-- Mathlib/NumberTheory/Transcendental/Lindemann/Init/AnalyticalPart.lean
theorem exp_polynomial_approx (f : ℤ[X]) (hf : f.eval 0 ≠ 0) :
    ∃ c,
      ∀ p > (eval 0 f).natAbs, p.Prime →
        ∃ n : ℤ, ¬ ↑p ∣ n ∧ ∃ gp : ℤ[X], gp.natDegree ≤ p * f.natDegree - 1 ∧
          ∀ {r : ℂ}, r ∈ f.aroots ℂ →
            ‖n • exp r - p • aeval r gp‖ ≤ c ^ p / (p - 1)!

Proof state:
case h.right.intro.intro.refine_2
f : ℤ[X]
hf : eval 0 f ≠ 0
c' : ℂ → ℝ
c'0 : ∀ (s : ℂ), c' s ≥ 0
Pp'_le : ∀ (s : ℂ) (p : ℕ), p ≠ 0 → ‖LindemannWeierstrass.P (map (algebraMap ℤ ℂ) (X ^ (p - 1) * f ^ p)) s‖ ≤ c' s ^ p
p : ℕ
p_gt : p > (eval 0 f).natAbs
prime_p : Nat.Prime p
gp' : ℤ[X]
h' : eval 0 (sumIDeriv (X ^ (p - 1) * f ^ p)) = (p - 1)! • eval 0 (f ^ p) + p ! • eval 0 gp'
gp : ℤ[X]
gp'_le : gp.natDegree ≤ (X ^ (p - 1) * f ^ p).natDegree - p
r : ℂ
hr : r ∈ f.aroots ℂ
h : (aeval r) (sumIDeriv (X ^ (p - 1) * f ^ p)) = ↑p ! * (aeval r) gp
⊢ ‖↑(eval 0 f ^ p + ↑p * eval 0 gp') * cexp r - ↑p * (aeval r) gp‖ ≤
    (if h : (Multiset.map c' (f.aroots ℂ)).toFinset.Nonempty then (Multiset.map c' (f.aroots ℂ)).toFinset.max' h
        else 0) ^
        p /
      ↑(p - 1)!

Tactic:
have :
    (↑(eval 0 f ^ p + p * eval 0 gp') * cexp r - p * (aeval r) gp) * (p - 1)! =
    ((eval 0 f ^ p * cexp r) * (p - 1)! +
      ↑(p * (p - 1)!) * (eval 0 gp' * cexp r - (aeval r) gp)) := by
  push_cast; ring

Result:
case h.right.intro.intro.refine_2
f : ℤ[X]
hf : eval 0 f ≠ 0
c' : ℂ → ℝ
c'0 : ∀ (s : ℂ), c' s ≥ 0
Pp'_le : ∀ (s : ℂ) (p : ℕ), p ≠ 0 → ‖LindemannWeierstrass.P (map (algebraMap ℤ ℂ) (X ^ (p - 1) * f ^ p)) s‖ ≤ c' s ^ p
p : ℕ
p_gt : p > (eval 0 f).natAbs
prime_p : Nat.Prime p
gp' : ℤ[X]
h' : eval 0 (sumIDeriv (X ^ (p - 1) * f ^ p)) = (p - 1)! • eval 0 (f ^ p) + p ! • eval 0 gp'
gp : ℤ[X]
gp'_le : gp.natDegree ≤ (X ^ (p - 1) * f ^ p).natDegree - p
r : ℂ
hr : r ∈ f.aroots ℂ
h : (aeval r) (sumIDeriv (X ^ (p - 1) * f ^ p)) = ↑p ! * (aeval r) gp
this :
  (↑(eval 0 f ^ p + ↑p * eval 0 gp') * cexp r - ↑p * (aeval r) gp) * ↑(p - 1)! =
    ↑(eval 0 f) ^ p * cexp r * ↑(p - 1)! + ↑(p * (p - 1)!) * (↑(eval 0 gp') * cexp r - (aeval r) gp)
⊢ ‖↑(eval 0 f ^ p + ↑p * eval 0 gp') * cexp r - ↑p * (aeval r) gp‖ ≤
    (if h : (Multiset.map c' (f.aroots ℂ)).toFinset.Nonempty then (Multiset.map c' (f.aroots ℂ)).toFinset.max' h
        else 0) ^
        p /
      ↑(p - 1)!

Proof state:
case h.intro.intro
f : ℤ[X]
hf : eval 0 f ≠ 0
c' : ℂ → ℝ
c'0 : ∀ (s : ℂ), c' s ≥ 0
Pp'_le : ∀ (s : ℂ) (p : ℕ), p ≠ 0 → ‖LindemannWeierstrass.P (map (algebraMap ℤ ℂ) (X ^ (p - 1) * f ^ p)) s‖ ≤ c' s ^ p
p : ℕ
p_gt : p > (eval 0 f).natAbs
prime_p : Nat.Prime p
gp' : ℤ[X]
h' : eval 0 (sumIDeriv (X ^ (p - 1) * f ^ p)) = (p - 1)! • eval 0 (f ^ p) + p ! • eval 0 gp'
⊢ ∃ n,
    ¬↑p ∣ n ∧
      ∃ gp,
        gp.natDegree ≤ p * f.natDegree - 1 ∧
          ∀ {r : ℂ},
            r ∈ f.aroots ℂ →
              ‖↑n * cexp r - ↑p * (aeval r) gp‖ ≤
                (if h : (Multiset.map c' (f.aroots ℂ)).toFinset.Nonempty then
                      (Multiset.map c' (f.aroots ℂ)).toFinset.max' h
                    else 0) ^
                    p /
                  ↑(p - 1)!

Tactic:
use f.eval 0 ^ p + p * gp'.eval 0

Result:
case h
f : ℤ[X]
hf : eval 0 f ≠ 0
c' : ℂ → ℝ
c'0 : ∀ (s : ℂ), c' s ≥ 0
Pp'_le : ∀ (s : ℂ) (p : ℕ), p ≠ 0 → ‖LindemannWeierstrass.P (map (algebraMap ℤ ℂ) (X ^ (p - 1) * f ^ p)) s‖ ≤ c' s ^ p
p : ℕ
p_gt : p > (eval 0 f).natAbs
prime_p : Nat.Prime p
gp' : ℤ[X]
h' : eval 0 (sumIDeriv (X ^ (p - 1) * f ^ p)) = (p - 1)! • eval 0 (f ^ p) + p ! • eval 0 gp'
⊢ ¬↑p ∣ eval 0 f ^ p + ↑p * eval 0 gp' ∧
    ∃ gp,
      gp.natDegree ≤ p * f.natDegree - 1 ∧
        ∀ {r : ℂ},
          r ∈ f.aroots ℂ →
            ‖↑(eval 0 f ^ p + ↑p * eval 0 gp') * cexp r - ↑p * (aeval r) gp‖ ≤
              (if h : (Multiset.map c' (f.aroots ℂ)).toFinset.Nonempty then
                    (Multiset.map c' (f.aroots ℂ)).toFinset.max' h
                  else 0) ^
                  p /
                ↑(p - 1)!

Proof state:
f : ℤ[X]
hf : eval 0 f ≠ 0
⊢ ∃ c,
    ∀ p > (eval 0 f).natAbs,
      Nat.Prime p →
        ∃ n,
          ¬↑p ∣ n ∧
            ∃ gp,
              gp.natDegree ≤ p * f.natDegree - 1 ∧
                ∀ {r : ℂ}, r ∈ f.aroots ℂ → ‖n • cexp r - p • (aeval r) gp‖ ≤ c ^ p / ↑(p - 1)!

Tactic:
simp_rw [nsmul_eq_mul, zsmul_eq_mul]

Result:
f : ℤ[X]
hf : eval 0 f ≠ 0
⊢ ∃ c,
    ∀ p > (eval 0 f).natAbs,
      Nat.Prime p →
        ∃ n,
          ¬↑p ∣ n ∧
            ∃ gp,
              gp.natDegree ≤ p * f.natDegree - 1 ∧
                ∀ {r : ℂ}, r ∈ f.aroots ℂ → ‖↑n * cexp r - ↑p * (aeval r) gp‖ ≤ c ^ p / ↑(p - 1)!

Proof state:
case h.right.intro.intro.refine_2
f : ℤ[X]
hf : eval 0 f ≠ 0
c' : ℂ → ℝ
c'0 : ∀ (s : ℂ), c' s ≥ 0
Pp'_le : ∀ (s : ℂ) (p : ℕ), p ≠ 0 → ‖LindemannWeierstrass.P (map (algebraMap ℤ ℂ) (X ^ (p - 1) * f ^ p)) s‖ ≤ c' s ^ p
p : ℕ
p_gt : p > (eval 0 f).natAbs
prime_p : Nat.Prime p
gp' : ℤ[X]
h' : eval 0 (sumIDeriv (X ^ (p - 1) * f ^ p)) = (p - 1)! • eval 0 (f ^ p) + p ! • eval 0 gp'
gp : ℤ[X]
gp'_le : gp.natDegree ≤ (X ^ (p - 1) * f ^ p).natDegree - p
r : ℂ
hr : r ∈ f.aroots ℂ
h : (aeval r) (sumIDeriv (X ^ (p - 1) * f ^ p)) = ↑p ! * (aeval r) gp
this✝ :
  (↑(eval 0 f ^ p + ↑p * eval 0 gp') * cexp r - ↑p * (aeval r) gp) * ↑(p - 1)! =
    ↑(eval 0 f) ^ p * cexp r * ↑(p - 1)! + ↑(p * (p - 1)!) * (↑(eval 0 gp') * cexp r - (aeval r) gp)
this :
  ↑(eval 0 f) ^ p * cexp r * ↑(p - 1)! +
      (↑p ! * (↑(eval 0 gp') * cexp r) - (aeval r) (sumIDeriv (X ^ (p - 1) * f ^ p))) =
    ↑((p - 1)! • eval 0 (f ^ p) + p ! • eval 0 gp') * cexp r - (aeval r) (sumIDeriv (X ^ (p - 1) * f ^ p))
aux : c' r ∈ (Multiset.map c' (f.aroots ℂ)).toFinset
⊢ c' r ≤
    if h : (Multiset.map c' (f.aroots ℂ)).toFinset.Nonempty then (Multiset.map c' (f.aroots ℂ)).toFinset.max' h else 0

Tactic:
have h : ((f.aroots ℂ).map c').toFinset.Nonempty := ⟨c' r, aux⟩

Result:
case h.right.intro.intro.refine_2
f : ℤ[X]
hf : eval 0 f ≠ 0
c' : ℂ → ℝ
c'0 : ∀ (s : ℂ), c' s ≥ 0
Pp'_le : ∀ (s : ℂ) (p : ℕ), p ≠ 0 → ‖LindemannWeierstrass.P (map (algebraMap ℤ ℂ) (X ^ (p - 1) * f ^ p)) s‖ ≤ c' s ^ p
p : ℕ
p_gt : p > (eval 0 f).natAbs
prime_p : Nat.Prime p
gp' : ℤ[X]
h' : eval 0 (sumIDeriv (X ^ (p - 1) * f ^ p)) = (p - 1)! • eval 0 (f ^ p) + p ! • eval 0 gp'
gp : ℤ[X]
gp'_le : gp.natDegree ≤ (X ^ (p - 1) * f ^ p).natDegree - p
r : ℂ
hr : r ∈ f.aroots ℂ
h✝ : (aeval r) (sumIDeriv (X ^ (p - 1) * f ^ p)) = ↑p ! * (aeval r) gp
this✝ :
  (↑(eval 0 f ^ p + ↑p * eval 0 gp') * cexp r - ↑p * (aeval r) gp) * ↑(p - 1)! =
    ↑(eval 0 f) ^ p * cexp r * ↑(p - 1)! + ↑(p * (p - 1)!) * (↑(eval 0 gp') * cexp r - (aeval r) gp)
this :
  ↑(eval 0 f) ^ p * cexp r * ↑(p - 1)! +
      (↑p ! * (↑(eval 0 gp') * cexp r) - (aeval r) (sumIDeriv (X ^ (p - 1) * f ^ p))) =
    ↑((p - 1)! • eval 0 (f ^ p) + p ! • eval 0 gp') * cexp r - (aeval r) (sumIDeriv (X ^ (p - 1) * f ^ p))
aux : c' r ∈ (Multiset.map c' (f.aroots ℂ)).toFinset
h : (Multiset.map c' (f.aroots ℂ)).toFinset.Nonempty
⊢ c' r ≤
    if h : (Multiset.map c' (f.aroots ℂ)).toFinset.Nonempty then (Multiset.map c' (f.aroots ℂ)).toFinset.max' h else 0