-- Mathlib/RingTheory/MvPolynomial/Groebner.lean
theorem div {ι : Type*} {b : ι → MvPolynomial σ R}
    (hb : ∀ i, IsUnit (m.leadingCoeff (b i))) (f : MvPolynomial σ R) :
    ∃ (g : ι →₀ (MvPolynomial σ R)) (r : MvPolynomial σ R),
      f = Finsupp.linearCombination _ b g + r ∧
        (∀ i, m.degree (b i * (g i)) ≼[m] m.degree f) ∧
        (∀ c ∈ r.support, ∀ i, ¬ (m.degree (b i) ≤ c))

Proof state:
case h.right.left
σ : Type u_1
m : MonomialOrder 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
hf : ∀ (i : ι), ¬m.degree (b i) ≤ m.degree f
g' : ι →₀ MvPolynomial σ R
r' : MvPolynomial σ R
H' :
  m.subLTerm f = (Finsupp.linearCombination (MvPolynomial σ R) b) g' + r' ∧
    (∀ (i : ι), m.toSyn (m.degree (b i * g' i)) ≤ m.toSyn (m.degree (m.subLTerm f))) ∧
      ∀ c ∈ r'.support, ∀ (i : ι), ¬m.degree (b i) ≤ c
⊢ ∀ (i : ι), m.toSyn (m.degree (b i * g' i)) ≤ m.toSyn (m.degree f)

Tactic:
exact fun b ↦ le_trans (H'.2.1 b) (degree_sub_LTerm_le f)

Result:
no goals

Proof state:
case pos
σ : Type u_1
m : MonomialOrder 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
i : ι
hb0 : m.degree (b i) = 0
j : ι
hj : j = i
⊢ m.toSyn (m.degree (b j * (Finsupp.single i (⋯.unit⁻¹ • f)) j)) ≤ m.toSyn (m.degree f)

Tactic:
apply le_trans degree_mul_le

Result:
case pos
σ : Type u_1
m : MonomialOrder 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
i : ι
hb0 : m.degree (b i) = 0
j : ι
hj : j = i
⊢ m.toSyn (m.degree (b j) + m.degree ((Finsupp.single i (⋯.unit⁻¹ • f)) j)) ≤ m.toSyn (m.degree f)

Proof state:
case pos
σ : Type u_1
m : MonomialOrder 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 : m.toSyn (m.degree (m.reduce ⋯ f)) < m.toSyn (m.degree f)
g' : ι →₀ MvPolynomial σ R
r' : MvPolynomial σ R
H' :
  m.reduce ⋯ f = (Finsupp.linearCombination (MvPolynomial σ R) b) g' + r' ∧
    (∀ (i_1 : ι), m.toSyn (m.degree (b i_1 * g' i_1)) ≤ m.toSyn (m.degree (m.reduce ⋯ f))) ∧
      ∀ c ∈ r'.support, ∀ (i : ι), ¬m.degree (b i) ≤ c
j : ι
hc : i = j
⊢ m.toSyn (m.degree (b i)) + m.toSyn (m.degree f - m.degree (b i)) = m.toSyn (m.degree f)

Tactic:
rw [← map_add, m.toSyn.injective.eq_iff]

Result:
case pos
σ : Type u_1
m : MonomialOrder 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 : m.toSyn (m.degree (m.reduce ⋯ f)) < m.toSyn (m.degree f)
g' : ι →₀ MvPolynomial σ R
r' : MvPolynomial σ R
H' :
  m.reduce ⋯ f = (Finsupp.linearCombination (MvPolynomial σ R) b) g' + r' ∧
    (∀ (i_1 : ι), m.toSyn (m.degree (b i_1 * g' i_1)) ≤ m.toSyn (m.degree (m.reduce ⋯ f))) ∧
      ∀ c ∈ r'.support, ∀ (i : ι), ¬m.degree (b i) ≤ c
j : ι
hc : i = j
⊢ m.degree (b i) + (m.degree f - m.degree (b i)) = m.degree f

Proof state:
σ : Type u_1
m : MonomialOrder 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
⊢ f = (Finsupp.linearCombination (MvPolynomial σ R) b) 0 + 0

Tactic:
simp [hf0]

Result:
no goals

Proof state:
case h.right
σ : Type u_1
m : MonomialOrder 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
i : ι
hb0 : m.degree (b i) = 0
⊢ (∀ (i_1 : ι), m.toSyn (m.degree (b i_1 * (Finsupp.single i (⋯.unit⁻¹ • f)) i_1)) ≤ m.toSyn (m.degree f)) ∧
    ∀ c ∈ support 0, ∀ (i : ι), ¬m.degree (b i) ≤ c

Tactic:
constructor

Result:
case h.right.left
σ : Type u_1
m : MonomialOrder 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
i : ι
hb0 : m.degree (b i) = 0
⊢ ∀ (i_1 : ι), m.toSyn (m.degree (b i_1 * (Finsupp.single i (⋯.unit⁻¹ • f)) i_1)) ≤ m.toSyn (m.degree f)

case h.right.right
σ : Type u_1
m : MonomialOrder 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
i : ι
hb0 : m.degree (b i) = 0
⊢ ∀ c ∈ support 0, ∀ (i : ι), ¬m.degree (b i) ≤ c

Proof state:
case h.right.left.left
σ : Type u_1
m : MonomialOrder 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 : m.toSyn (m.degree (m.reduce ⋯ f)) < m.toSyn (m.degree f)
g' : ι →₀ MvPolynomial σ R
r' : MvPolynomial σ R
H' :
  m.reduce ⋯ f = (Finsupp.linearCombination (MvPolynomial σ R) b) g' + r' ∧
    (∀ (i_1 : ι), m.toSyn (m.degree (b i_1 * g' i_1)) ≤ m.toSyn (m.degree (m.reduce ⋯ f))) ∧
      ∀ c ∈ r'.support, ∀ (i : ι), ¬m.degree (b i) ≤ c
j : ι
⊢ m.toSyn (m.degree (b j * g' j)) ≤ m.toSyn (m.degree f)

Tactic:
exact le_trans (H'.2.1 _) (le_of_lt deg_reduce)

Result:
no goals

Proof state:
case pos.intro
σ : Type u_1
m : MonomialOrder 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
⊢ ∃ g r,
    f = (Finsupp.linearCombination (MvPolynomial σ R) b) g + r ∧
      (∀ (i : ι), m.toSyn (m.degree (b i * g i)) ≤ m.toSyn (m.degree f)) ∧
        ∀ c ∈ r.support, ∀ (i : ι), ¬m.degree (b i) ≤ c

Tactic:
have deg_reduce : m.degree (m.reduce (hb i) f) ≺[m] m.degree f := by
  apply degree_reduce_lt (hb i) hf
  intro hf0'
  apply hb' i
  simpa [hf0'] using hf

Result:
case pos.intro
σ : Type u_1
m : MonomialOrder 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 : m.toSyn (m.degree (m.reduce ⋯ f)) < m.toSyn (m.degree f)
⊢ ∃ g r,
    f = (Finsupp.linearCombination (MvPolynomial σ R) b) g + r ∧
      (∀ (i : ι), m.toSyn (m.degree (b i * g i)) ≤ m.toSyn (m.degree f)) ∧
        ∀ c ∈ r.support, ∀ (i : ι), ¬m.degree (b i) ≤ c