-- Module: Mathlib.RingTheory.MvPolynomial.Groebner
Goal:
σ : Type u_1
m : MonomialOrder.{u_1, u_4} σ
R : Type u_2
inst : CommRing R
ι : Type u_3
b : ι → MvPolynomial σ R
hb : ∀ (i : ι), IsUnit (m.leadingCoeff (b i))
f : MvPolynomial σ R
hb' : ∀ (i : ι), m.degree (b i) ≠ 0
hf0 : ¬f = 0
i : ι
hf : m.degree (b i) ≤ m.degree f
deg_reduce :
  LT.lt.{u_4} (α := m.syn) ((m.toSyn : (a : σ →₀ ℕ) → m.syn) (m.degree (m.reduce ⋯ f)) : m.syn)
    ((m.toSyn : (a : σ →₀ ℕ) → m.syn) (m.degree f) : m.syn)
g' : ι →₀ MvPolynomial σ R
r' : MvPolynomial σ R
H' :
  m.reduce ⋯ f =
      HAdd.hAdd (α := MvPolynomial σ R)
        ((Finsupp.linearCombination (MvPolynomial σ R) b : (a : ι →₀ MvPolynomial σ R) → MvPolynomial σ R) g' :
          MvPolynomial σ R)
        r' ∧
    (∀ (i_1 : ι),
        LE.le.{u_4} (α := m.syn)
          ((m.toSyn : (a : σ →₀ ℕ) → m.syn)
              (m.degree
                (HMul.hMul (β := MvPolynomial σ R) (b i_1)
                  ((g' : (a : ι) → MvPolynomial σ R) i_1 : MvPolynomial σ R))) :
            m.syn)
          ((m.toSyn : (a : σ →₀ ℕ) → m.syn) (m.degree (m.reduce ⋯ f)) : m.syn)) ∧
      ∀ c ∈ r'.support, ∀ (i : ι), ¬m.degree (b i) ≤ c
j : ι
h h₁ h₂ :
  LE.le.{u_4} (α := m.syn)
    ((m.toSyn : (a : σ →₀ ℕ) → m.syn)
        (m.degree
          (HMul.hMul (β := MvPolynomial σ R) (b j)
            (if i = j then
                (monomial (m.degree f - m.degree (b i)) : (a : R) → MvPolynomial σ R)
                  ((↑⋯.unit⁻¹ : R) * m.leadingCoeff f)
              else 0 :
              MvPolynomial σ R))) :
      m.syn)
    ((m.toSyn : (a : σ →₀ ℕ) → m.syn) (m.degree f) : m.syn)
pos :
  ∀ (hc : i = j),
    LE.le.{u_4} (α := m.syn)
      ((m.toSyn : (a : σ →₀ ℕ) → m.syn)
          (m.degree
            (HMul.hMul (β := MvPolynomial σ R) (b j)
              ((monomial (m.degree f - m.degree (b i)) : (a : R) → MvPolynomial σ R)
                  ((↑⋯.unit⁻¹ : R) * m.leadingCoeff f) :
                MvPolynomial σ R))) :
        m.syn)
      ((m.toSyn : (a : σ →₀ ℕ) → m.syn) (m.degree f) : m.syn)
neg neg₁ :
  ∀ (hc : ¬i = j),
    LE.le.{u_4} (α := m.syn) ((m.toSyn : (a : σ →₀ ℕ) → m.syn) (m.degree (b j * 0)) : m.syn)
      ((m.toSyn : (a : σ →₀ ℕ) → m.syn) (m.degree f) : m.syn)
⊢ LE.le.{u_4} (α := m.syn)
    ((m.toSyn : (a : σ →₀ ℕ) → m.syn)
        (m.degree
          (HMul.hMul (β := MvPolynomial σ R) (b j)
            (((Finsupp.single i
                      ((monomial (m.degree f - m.degree (b i)) : (a : R) → MvPolynomial σ R)
                        ((↑⋯.unit⁻¹ : R) * m.leadingCoeff f)) :
                    ι →₀ MvPolynomial σ R) :
                  (a : ι) → MvPolynomial σ R)
                j :
              MvPolynomial σ R))) :
      m.syn)
    ((m.toSyn : (a : σ →₀ ℕ) → m.syn) (m.degree f) : m.syn)

Tactic:
classical
        rw [Finsupp.single_apply]
        split_ifs with hc
        · subst j
          grw [degree_mul_le, map_add, degree_monomial_le, ← map_add, add_tsub_cancel_of_le hf]
        · simp only [mul_zero, degree_zero, map_zero]
          exact bot_le

Premises used: map_zero, map_add, MonomialOrder.degree_mul_le, MulZeroClass.mul_zero, MonomialOrder.degree_zero, Finsupp.single_apply, MonomialOrder.degree_monomial_le, bot_le, add_tsub_cancel_of_le