-- 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 neg
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
⊢ ∃ 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:
by_cases hf : ∃ i, m.degree (b i) ≤ m.degree f

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
hf : ∃ i, 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

case neg
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, 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

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
⊢ ∃ 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:
by_cases hb' : ∃ i, m.degree (b i) = 0

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, m.degree (b i) = 0
⊢ ∃ 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

case neg
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, m.degree (b i) = 0
⊢ ∃ 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

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
⊢ ∀ c ∈ support 0, ∀ (i : ι), ¬m.degree (b i) ≤ c

Tactic:
simp

Result:
no goals

Proof state:
case h.e'_1.h.e'_4
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
c : σ →₀ ℕ
hc : c ∈ (r' + (monomial (m.degree f)) (m.leadingCoeff f)).support
i : ι
hc' : c ∉ r'.support
this : c ∉ r'.support → c ∈ ((monomial (m.degree f)) (m.leadingCoeff f)).support
⊢ c = m.degree f

Tactic:
simpa only [Finset.mem_singleton] using support_monomial_subset (this hc')

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
hb' : ∀ (i : ι), m.degree (b✝ i) ≠ 0
hf0 : ¬f = 0
hf : ∀ (i : ι), ¬m.degree (b✝ i) ≤ m.degree f
hf'0 : m.subLTerm f = 0
b : ι
⊢ 0 ≤ m.toSyn (m.degree (m.subLTerm f))

Tactic:
exact bot_le

Result:
no goals

Proof state:
case h.e'_1.h.e'_4
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
c : σ →₀ ℕ
hc : c ∈ (r' + (monomial (m.degree f)) (m.leadingCoeff f)).support
i : ι
hc' : c ∉ r'.support
this : c ∈ r'.support ∪ ((monomial (m.degree f)) (m.leadingCoeff f)).support
⊢ c = m.degree f

Tactic:
rw [Finset.mem_union, Classical.or_iff_not_imp_left] at this

Result:
case h.e'_1.h.e'_4
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
c : σ →₀ ℕ
hc : c ∈ (r' + (monomial (m.degree f)) (m.leadingCoeff f)).support
i : ι
hc' : c ∉ r'.support
this : c ∉ r'.support → c ∈ ((monomial (m.degree f)) (m.leadingCoeff f)).support
⊢ c = m.degree f

Proof state:
case neg
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 j * 0)) ≤ m.toSyn (m.degree f)

Tactic:
simp only [mul_zero, degree_zero, map_zero]

Result:
case neg
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
⊢ 0 ≤ m.toSyn (m.degree f)