Module: Mathlib.RingTheory.MvPolynomial.MonomialOrder

Goal state before:
pp: case inr.inl
σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
p₂ q₁ q₂ : MvPolynomial σ R
this :
  ∀ {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [inst : CommRing R] [NoZeroDivisors R]
    (p₁ p₂ q₁ q₂ : MvPolynomial σ R),
    p₁ ≠ 0 ∧ p₂ ≠ 0 ∧ q₁ ≠ 0 ∧ q₂ ≠ 0 →
      m.sPolynomial (m.leadingTerm p₁ * q₁) (m.leadingTerm p₂ * q₂) =
        (monomial (m.degree (p₁ * q₁) ⊔ m.degree (p₂ * q₂) - m.degree q₁ ⊔ m.degree q₂))
            (m.leadingCoeff p₁ * m.leadingCoeff p₂) *
          m.sPolynomial q₁ q₂
⊢ m.sPolynomial (m.leadingTerm 0 * q₁) (m.leadingTerm p₂ * q₂) =
    (monomial (m.degree (0 * q₁) ⊔ m.degree (p₂ * q₂) - m.degree q₁ ⊔ m.degree q₂))
        (m.leadingCoeff 0 * m.leadingCoeff p₂) *
      m.sPolynomial q₁ q₂

pp: case inr.inr.inl
σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
p₁ q₁ q₂ : MvPolynomial σ R
this :
  ∀ {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [inst : CommRing R] [NoZeroDivisors R]
    (p₁ p₂ q₁ q₂ : MvPolynomial σ R),
    p₁ ≠ 0 ∧ p₂ ≠ 0 ∧ q₁ ≠ 0 ∧ q₂ ≠ 0 →
      m.sPolynomial (m.leadingTerm p₁ * q₁) (m.leadingTerm p₂ * q₂) =
        (monomial (m.degree (p₁ * q₁) ⊔ m.degree (p₂ * q₂) - m.degree q₁ ⊔ m.degree q₂))
            (m.leadingCoeff p₁ * m.leadingCoeff p₂) *
          m.sPolynomial q₁ q₂
⊢ m.sPolynomial (m.leadingTerm p₁ * q₁) (m.leadingTerm 0 * q₂) =
    (monomial (m.degree (p₁ * q₁) ⊔ m.degree (0 * q₂) - m.degree q₁ ⊔ m.degree q₂))
        (m.leadingCoeff p₁ * m.leadingCoeff 0) *
      m.sPolynomial q₁ q₂

pp: case inr.inr.inr.inl
σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
p₁ p₂ q₂ : MvPolynomial σ R
this :
  ∀ {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [inst : CommRing R] [NoZeroDivisors R]
    (p₁ p₂ q₁ q₂ : MvPolynomial σ R),
    p₁ ≠ 0 ∧ p₂ ≠ 0 ∧ q₁ ≠ 0 ∧ q₂ ≠ 0 →
      m.sPolynomial (m.leadingTerm p₁ * q₁) (m.leadingTerm p₂ * q₂) =
        (monomial (m.degree (p₁ * q₁) ⊔ m.degree (p₂ * q₂) - m.degree q₁ ⊔ m.degree q₂))
            (m.leadingCoeff p₁ * m.leadingCoeff p₂) *
          m.sPolynomial q₁ q₂
⊢ m.sPolynomial (m.leadingTerm p₁ * 0) (m.leadingTerm p₂ * q₂) =
    (monomial (m.degree (p₁ * 0) ⊔ m.degree (p₂ * q₂) - m.degree 0 ⊔ m.degree q₂))
        (m.leadingCoeff p₁ * m.leadingCoeff p₂) *
      m.sPolynomial 0 q₂

pp: case inr.inr.inr.inr
σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
p₁ p₂ q₁ : MvPolynomial σ R
this :
  ∀ {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [inst : CommRing R] [NoZeroDivisors R]
    (p₁ p₂ q₁ q₂ : MvPolynomial σ R),
    p₁ ≠ 0 ∧ p₂ ≠ 0 ∧ q₁ ≠ 0 ∧ q₂ ≠ 0 →
      m.sPolynomial (m.leadingTerm p₁ * q₁) (m.leadingTerm p₂ * q₂) =
        (monomial (m.degree (p₁ * q₁) ⊔ m.degree (p₂ * q₂) - m.degree q₁ ⊔ m.degree q₂))
            (m.leadingCoeff p₁ * m.leadingCoeff p₂) *
          m.sPolynomial q₁ q₂
⊢ m.sPolynomial (m.leadingTerm p₁ * q₁) (m.leadingTerm p₂ * 0) =
    (monomial (m.degree (p₁ * q₁) ⊔ m.degree (p₂ * 0) - m.degree q₁ ⊔ m.degree 0))
        (m.leadingCoeff p₁ * m.leadingCoeff p₂) *
      m.sPolynomial q₁ 0

Tactic:
<;>

Goal state after:
case inr.inl
σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
p₂ q₁ q₂ : MvPolynomial σ R
this :
  ∀ {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [inst : CommRing R] [NoZeroDivisors R]
    (p₁ p₂ q₁ q₂ : MvPolynomial σ R),
    p₁ ≠ 0 ∧ p₂ ≠ 0 ∧ q₁ ≠ 0 ∧ q₂ ≠ 0 →
      m.sPolynomial (m.leadingTerm p₁ * q₁) (m.leadingTerm p₂ * q₂) =
        (monomial (m.degree (p₁ * q₁) ⊔ m.degree (p₂ * q₂) - m.degree q₁ ⊔ m.degree q₂))
            (m.leadingCoeff p₁ * m.leadingCoeff p₂) *
          m.sPolynomial q₁ q₂
⊢ m.sPolynomial (m.leadingTerm 0 * q₁) (m.leadingTerm p₂ * q₂) =
    (monomial (m.degree (0 * q₁) ⊔ m.degree (p₂ * q₂) - m.degree q₁ ⊔ m.degree q₂))
        (m.leadingCoeff 0 * m.leadingCoeff p₂) *
      m.sPolynomial q₁ q₂

case inr.inr.inl
σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
p₁ q₁ q₂ : MvPolynomial σ R
this :
  ∀ {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [inst : CommRing R] [NoZeroDivisors R]
    (p₁ p₂ q₁ q₂ : MvPolynomial σ R),
    p₁ ≠ 0 ∧ p₂ ≠ 0 ∧ q₁ ≠ 0 ∧ q₂ ≠ 0 →
      m.sPolynomial (m.leadingTerm p₁ * q₁) (m.leadingTerm p₂ * q₂) =
        (monomial (m.degree (p₁ * q₁) ⊔ m.degree (p₂ * q₂) - m.degree q₁ ⊔ m.degree q₂))
            (m.leadingCoeff p₁ * m.leadingCoeff p₂) *
          m.sPolynomial q₁ q₂
⊢ m.sPolynomial (m.leadingTerm p₁ * q₁) (m.leadingTerm 0 * q₂) =
    (monomial (m.degree (p₁ * q₁) ⊔ m.degree (0 * q₂) - m.degree q₁ ⊔ m.degree q₂))
        (m.leadingCoeff p₁ * m.leadingCoeff 0) *
      m.sPolynomial q₁ q₂

case inr.inr.inr.inl
σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
p₁ p₂ q₂ : MvPolynomial σ R
this :
  ∀ {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [inst : CommRing R] [NoZeroDivisors R]
    (p₁ p₂ q₁ q₂ : MvPolynomial σ R),
    p₁ ≠ 0 ∧ p₂ ≠ 0 ∧ q₁ ≠ 0 ∧ q₂ ≠ 0 →
      m.sPolynomial (m.leadingTerm p₁ * q₁) (m.leadingTerm p₂ * q₂) =
        (monomial (m.degree (p₁ * q₁) ⊔ m.degree (p₂ * q₂) - m.degree q₁ ⊔ m.degree q₂))
            (m.leadingCoeff p₁ * m.leadingCoeff p₂) *
          m.sPolynomial q₁ q₂
⊢ m.sPolynomial (m.leadingTerm p₁ * 0) (m.leadingTerm p₂ * q₂) =
    (monomial (m.degree (p₁ * 0) ⊔ m.degree (p₂ * q₂) - m.degree 0 ⊔ m.degree q₂))
        (m.leadingCoeff p₁ * m.leadingCoeff p₂) *
      m.sPolynomial 0 q₂

case inr.inr.inr.inr
σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
p₁ p₂ q₁ : MvPolynomial σ R
this :
  ∀ {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [inst : CommRing R] [NoZeroDivisors R]
    (p₁ p₂ q₁ q₂ : MvPolynomial σ R),
    p₁ ≠ 0 ∧ p₂ ≠ 0 ∧ q₁ ≠ 0 ∧ q₂ ≠ 0 →
      m.sPolynomial (m.leadingTerm p₁ * q₁) (m.leadingTerm p₂ * q₂) =
        (monomial (m.degree (p₁ * q₁) ⊔ m.degree (p₂ * q₂) - m.degree q₁ ⊔ m.degree q₂))
            (m.leadingCoeff p₁ * m.leadingCoeff p₂) *
          m.sPolynomial q₁ q₂
⊢ m.sPolynomial (m.leadingTerm p₁ * q₁) (m.leadingTerm p₂ * 0) =
    (monomial (m.degree (p₁ * q₁) ⊔ m.degree (p₂ * 0) - m.degree q₁ ⊔ m.degree 0))
        (m.leadingCoeff p₁ * m.leadingCoeff p₂) *
      m.sPolynomial q₁ 0