theorem exists_pos_real_of_irrational_root {α : ℝ} (ha : Irrational α) {f : ℤ[X]} (f0 : f ≠ 0)
    (fa : eval α (map (algebraMap ℤ ℝ) f) = 0) :
    ∃ A : ℝ, 0 < A ∧ ∀ a : ℤ, ∀ b : ℕ,
      (1 : ℝ) ≤ ((b : ℝ) + 1) ^ f.natDegree * (|α - a / (b + 1)| * A) := by
  -- `fR` is `f` viewed as a polynomial with `ℝ` coefficients.
  set fR : ℝ[X] := map (algebraMap ℤ ℝ) f
  -- `fR` is non-zero, since `f` is non-zero.
  obtain fR0 : fR ≠ 0 := fun fR0 =>
    (map_injective (algebraMap ℤ ℝ) fun _ _ A => Int.cast_inj.mp A).ne f0
      (fR0.trans (Polynomial.map_zero _).symm)
  -- reformulating assumption `fa`: `α` is a root of `fR`.
  have ar : α ∈ (fR.roots.toFinset : Set ℝ) :=
    Finset.mem_coe.mpr (Multiset.mem_toFinset.mpr ((mem_roots fR0).mpr (IsRoot.def.mpr fa)))
  -- Since the polynomial `fR` has finitely many roots, there is a closed interval centered at `α`
  -- such that `α` is the only root of `fR` in the interval.
  obtain ⟨ζ, z0, U⟩ : ∃ ζ > 0, closedBall α ζ ∩ fR.roots.toFinset = {α} :=
    @exists_closedBall_inter_eq_singleton_of_discrete _ _ _ discrete_of_t1_of_finite _ ar
  -- Since `fR` is continuous, it is bounded on the interval above.
  obtain ⟨xm, -, hM⟩ : ∃ xm : ℝ, xm ∈ Icc (α - ζ) (α + ζ) ∧
      IsMaxOn (|fR.derivative.eval ·|) (Icc (α - ζ) (α + ζ)) xm :=
    IsCompact.exists_isMaxOn isCompact_Icc
      ⟨α, (sub_lt_self α z0).le, (lt_add_of_pos_right α z0).le⟩
      (continuous_abs.comp fR.derivative.continuous_aeval).continuousOn
  -- Use the key lemma `exists_one_le_pow_mul_dist`: we are left to show that ...
  refine'
    @exists_one_le_pow_mul_dist ℤ ℕ ℝ _ _ _ (fun y => fR.eval y) α ζ |fR.derivative.eval xm| _ z0
      (fun y hy => _) fun z a hq => _
  -- 1: the denominators are positive -- essentially by definition;
  · exact fun a => one_le_pow_of_one_le ((le_add_iff_nonneg_left 1).mpr a.cast_nonneg) _
  -- 2: the polynomial `fR` is Lipschitz at `α` -- as its derivative continuous;
  · rw [mul_comm]
    rw [Real.closedBall_eq_Icc] at hy
    -- apply the Mean Value Theorem: the bound on the derivative comes from differentiability.
    refine'
      Convex.norm_image_sub_le_of_norm_deriv_le (fun _ _ => fR.differentiableAt)
        (fun y h => by rw [fR.deriv]; exact hM h) (convex_Icc _ _) hy (mem_Icc_iff_abs_le.mp _)
    exact @mem_closedBall_self ℝ _ α ζ (le_of_lt z0)
  -- 3: the weird inequality of Liouville type with powers of the denominators.
  · show 1 ≤ (a + 1 : ℝ) ^ f.natDegree * |eval α fR - eval ((z : ℝ) / (a + 1)) fR|
    rw [fa, zero_sub, abs_neg]
    rw [show (a + 1 : ℝ) = ((a + 1 : ℕ) : ℤ) by norm_cast] at hq ⊢
    -- key observation: the right-hand side of the inequality is an *integer*.  Therefore,
    -- if its absolute value is not at least one, then it vanishes.  Proceed by contradiction
    refine' one_le_pow_mul_abs_eval_div (Int.coe_nat_succ_pos a) fun hy => _
    -- As the evaluation of the polynomial vanishes, we found a root of `fR` that is rational.
    -- We know that `α` is the only root of `fR` in our interval, and `α` is irrational:
    -- follow your nose.
    refine' (irrational_iff_ne_rational α).mp ha z (a + 1) (mem_singleton_iff.mp _).symm
    refine' U.subset _
    

Proof state:
case intro.intro.intro.intro.refine'_3
α : ℝ
ha : Irrational α
f : ℤ[X]
f0 : f ≠ 0
fR : ℝ[X] := map (algebraMap ℤ ℝ) f
fa : eval α fR = 0
fR0 : fR ≠ 0
ar : α ∈ ↑(Multiset.toFinset (roots fR))
ζ : ℝ
z0 : ζ > 0
U : closedBall α ζ ∩ ↑(Multiset.toFinset (roots fR)) = {α}
xm : ℝ
hM : IsMaxOn (fun x => |eval x (derivative fR)|) (Icc (α - ζ) (α + ζ)) xm
z : ℤ
a : ℕ
hq : ↑z / ↑↑(a + 1) ∈ closedBall α ζ
hy : eval (↑z / ↑↑(a + 1)) (map (algebraMap ℤ ℝ) f) = 0
⊢ ↑z / ↑(↑a + 1) ∈ closedBall α ζ ∩ ↑(Multiset.toFinset (roots fR))

Next tactic:
refine' ⟨hq, Finset.mem_coe.mp (Multiset.mem_toFinset.mpr _)⟩