : ∀ b ≥ n₀, ∀ u ∈ Set.Icc (1 / 2 * b) b, f u = c₂ * f b
n : ℕ
hn : n ≥ 1
hyp_ind : ∀ z ∈ Set.Ico (max n₀ 2) (2 ^ n * max n₀ 2), f z = f (max n₀ 2)
z : ℝ
hz : z ∈ Set.Ico (2 ^ n * max n₀ 2) (2 ^ (n + 1) * max n₀ 2)
z_nonneg : 0 ≤ z
le_2n : max n₀ 2 ≤ 2 ^ n * max n₀ 2
n₀_le_z : n₀ ≤ z
fz_eq_c₂fz : f z = c₂ * f z
z_to_half_z' : f (1 / 2 * z) = c₂ * f z
z_to_half_z : f (1 / 2 * z) = f z
⊢ f (1 / 2 * z) = f (max n₀ 2)

Next tactic:
refine hyp_ind (1/2 * z) ⟨?lb, ?ub⟩