-- Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Degree.lean
private lemma natDegree_coeff_Φ_ofNat (n : ℕ) :
    (W.Φ n).natDegree ≤ n ^ 2 ∧ (W.Φ n).coeff (n ^ 2) = 1

Proof state:
case succ.succ.left
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
dm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).natDegree ≤ m + n :=
  fun {m n} {p q} => natDegree_mul_le_of_le
dp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n} {p} => natDegree_pow_le_of_le n
cm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).coeff (m + n) = p.coeff m * q.coeff n :=
  fun {m n} {p q} => coeff_mul_of_natDegree_le
h : ∀ {n : ℕ},
  (W.preΨ' n).natDegree ≤ WeierstrassCurve.expDegree n ∧
    (W.preΨ' n).coeff (WeierstrassCurve.expDegree n) = ↑(WeierstrassCurve.expCoeff n) :=
  fun {n} => WeierstrassCurve.natDegree_coeff_preΨ' W n
n : ℕ
hd : (n + 1 + 1) ^ 2 = 1 + 2 * WeierstrassCurve.expDegree (n + 2) + if Even (n + 1) then 0 else 3
hd' :
  (n + 1 + 1) ^ 2 =
    WeierstrassCurve.expDegree (n + 3) + WeierstrassCurve.expDegree (n + 1) + if Even (n + 1) then 3 else 0
hc :
  1 =
    (1 * WeierstrassCurve.expCoeff (n + 2) ^ 2 * if Even (n + 1) then 1 else 4) -
      WeierstrassCurve.expCoeff (n + 3) * WeierstrassCurve.expCoeff (n + 1) * if Even (n + 1) then 4 else 1
⊢ ((X * W.preΨ' (n + 1 + 1) ^ 2 * if Even (n + 1) then 1 else W.Ψ₂Sq) -
        W.preΨ' (n + 1 + 2) * W.preΨ' (n + 1) * if Even (n + 1) then W.Ψ₂Sq else 1).natDegree ≤
    (n + 1 + 1) ^ 2

Tactic:
nth_rw 1 [← max_self <| (_ + _) ^ 2, hd, hd']

Result:
case succ.succ.left
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
dm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).natDegree ≤ m + n :=
  fun {m n} {p q} => natDegree_mul_le_of_le
dp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n} {p} => natDegree_pow_le_of_le n
cm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).coeff (m + n) = p.coeff m * q.coeff n :=
  fun {m n} {p q} => coeff_mul_of_natDegree_le
h : ∀ {n : ℕ},
  (W.preΨ' n).natDegree ≤ WeierstrassCurve.expDegree n ∧
    (W.preΨ' n).coeff (WeierstrassCurve.expDegree n) = ↑(WeierstrassCurve.expCoeff n) :=
  fun {n} => WeierstrassCurve.natDegree_coeff_preΨ' W n
n : ℕ
hd : (n + 1 + 1) ^ 2 = 1 + 2 * WeierstrassCurve.expDegree (n + 2) + if Even (n + 1) then 0 else 3
hd' :
  (n + 1 + 1) ^ 2 =
    WeierstrassCurve.expDegree (n + 3) + WeierstrassCurve.expDegree (n + 1) + if Even (n + 1) then 3 else 0
hc :
  1 =
    (1 * WeierstrassCurve.expCoeff (n + 2) ^ 2 * if Even (n + 1) then 1 else 4) -
      WeierstrassCurve.expCoeff (n + 3) * WeierstrassCurve.expCoeff (n + 1) * if Even (n + 1) then 4 else 1
⊢ ((X * W.preΨ' (n + 1 + 1) ^ 2 * if Even (n + 1) then 1 else W.Ψ₂Sq) -
        W.preΨ' (n + 1 + 2) * W.preΨ' (n + 1) * if Even (n + 1) then W.Ψ₂Sq else 1).natDegree ≤
    (1 + 2 * WeierstrassCurve.expDegree (n + 2) + if Even (n + 1) then 0 else 3) ⊔
      (WeierstrassCurve.expDegree (n + 3) + WeierstrassCurve.expDegree (n + 1) + if Even (n + 1) then 3 else 0)

Proof state:
case succ.succ.left
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
dm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).natDegree ≤ m + n :=
  fun {m n} {p q} => natDegree_mul_le_of_le
dp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n} {p} => natDegree_pow_le_of_le n
cm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).coeff (m + n) = p.coeff m * q.coeff n :=
  fun {m n} {p q} => coeff_mul_of_natDegree_le
h : ∀ {n : ℕ},
  (W.preΨ' n).natDegree ≤ WeierstrassCurve.expDegree n ∧
    (W.preΨ' n).coeff (WeierstrassCurve.expDegree n) = ↑(WeierstrassCurve.expCoeff n) :=
  fun {n} => WeierstrassCurve.natDegree_coeff_preΨ' W n
n : ℕ
hd : (n + 1 + 1) ^ 2 = 1 + 2 * WeierstrassCurve.expDegree (n + 2) + if Even (n + 1) then 0 else 3
hd' :
  (n + 1 + 1) ^ 2 =
    WeierstrassCurve.expDegree (n + 3) + WeierstrassCurve.expDegree (n + 1) + if Even (n + 1) then 3 else 0
hc :
  1 =
    (1 * WeierstrassCurve.expCoeff (n + 2) ^ 2 * if Even (n + 1) then 1 else 4) -
      WeierstrassCurve.expCoeff (n + 3) * WeierstrassCurve.expCoeff (n + 1) * if Even (n + 1) then 4 else 1
⊢ ((X * W.preΨ' (n + 1 + 1) ^ 2 * if Even (n + 1) then 1 else W.Ψ₂Sq) -
        W.preΨ' (n + 1 + 2) * W.preΨ' (n + 1) * if Even (n + 1) then W.Ψ₂Sq else 1).natDegree ≤
    (1 + 2 * WeierstrassCurve.expDegree (n + 2) + if Even (n + 1) then 0 else 3) ⊔
      (WeierstrassCurve.expDegree (n + 3) + WeierstrassCurve.expDegree (n + 1) + if Even (n + 1) then 3 else 0)

Tactic:
refine natDegree_sub_le_of_le (dm (dm natDegree_X_le (dp h.1)) ?_) (dm (dm h.1 h.1) ?_)

Result:
case succ.succ.left.refine_1
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
dm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).natDegree ≤ m + n :=
  fun {m n} {p q} => natDegree_mul_le_of_le
dp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n} {p} => natDegree_pow_le_of_le n
cm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).coeff (m + n) = p.coeff m * q.coeff n :=
  fun {m n} {p q} => coeff_mul_of_natDegree_le
h : ∀ {n : ℕ},
  (W.preΨ' n).natDegree ≤ WeierstrassCurve.expDegree n ∧
    (W.preΨ' n).coeff (WeierstrassCurve.expDegree n) = ↑(WeierstrassCurve.expCoeff n) :=
  fun {n} => WeierstrassCurve.natDegree_coeff_preΨ' W n
n : ℕ
hd : (n + 1 + 1) ^ 2 = 1 + 2 * WeierstrassCurve.expDegree (n + 2) + if Even (n + 1) then 0 else 3
hd' :
  (n + 1 + 1) ^ 2 =
    WeierstrassCurve.expDegree (n + 3) + WeierstrassCurve.expDegree (n + 1) + if Even (n + 1) then 3 else 0
hc :
  1 =
    (1 * WeierstrassCurve.expCoeff (n + 2) ^ 2 * if Even (n + 1) then 1 else 4) -
      WeierstrassCurve.expCoeff (n + 3) * WeierstrassCurve.expCoeff (n + 1) * if Even (n + 1) then 4 else 1
⊢ (if Even (n + 1) then 1 else W.Ψ₂Sq).natDegree ≤ if Even (n + 1) then 0 else 3

case succ.succ.left.refine_2
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
dm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).natDegree ≤ m + n :=
  fun {m n} {p q} => natDegree_mul_le_of_le
dp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n} {p} => natDegree_pow_le_of_le n
cm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).coeff (m + n) = p.coeff m * q.coeff n :=
  fun {m n} {p q} => coeff_mul_of_natDegree_le
h : ∀ {n : ℕ},
  (W.preΨ' n).natDegree ≤ WeierstrassCurve.expDegree n ∧
    (W.preΨ' n).coeff (WeierstrassCurve.expDegree n) = ↑(WeierstrassCurve.expCoeff n) :=
  fun {n} => WeierstrassCurve.natDegree_coeff_preΨ' W n
n : ℕ
hd : (n + 1 + 1) ^ 2 = 1 + 2 * WeierstrassCurve.expDegree (n + 2) + if Even (n + 1) then 0 else 3
hd' :
  (n + 1 + 1) ^ 2 =
    WeierstrassCurve.expDegree (n + 3) + WeierstrassCurve.expDegree (n + 1) + if Even (n + 1) then 3 else 0
hc :
  1 =
    (1 * WeierstrassCurve.expCoeff (n + 2) ^ 2 * if Even (n + 1) then 1 else 4) -
      WeierstrassCurve.expCoeff (n + 3) * WeierstrassCurve.expCoeff (n + 1) * if Even (n + 1) then 4 else 1
⊢ (if Even (n + 1) then W.Ψ₂Sq else 1).natDegree ≤ if Even (n + 1) then 3 else 0