/-- If `A` is a family of enough low-degree polynomials over a finite field,
there is a pair of elements in `A` (with different indices but not necessarily
distinct), such that the difference of their remainders is close together. -/
theorem exists_approx_polynomial {b : Fq[X]} (hb : b ≠ 0) {ε : ℝ} (hε : 0 < ε)
    (A : Fin (Fintype.card Fq ^ ⌈-log ε / log (Fintype.card Fq)⌉₊).succ → Fq[X]) :
    ∃ i₀ i₁, i₀ ≠ i₁ ∧ (cardPowDegree (A i₁ % b - A i₀ % b) : ℝ) < cardPowDegree b • ε := by
  have hbε : 0 < cardPowDegree b • ε := by
    rw [Algebra.smul_def, eq_intCast]
    exact mul_pos (Int.cast_pos.mpr (AbsoluteValue.pos _ hb)) hε
  have one_lt_q : 1 < Fintype.card Fq := Fintype.one_lt_card
  have one_lt_q' : (1 : ℝ) < Fintype.card Fq := by assumption_mod_cast
  have q_pos : 0 < Fintype.card Fq := by linarith
  have q_pos' : (0 : ℝ) < Fintype.card Fq := by assumption_mod_cast
  -- If `b` is already small enough, then the remainders are equal and we are done.
  by_cases le_b : b.natDegree ≤ ⌈-log ε / log (Fintype.card Fq)⌉₊
  · obtain ⟨i₀, i₁, i_ne, mod_eq⟩ :=
      exists_eq_polynomial le_rfl b le_b (fun i => A i % b) fun i => EuclideanDomain.mod_lt (A i) hb
    refine' ⟨i₀, i₁, i_ne, _⟩
    rwa [mod_eq, sub_self, map_zero, Int.cast_zero]
  -- Otherwise, it suffices to choose two elements whose difference is of small enough degree.
  rw [not_le] at le_b
  obtain ⟨i₀, i₁, i_ne, deg_lt⟩ := exists_approx_polynomial_aux le_rfl b (fun i => A i % b) fun i =>
    EuclideanDomain.mod_lt (A i) hb
  use i₀, i₁, i_ne
  -- Again, if the remainders are equal we are done.
  by_cases h : A i₁ % b = A i₀ % b
  · rwa [h, sub_self, map_zero, Int.cast_zero]
  have h' : A i₁ % b - A i₀ % b ≠ 0 := mt sub_eq_zero.mp h
  -- If the remainders are not equal, we'll show their difference is of small degree.
  -- In particular, we'll show the degree is less than the following:
  suffices (natDegree (A i₁ % b - A i₀ % b) : ℝ) < b.natDegree + log ε / log (Fintype.card Fq) by
    rwa [← Real.log_lt_log_iff (Int.cast_pos.mpr (cardPowDegree.pos h')) hbε,
      cardPowDegree_nonzero _ h', cardPowDegree_nonzero _ hb, Algebra.smul_def, eq_intCast,
      Int.cast_pow, Int.cast_ofNat, Int.cast_pow, Int.cast_ofNat,
      log_mul (pow_ne_zero _ q_pos'.ne') hε.ne', ← rpow_nat_cast, ← rpow_nat_cast, log_rpow q_pos',
      log_rpow q_pos', ← lt_div_iff (log_pos one_lt_q'), add_div,
      mul_div_cancel _ (log_pos one_lt_q').ne']
  -- And that result follows from manipulating the result from `exists_approx_polynomial_aux`
  -- to turn the `-⌈-stuff⌉₊` into `+ stuff`.
  refine' lt_of_lt_of_le (Nat.cast_lt.mpr (WithBot.coe_lt_coe.mp _)) _
  swap
  · convert deg_lt
    rw [degree_eq_natDegree h']; rfl
  

Proof state:
case neg.refine'_3
Fq : Type u_1
inst✝¹ : Fintype Fq
inst✝ : Field Fq
b : Fq[X]
hb : b ≠ 0
ε : ℝ
hε : 0 < ε
A : Fin (Nat.succ (Fintype.card Fq ^ ⌈-log ε / log ↑(Fintype.card Fq)⌉₊)) → Fq[X]
hbε : 0 < cardPowDegree b • ε
one_lt_q : 1 < Fintype.card Fq
one_lt_q' : 1 < ↑(Fintype.card Fq)
q_pos : 0 < Fintype.card Fq
q_pos' : 0 < ↑(Fintype.card Fq)
le_b : ⌈-log ε / log ↑(Fintype.card Fq)⌉₊ < natDegree b
i₀ i₁ : Fin (Nat.succ (Fintype.card Fq ^ ⌈-log ε / log ↑(Fintype.card Fq)⌉₊))
i_ne : i₀ ≠ i₁
deg_lt : degree (A i₁ % b - A i₀ % b) < ↑(natDegree b - ⌈-log ε / log ↑(Fintype.card Fq)⌉₊)
h : ¬A i₁ % b = A i₀ % b
h' : A i₁ % b - A i₀ % b ≠ 0
⊢ ↑(natDegree b - ⌈-log ε / log ↑(Fintype.card Fq)⌉₊) ≤ ↑(natDegree b) + log ε / log ↑(Fintype.card Fq)

Next tactic:
rw [← sub_neg_eq_add, neg_div]