Module: Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree

Goal state before:
pp: case odd.right
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_add_eq_of_natDegree_le
cp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ n → (p ^ m).coeff (m * n) = p.coeff n ^ m :=
  fun {m n} {p} ↦ coeff_pow_of_natDegree_le
m : ℕ
h₁ : (W.preΨ' (m + 1)).natDegree ≤ expDegree (m + 1) ∧ (W.preΨ' (m + 1)).coeff (expDegree (m + 1)) = ↑(expCoeff (m + 1))
h₂ : (W.preΨ' (m + 2)).natDegree ≤ expDegree (m + 2) ∧ (W.preΨ' (m + 2)).coeff (expDegree (m + 2)) = ↑(expCoeff (m + 2))
h₃ : (W.preΨ' (m + 3)).natDegree ≤ expDegree (m + 3) ∧ (W.preΨ' (m + 3)).coeff (expDegree (m + 3)) = ↑(expCoeff (m + 3))
h₄ : (W.preΨ' (m + 4)).natDegree ≤ expDegree (m + 4) ∧ (W.preΨ' (m + 4)).coeff (expDegree (m + 4)) = ↑(expCoeff (m + 4))
⊢ ((↑(expCoeff (m + 4)) * ↑(expCoeff (m + 2)) ^ 3 * if Even m then 4 ^ 2 else 1) -
      ↑(expCoeff (m + 1)) * ↑(expCoeff (m + 3)) ^ 3 * if Even m then 1 else 4 ^ 2) =
    ↑(expCoeff (2 * (m + 2) + 1))

pp: case odd.right
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_add_eq_of_natDegree_le
cp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ n → (p ^ m).coeff (m * n) = p.coeff n ^ m :=
  fun {m n} {p} ↦ coeff_pow_of_natDegree_le
m : ℕ
h₁ : (W.preΨ' (m + 1)).natDegree ≤ expDegree (m + 1) ∧ (W.preΨ' (m + 1)).coeff (expDegree (m + 1)) = ↑(expCoeff (m + 1))
h₂ : (W.preΨ' (m + 2)).natDegree ≤ expDegree (m + 2) ∧ (W.preΨ' (m + 2)).coeff (expDegree (m + 2)) = ↑(expCoeff (m + 2))
h₃ : (W.preΨ' (m + 3)).natDegree ≤ expDegree (m + 3) ∧ (W.preΨ' (m + 3)).coeff (expDegree (m + 3)) = ↑(expCoeff (m + 3))
h₄ : (W.preΨ' (m + 4)).natDegree ≤ expDegree (m + 4) ∧ (W.preΨ' (m + 4)).coeff (expDegree (m + 4)) = ↑(expCoeff (m + 4))
⊢ (if Even m then 1 else W.Ψ₂Sq ^ 2).natDegree ≤ if Even m then 0 else 2 * 3

pp: case odd.right
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_add_eq_of_natDegree_le
cp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ n → (p ^ m).coeff (m * n) = p.coeff n ^ m :=
  fun {m n} {p} ↦ coeff_pow_of_natDegree_le
m : ℕ
h₁ : (W.preΨ' (m + 1)).natDegree ≤ expDegree (m + 1) ∧ (W.preΨ' (m + 1)).coeff (expDegree (m + 1)) = ↑(expCoeff (m + 1))
h₂ : (W.preΨ' (m + 2)).natDegree ≤ expDegree (m + 2) ∧ (W.preΨ' (m + 2)).coeff (expDegree (m + 2)) = ↑(expCoeff (m + 2))
h₃ : (W.preΨ' (m + 3)).natDegree ≤ expDegree (m + 3) ∧ (W.preΨ' (m + 3)).coeff (expDegree (m + 3)) = ↑(expCoeff (m + 3))
h₄ : (W.preΨ' (m + 4)).natDegree ≤ expDegree (m + 4) ∧ (W.preΨ' (m + 4)).coeff (expDegree (m + 4)) = ↑(expCoeff (m + 4))
⊢ (if Even m then W.Ψ₂Sq ^ 2 else 1).natDegree ≤ if Even m then 2 * 3 else 0

Tactic:
(expCoeff_rec m).2

Goal state after:
case odd.right
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_add_eq_of_natDegree_le
cp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ n → (p ^ m).coeff (m * n) = p.coeff n ^ m :=
  fun {m n} {p} ↦ coeff_pow_of_natDegree_le
m : ℕ
h₁ : (W.preΨ' (m + 1)).natDegree ≤ expDegree (m + 1) ∧ (W.preΨ' (m + 1)).coeff (expDegree (m + 1)) = ↑(expCoeff (m + 1))
h₂ : (W.preΨ' (m + 2)).natDegree ≤ expDegree (m + 2) ∧ (W.preΨ' (m + 2)).coeff (expDegree (m + 2)) = ↑(expCoeff (m + 2))
h₃ : (W.preΨ' (m + 3)).natDegree ≤ expDegree (m + 3) ∧ (W.preΨ' (m + 3)).coeff (expDegree (m + 3)) = ↑(expCoeff (m + 3))
h₄ : (W.preΨ' (m + 4)).natDegree ≤ expDegree (m + 4) ∧ (W.preΨ' (m + 4)).coeff (expDegree (m + 4)) = ↑(expCoeff (m + 4))
⊢ ((↑(expCoeff (m + 4)) * ↑(expCoeff (m + 2)) ^ 3 * if Even m then 4 ^ 2 else 1) -
      ↑(expCoeff (m + 1)) * ↑(expCoeff (m + 3)) ^ 3 * if Even m then 1 else 4 ^ 2) =
    ↑((expCoeff (m + 4) * expCoeff (m + 2) ^ 3 * if Even m then 4 ^ 2 else 1) -
        expCoeff (m + 1) * expCoeff (m + 3) ^ 3 * if Even m then 1 else 4 ^ 2)

case odd.right
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_add_eq_of_natDegree_le
cp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ n → (p ^ m).coeff (m * n) = p.coeff n ^ m :=
  fun {m n} {p} ↦ coeff_pow_of_natDegree_le
m : ℕ
h₁ : (W.preΨ' (m + 1)).natDegree ≤ expDegree (m + 1) ∧ (W.preΨ' (m + 1)).coeff (expDegree (m + 1)) = ↑(expCoeff (m + 1))
h₂ : (W.preΨ' (m + 2)).natDegree ≤ expDegree (m + 2) ∧ (W.preΨ' (m + 2)).coeff (expDegree (m + 2)) = ↑(expCoeff (m + 2))
h₃ : (W.preΨ' (m + 3)).natDegree ≤ expDegree (m + 3) ∧ (W.preΨ' (m + 3)).coeff (expDegree (m + 3)) = ↑(expCoeff (m + 3))
h₄ : (W.preΨ' (m + 4)).natDegree ≤ expDegree (m + 4) ∧ (W.preΨ' (m + 4)).coeff (expDegree (m + 4)) = ↑(expCoeff (m + 4))
⊢ (if Even m then 1 else W.Ψ₂Sq ^ 2).natDegree ≤ if Even m then 0 else 2 * 3

case odd.right
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_add_eq_of_natDegree_le
cp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ n → (p ^ m).coeff (m * n) = p.coeff n ^ m :=
  fun {m n} {p} ↦ coeff_pow_of_natDegree_le
m : ℕ
h₁ : (W.preΨ' (m + 1)).natDegree ≤ expDegree (m + 1) ∧ (W.preΨ' (m + 1)).coeff (expDegree (m + 1)) = ↑(expCoeff (m + 1))
h₂ : (W.preΨ' (m + 2)).natDegree ≤ expDegree (m + 2) ∧ (W.preΨ' (m + 2)).coeff (expDegree (m + 2)) = ↑(expCoeff (m + 2))
h₃ : (W.preΨ' (m + 3)).natDegree ≤ expDegree (m + 3) ∧ (W.preΨ' (m + 3)).coeff (expDegree (m + 3)) = ↑(expCoeff (m + 3))
h₄ : (W.preΨ' (m + 4)).natDegree ≤ expDegree (m + 4) ∧ (W.preΨ' (m + 4)).coeff (expDegree (m + 4)) = ↑(expCoeff (m + 4))
⊢ (if Even m then W.Ψ₂Sq ^ 2 else 1).natDegree ≤ if Even m then 2 * 3 else 0