-- 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
h :
  f =
    HAdd.hAdd (α := MvPolynomial σ R)
      ((Finsupp.linearCombination (MvPolynomial σ R) b : (a : ι →₀ MvPolynomial σ R) → MvPolynomial σ R)
          (HAdd.hAdd (β := ι →₀ MvPolynomial σ R) g'
            (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)) :
        MvPolynomial σ R)
      r'
h₁ :
  (∀ (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)
                ((HAdd.hAdd (β := ι →₀ MvPolynomial σ R) g'
                        (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)
                    i_1 :
                  MvPolynomial σ R))) :
          m.syn)
        ((m.toSyn : (a : σ →₀ ℕ) → m.syn) (m.degree f) : m.syn)) ∧
    ∀ c ∈ r'.support, ∀ (i : ι), ¬m.degree (b i) ≤ c
⊢ f =
      HAdd.hAdd (α := MvPolynomial σ R)
        ((Finsupp.linearCombination (MvPolynomial σ R) b : (a : ι →₀ MvPolynomial σ R) → MvPolynomial σ R)
            (HAdd.hAdd (β := ι →₀ MvPolynomial σ R) g'
              (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)) :
          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)
                  ((HAdd.hAdd (β := ι →₀ MvPolynomial σ R) g'
                          (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)
                      i_1 :
                    MvPolynomial σ R))) :
            m.syn)
          ((m.toSyn : (a : σ →₀ ℕ) → m.syn) (m.degree f) : m.syn)) ∧
      ∀ c ∈ r'.support, ∀ (i : ι), ¬m.degree (b i) ≤ c

Tactic:
constructor