awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.RingTheory.Polynomial.Basic
Goal:
R : Type u
inst_1 : CommRing R
inst : IsNoetherianRing R
I : Ideal R[X]
M : Submodule R R := ⋯.min (r := fun (x1 x2 : Submodule R R) ↦ x1 > x2) (Set.range I.leadingCoeffNth) ⋯
hm : M ∈ Set.range I.leadingCoeffNth
N : ℕ
HN : I.leadingCoeffNth N = M
s : Finset R[X]
hs : Submodule.span R (↑s : Set R[X]) = I.degreeLE (↑N : WithBot ℕ)
hm2 : ∀ (k : ℕ), I.leadingCoeffNth k ≤ M
hs2 : ∀ {x : R[X]}, x ∈ I.degreeLE (↑N : WithBot ℕ) → Membership.mem (γ := Ideal R[X]) (Ideal.span (↑s : Set R[X])) x
this_1 : Eq (α := Submodule R[X] R[X]) (Submodule.span R[X] (↑s : Set R[X])) (Ideal.span (↑s : Set R[X]))
k : ℕ
ih :
  ∀ m < k,
    ∀ ⦃p : R[X]⦄ (hp : p ∈ I) (hn : p.natDegree = m),
      Membership.mem (γ := Submodule R[X] R[X]) (Ideal.span (↑s : Set R[X])) p
p : R[X]
hp : p ∈ I
hn : p.natDegree = k
h : N < k
hp0 : p ≠ 0
this_2 : Ne.{u + 1} (α := R) 0 1
this : Nontrivial R
⊢ p.leadingCoeff ∈ M

Tactic:
exact hm2 k ((I.mem_leadingCoeffNth _ _).2
              ⟨_, hp, hn ▸ Polynomial.degree_le_natDegree, rfl⟩)

Premises used: Iff.mpr, Eq.rec, Polynomial.degree_le_natDegree, Ideal.mem_leadingCoeffNth, And.intro, Exists.intro, rfl
-- Module: Mathlib.RingTheory.Polynomial.Basic
Goal:
R : Type u
inst_1 : CommRing R
inst : IsNoetherianRing R
I : Ideal R[X]
M : Submodule R R := ⋯.min (r := fun (x1 x2 : Submodule R R) ↦ x1 > x2) (Set.range I.leadingCoeffNth) ⋯
hm : M ∈ Set.range I.leadingCoeffNth
N : ℕ
HN : I.leadingCoeffNth N = M
s : Finset R[X]
hs : Submodule.span R (↑s : Set R[X]) = I.degreeLE (↑N : WithBot ℕ)
hm2 : ∀ (k : ℕ), I.leadingCoeffNth k ≤ M
hs2 : ∀ {x : R[X]}, x ∈ I.degreeLE (↑N : WithBot ℕ) → Membership.mem (γ := Ideal R[X]) (Ideal.span (↑s : Set R[X])) x
this_1 : Eq (α := Submodule R[X] R[X]) (Submodule.span R[X] (↑s : Set R[X])) (Ideal.span (↑s : Set R[X]))
k : ℕ
ih :
  ∀ m < k,
    ∀ ⦃p : R[X]⦄ (hp : p ∈ I) (hn : p.natDegree = m),
      Membership.mem (γ := Submodule R[X] R[X]) (Ideal.span (↑s : Set R[X])) p
p : R[X]
hp : p ∈ I
hn : p.natDegree = k
h : N < k
hp0 : p ≠ 0
this_2 : Ne.{u + 1} (α := R) 0 1
this : Nontrivial R
⊢ p.leadingCoeff ∈ I.leadingCoeffNth N

Tactic:
have : p.leadingCoeff ∈ I.leadingCoeffNth N := by
            rw [HN]
            exact hm2 k ((I.mem_leadingCoeffNth _ _).2
              ⟨_, hp, hn ▸ Polynomial.degree_le_natDegree, rfl⟩)
-- Module: Mathlib.RingTheory.Polynomial.Basic
Goal:
R : Type u
inst_1 : CommRing R
inst : IsNoetherianRing R
I : Ideal R[X]
M : Submodule R R := ⋯.min (r := fun (x1 x2 : Submodule R R) ↦ x1 > x2) (Set.range I.leadingCoeffNth) ⋯
hm : M ∈ Set.range I.leadingCoeffNth
N : ℕ
HN : I.leadingCoeffNth N = M
s : Finset R[X]
hs : Submodule.span R (↑s : Set R[X]) = I.degreeLE (↑N : WithBot ℕ)
hm2 : ∀ (k : ℕ), I.leadingCoeffNth k ≤ M
hs2 : ∀ {x : R[X]}, x ∈ I.degreeLE (↑N : WithBot ℕ) → Membership.mem (γ := Ideal R[X]) (Ideal.span (↑s : Set R[X])) x
this_1 : Eq (α := Submodule R[X] R[X]) (Submodule.span R[X] (↑s : Set R[X])) (Ideal.span (↑s : Set R[X]))
k : ℕ
ih :
  ∀ m < k,
    ∀ ⦃p : R[X]⦄ (hp : p ∈ I) (hn : p.natDegree = m),
      Membership.mem (γ := Submodule R[X] R[X]) (Ideal.span (↑s : Set R[X])) p
p : R[X]
hp : p ∈ I
hn : p.natDegree = k
h : N < k
hp0 : p ≠ 0
this_2 : Ne.{u + 1} (α := R) 0 1
this_3 : Nontrivial R
this : p.leadingCoeff ∈ I.leadingCoeffNth N
⊢ ∃ p_1 ∈ I, p_1.degree ≤ (↑N : WithBot ℕ) ∧ p_1.leadingCoeff = p.leadingCoeff

Tactic:
rw [I.mem_leadingCoeffNth] at this

Premises used: Ideal.mem_leadingCoeffNth
-- Module: Mathlib.RingTheory.Polynomial.Basic
Goal:
R : Type u
inst_1 : CommRing R
inst : IsNoetherianRing R
I : Ideal R[X]
M : Submodule R R := ⋯.min (r := fun (x1 x2 : Submodule R R) ↦ x1 > x2) (Set.range I.leadingCoeffNth) ⋯
hm : M ∈ Set.range I.leadingCoeffNth
N : ℕ
HN : I.leadingCoeffNth N = M
s : Finset R[X]
hs : Submodule.span R (↑s : Set R[X]) = I.degreeLE (↑N : WithBot ℕ)
hm2 : ∀ (k : ℕ), I.leadingCoeffNth k ≤ M
hs2 : ∀ {x : R[X]}, x ∈ I.degreeLE (↑N : WithBot ℕ) → Membership.mem (γ := Ideal R[X]) (Ideal.span (↑s : Set R[X])) x
this_1 : Eq (α := Submodule R[X] R[X]) (Submodule.span R[X] (↑s : Set R[X])) (Ideal.span (↑s : Set R[X]))
k : ℕ
ih :
  ∀ m < k,
    ∀ ⦃p : R[X]⦄ (hp : p ∈ I) (hn : p.natDegree = m),
      Membership.mem (γ := Submodule R[X] R[X]) (Ideal.span (↑s : Set R[X])) p
p : R[X]
hp : p ∈ I
hn : p.natDegree = k
h : N < k
hp0 : p ≠ 0
this_2 : Ne.{u + 1} (α := R) 0 1
this_3 : Nontrivial R
this : ∃ p_1 ∈ I, p_1.degree ≤ (↑N : WithBot ℕ) ∧ p_1.leadingCoeff = p.leadingCoeff
h₁ :
  ∀ (q : R[X]) (hq : q ∈ I) (hdq : q.degree ≤ (↑N : WithBot ℕ)) (hlqp : q.leadingCoeff = p.leadingCoeff),
    Membership.mem (γ := Submodule R[X] R[X]) (Ideal.span (↑s : Set R[X])) p
⊢ Membership.mem (γ := Submodule R[X] R[X]) (Ideal.span (↑s : Set R[X])) p

Tactic:
rcases this with ⟨q, hq, hdq, hlqp⟩
-- Module: Mathlib.RingTheory.Polynomial.Basic
Goal:
R : Type u
inst_1 : CommRing R
inst : IsNoetherianRing R
I : Ideal R[X]
M : Submodule R R := ⋯.min (r := fun (x1 x2 : Submodule R R) ↦ x1 > x2) (Set.range I.leadingCoeffNth) ⋯
hm : M ∈ Set.range I.leadingCoeffNth
N : ℕ
HN : I.leadingCoeffNth N = M
s : Finset R[X]
hs : Submodule.span R (↑s : Set R[X]) = I.degreeLE (↑N : WithBot ℕ)
hm2 : ∀ (k : ℕ), I.leadingCoeffNth k ≤ M
hs2 : ∀ {x : R[X]}, x ∈ I.degreeLE (↑N : WithBot ℕ) → Membership.mem (γ := Ideal R[X]) (Ideal.span (↑s : Set R[X])) x
this_1 : Eq (α := Submodule R[X] R[X]) (Submodule.span R[X] (↑s : Set R[X])) (Ideal.span (↑s : Set R[X]))
k : ℕ
ih :
  ∀ m < k,
    ∀ ⦃p : R[X]⦄ (hp : p ∈ I) (hn : p.natDegree = m),
      Membership.mem (γ := Submodule R[X] R[X]) (Ideal.span (↑s : Set R[X])) p
p : R[X]
hp : p ∈ I
hn : p.natDegree = k
h : N < k
hp0 : p ≠ 0
this_2 : Ne.{u + 1} (α := R) 0 1
this : Nontrivial R
q : R[X]
hq : q ∈ I
hdq : q.degree ≤ (↑N : WithBot ℕ)
hlqp : q.leadingCoeff = p.leadingCoeff
H : q = 0
⊢ q.leadingCoeff = 0

Tactic:
rw [← Polynomial.leadingCoeff_eq_zero] at H

Premises used: Polynomial.leadingCoeff_eq_zero
-- Module: Mathlib.RingTheory.Polynomial.Basic
Goal:
R : Type u
inst_1 : CommRing R
inst : IsNoetherianRing R
I : Ideal R[X]
M : Submodule R R := ⋯.min (r := fun (x1 x2 : Submodule R R) ↦ x1 > x2) (Set.range I.leadingCoeffNth) ⋯
hm : M ∈ Set.range I.leadingCoeffNth
N : ℕ
HN : I.leadingCoeffNth N = M
s : Finset R[X]
hs : Submodule.span R (↑s : Set R[X]) = I.degreeLE (↑N : WithBot ℕ)
hm2 : ∀ (k : ℕ), I.leadingCoeffNth k ≤ M
hs2 : ∀ {x : R[X]}, x ∈ I.degreeLE (↑N : WithBot ℕ) → Membership.mem (γ := Ideal R[X]) (Ideal.span (↑s : Set R[X])) x
this_1 : Eq (α := Submodule R[X] R[X]) (Submodule.span R[X] (↑s : Set R[X])) (Ideal.span (↑s : Set R[X]))
k : ℕ
ih :
  ∀ m < k,
    ∀ ⦃p : R[X]⦄ (hp : p ∈ I) (hn : p.natDegree = m),
      Membership.mem (γ := Submodule R[X] R[X]) (Ideal.span (↑s : Set R[X])) p
p : R[X]
hp : p ∈ I
hn : p.natDegree = k
h : N < k
hp0 : p ≠ 0
this_2 : Ne.{u + 1} (α := R) 0 1
this : Nontrivial R
q : R[X]
hq : q ∈ I
hdq : q.degree ≤ (↑N : WithBot ℕ)
hlqp : q.leadingCoeff = p.leadingCoeff
H : q.leadingCoeff = 0
⊢ p = 0

Tactic:
rw [hlqp, Polynomial.leadingCoeff_eq_zero] at H

Premises used: Polynomial.leadingCoeff_eq_zero
-- Module: Mathlib.RingTheory.Polynomial.Basic
Goal:
R : Type u
inst_1 : CommRing R
inst : IsNoetherianRing R
I : Ideal R[X]
M : Submodule R R := ⋯.min (r := fun (x1 x2 : Submodule R R) ↦ x1 > x2) (Set.range I.leadingCoeffNth) ⋯
hm : M ∈ Set.range I.leadingCoeffNth
N : ℕ
HN : I.leadingCoeffNth N = M
s : Finset R[X]
hs : Submodule.span R (↑s : Set R[X]) = I.degreeLE (↑N : WithBot ℕ)
hm2 : ∀ (k : ℕ), I.leadingCoeffNth k ≤ M
hs2 : ∀ {x : R[X]}, x ∈ I.degreeLE (↑N : WithBot ℕ) → Membership.mem (γ := Ideal R[X]) (Ideal.span (↑s : Set R[X])) x
this_1 : Eq (α := Submodule R[X] R[X]) (Submodule.span R[X] (↑s : Set R[X])) (Ideal.span (↑s : Set R[X]))
k : ℕ
ih :
  ∀ m < k,
    ∀ ⦃p : R[X]⦄ (hp : p ∈ I) (hn : p.natDegree = m),
      Membership.mem (γ := Submodule R[X] R[X]) (Ideal.span (↑s : Set R[X])) p
p : R[X]
hp : p ∈ I
hn : p.natDegree = k
h : N < k
hp0 : p ≠ 0
this_2 : Ne.{u + 1} (α := R) 0 1
this : Nontrivial R
q : R[X]
hq : q ∈ I
hdq : q.degree ≤ (↑N : WithBot ℕ)
hlqp : q.leadingCoeff = p.leadingCoeff
H : p = 0
⊢ False

Tactic:
hp0 H
-- Module: Mathlib.RingTheory.Polynomial.Basic
Goal:
R : Type u
inst_1 : CommRing R
inst : IsNoetherianRing R
I : Ideal R[X]
M : Submodule R R := ⋯.min (r := fun (x1 x2 : Submodule R R) ↦ x1 > x2) (Set.range I.leadingCoeffNth) ⋯
hm : M ∈ Set.range I.leadingCoeffNth
N : ℕ
HN : I.leadingCoeffNth N = M
s : Finset R[X]
hs : Submodule.span R (↑s : Set R[X]) = I.degreeLE (↑N : WithBot ℕ)
hm2 : ∀ (k : ℕ), I.leadingCoeffNth k ≤ M
hs2 : ∀ {x : R[X]}, x ∈ I.degreeLE (↑N : WithBot ℕ) → Membership.mem (γ := Ideal R[X]) (Ideal.span (↑s : Set R[X])) x
this_1 : Eq (α := Submodule R[X] R[X]) (Submodule.span R[X] (↑s : Set R[X])) (Ideal.span (↑s : Set R[X]))
k : ℕ
ih :
  ∀ m < k,
    ∀ ⦃p : R[X]⦄ (hp : p ∈ I) (hn : p.natDegree = m),
      Membership.mem (γ := Submodule R[X] R[X]) (Ideal.span (↑s : Set R[X])) p
p : R[X]
hp : p ∈ I
hn : p.natDegree = k
h : N < k
hp0 : p ≠ 0
this_2 : Ne.{u + 1} (α := R) 0 1
this : Nontrivial R
q : R[X]
hq : q ∈ I
hdq : q.degree ≤ (↑N : WithBot ℕ)
hlqp : q.leadingCoeff = p.leadingCoeff
H : p = 0
⊢ False

Tactic:
exact hp0 H
-- Module: Mathlib.RingTheory.Polynomial.Basic
Goal:
R : Type u
inst_1 : CommRing R
inst : IsNoetherianRing R
I : Ideal R[X]
M : Submodule R R := ⋯.min (r := fun (x1 x2 : Submodule R R) ↦ x1 > x2) (Set.range I.leadingCoeffNth) ⋯
hm : M ∈ Set.range I.leadingCoeffNth
N : ℕ
HN : I.leadingCoeffNth N = M
s : Finset R[X]
hs : Submodule.span R (↑s : Set R[X]) = I.degreeLE (↑N : WithBot ℕ)
hm2 : ∀ (k : ℕ), I.leadingCoeffNth k ≤ M
hs2 : ∀ {x : R[X]}, x ∈ I.degreeLE (↑N : WithBot ℕ) → Membership.mem (γ := Ideal R[X]) (Ideal.span (↑s : Set R[X])) x
this_1 : Eq (α := Submodule R[X] R[X]) (Submodule.span R[X] (↑s : Set R[X])) (Ideal.span (↑s : Set R[X]))
k : ℕ
ih :
  ∀ m < k,
    ∀ ⦃p : R[X]⦄ (hp : p ∈ I) (hn : p.natDegree = m),
      Membership.mem (γ := Submodule R[X] R[X]) (Ideal.span (↑s : Set R[X])) p
p : R[X]
hp : p ∈ I
hn : p.natDegree = k
h : N < k
hp0 : p ≠ 0
this_2 : Ne.{u + 1} (α := R) 0 1
this : Nontrivial R
q : R[X]
hq : q ∈ I
hdq : q.degree ≤ (↑N : WithBot ℕ)
hlqp : q.leadingCoeff = p.leadingCoeff
⊢ q ≠ 0

Tactic:
have hq0 : q ≠ 0 := by
            intro H
            rw [← Polynomial.leadingCoeff_eq_zero] at H
            rw [hlqp, Polynomial.leadingCoeff_eq_zero] at H
            exact hp0 H
-- Module: Mathlib.RingTheory.Polynomial.Basic
Goal:
R : Type u
inst_1 : CommRing R
inst : IsNoetherianRing R
I : Ideal R[X]
M : Submodule R R := ⋯.min (r := fun (x1 x2 : Submodule R R) ↦ x1 > x2) (Set.range I.leadingCoeffNth) ⋯
hm : M ∈ Set.range I.leadingCoeffNth
N : ℕ
HN : I.leadingCoeffNth N = M
s : Finset R[X]
hs : Submodule.span R (↑s : Set R[X]) = I.degreeLE (↑N : WithBot ℕ)
hm2 : ∀ (k : ℕ), I.leadingCoeffNth k ≤ M
hs2 : ∀ {x : R[X]}, x ∈ I.degreeLE (↑N : WithBot ℕ) → Membership.mem (γ := Ideal R[X]) (Ideal.span (↑s : Set R[X])) x
this_1 : Eq (α := Submodule R[X] R[X]) (Submodule.span R[X] (↑s : Set R[X])) (Ideal.span (↑s : Set R[X]))
k : ℕ
ih :
  ∀ m < k,
    ∀ ⦃p : R[X]⦄ (hp : p ∈ I) (hn : p.natDegree = m),
      Membership.mem (γ := Submodule R[X] R[X]) (Ideal.span (↑s : Set R[X])) p
p : R[X]
hp : p ∈ I
hn : p.natDegree = k
h : N < k
hp0 : p ≠ 0
this_2 : Ne.{u + 1} (α := R) 0 1
this : Nontrivial R
q : R[X]
hq : q ∈ I
hdq : q.degree ≤ (↑N : WithBot ℕ)
hlqp : q.leadingCoeff = p.leadingCoeff
hq0 : q ≠ 0
rw : p.degree = q.degree + (↑(k - q.natDegree) : WithBot ℕ)
rw₁ : q.leadingCoeff * (HPow.hPow (α := R[X]) X (k - q.natDegree)).leadingCoeff ≠ 0
⊢ p.degree = (q * HPow.hPow (α := R[X]) X (k - q.natDegree)).degree

Tactic:
rw [Polynomial.degree_mul', Polynomial.degree_X_pow]

Premises used: Polynomial.degree_mul', Polynomial.degree_X_pow
-- Module: Mathlib.RingTheory.Polynomial.Basic
Goal:
R : Type u
inst_1 : CommRing R
inst : IsNoetherianRing R
I : Ideal R[X]
M : Submodule R R := ⋯.min (r := fun (x1 x2 : Submodule R R) ↦ x1 > x2) (Set.range I.leadingCoeffNth) ⋯
hm : M ∈ Set.range I.leadingCoeffNth
N : ℕ
HN : I.leadingCoeffNth N = M
s : Finset R[X]
hs : Submodule.span R (↑s : Set R[X]) = I.degreeLE (↑N : WithBot ℕ)
hm2 : ∀ (k : ℕ), I.leadingCoeffNth k ≤ M
hs2 : ∀ {x : R[X]}, x ∈ I.degreeLE (↑N : WithBot ℕ) → Membership.mem (γ := Ideal R[X]) (Ideal.span (↑s : Set R[X])) x
this_1 : Eq (α := Submodule R[X] R[X]) (Submodule.span R[X] (↑s : Set R[X])) (Ideal.span (↑s : Set R[X]))
k : ℕ
ih :
  ∀ m < k,
    ∀ ⦃p : R[X]⦄ (hp : p ∈ I) (hn : p.natDegree = m),
      Membership.mem (γ := Submodule R[X] R[X]) (Ideal.span (↑s : Set R[X])) p
p : R[X]
hp : p ∈ I
hn : p.natDegree = k
h : N < k
hp0 : p ≠ 0
this_2 : Ne.{u + 1} (α := R) 0 1
this : Nontrivial R
q : R[X]
hq : q ∈ I
hdq : q.degree ≤ (↑N : WithBot ℕ)
hlqp : q.leadingCoeff = p.leadingCoeff
hq0 : q ≠ 0
rw : (↑p.natDegree : WithBot ℕ) = HAdd.hAdd (α := WithBot ℕ) (↑q.natDegree : WithBot ℕ) (↑(k - q.natDegree) : WithBot ℕ)
⊢ p.degree = q.degree + (↑(k - q.natDegree) : WithBot ℕ)

Tactic:
rw [Polynomial.degree_eq_natDegree hp0, Polynomial.degree_eq_natDegree hq0]

Premises used: Polynomial.degree_eq_natDegree
-- Module: Mathlib.RingTheory.Polynomial.Basic
Goal:
R : Type u
inst_1 : CommRing R
inst : IsNoetherianRing R
I : Ideal R[X]
M : Submodule R R := ⋯.min (r := fun (x1 x2 : Submodule R R) ↦ x1 > x2) (Set.range I.leadingCoeffNth) ⋯
hm : M ∈ Set.range I.leadingCoeffNth
N : ℕ
HN : I.leadingCoeffNth N = M
s : Finset R[X]
hs : Submodule.span R (↑s : Set R[X]) = I.degreeLE (↑N : WithBot ℕ)
hm2 : ∀ (k : ℕ), I.leadingCoeffNth k ≤ M
hs2 : ∀ {x : R[X]}, x ∈ I.degreeLE (↑N : WithBot ℕ) → Membership.mem (γ := Ideal R[X]) (Ideal.span (↑s : Set R[X])) x
this_1 : Eq (α := Submodule R[X] R[X]) (Submodule.span R[X] (↑s : Set R[X])) (Ideal.span (↑s : Set R[X]))
k : ℕ
ih :
  ∀ m < k,
    ∀ ⦃p : R[X]⦄ (hp : p ∈ I) (hn : p.natDegree = m),
      Membership.mem (γ := Submodule R[X] R[X]) (Ideal.span (↑s : Set R[X])) p
p : R[X]
hp : p ∈ I
hn : p.natDegree = k
h : N < k
hp0 : p ≠ 0
this_2 : Ne.{u + 1} (α := R) 0 1
this : Nontrivial R
q : R[X]
hq : q ∈ I
hdq : q.degree ≤ (↑N : WithBot ℕ)
hlqp : q.leadingCoeff = p.leadingCoeff
hq0 : q ≠ 0
rw : q.natDegree ≤ k
⊢ (↑p.natDegree : WithBot ℕ) = HAdd.hAdd (α := WithBot ℕ) (↑q.natDegree : WithBot ℕ) (↑(k - q.natDegree) : WithBot ℕ)

Tactic:
rw [← Nat.cast_add, add_tsub_cancel_of_le, hn]

Premises used: Nat.cast_add, add_tsub_cancel_of_le
-- Module: Mathlib.RingTheory.Polynomial.Basic
Goal:
R : Type u
inst_1 : CommRing R
inst : IsNoetherianRing R
I : Ideal R[X]
M : Submodule R R := ⋯.min (r := fun (x1 x2 : Submodule R R) ↦ x1 > x2) (Set.range I.leadingCoeffNth) ⋯
hm : M ∈ Set.range I.leadingCoeffNth
N : ℕ
HN : I.leadingCoeffNth N = M
s : Finset R[X]
hs : Submodule.span R (↑s : Set R[X]) = I.degreeLE (↑N : WithBot ℕ)
hm2 : ∀ (k : ℕ), I.leadingCoeffNth k ≤ M
hs2 : ∀ {x : R[X]}, x ∈ I.degreeLE (↑N : WithBot ℕ) → Membership.mem (γ := Ideal R[X]) (Ideal.span (↑s : Set R[X])) x
this_1 : Eq (α := Submodule R[X] R[X]) (Submodule.span R[X] (↑s : Set R[X])) (Ideal.span (↑s : Set R[X]))
k : ℕ
ih :
  ∀ m < k,
    ∀ ⦃p : R[X]⦄ (hp : p ∈ I) (hn : p.natDegree = m),
      Membership.mem (γ := Submodule R[X] R[X]) (Ideal.span (↑s : Set R[X])) p
p : R[X]
hp : p ∈ I
hn : p.natDegree = k
h : N < k
hp0 : p ≠ 0
this_2 : Ne.{u + 1} (α := R) 0 1
this : Nontrivial R
q : R[X]
hq : q ∈ I
hdq : q.degree ≤ (↑N : WithBot ℕ)
hlqp : q.leadingCoeff = p.leadingCoeff
hq0 : q ≠ 0
⊢ q.natDegree ≤ k

Tactic:
le_trans (Polynomial.natDegree_le_of_degree_le hdq) (le_of_lt h)

Premises used: le_of_lt, Polynomial.natDegree_le_of_degree_le, le_trans
-- Module: Mathlib.RingTheory.Polynomial.Basic
Goal:
R : Type u
inst_1 : CommRing R
inst : IsNoetherianRing R
I : Ideal R[X]
M : Submodule R R := ⋯.min (r := fun (x1 x2 : Submodule R R) ↦ x1 > x2) (Set.range I.leadingCoeffNth) ⋯
hm : M ∈ Set.range I.leadingCoeffNth
N : ℕ
HN : I.leadingCoeffNth N = M
s : Finset R[X]
hs : Submodule.span R (↑s : Set R[X]) = I.degreeLE (↑N : WithBot ℕ)
hm2 : ∀ (k : ℕ), I.leadingCoeffNth k ≤ M
hs2 : ∀ {x : R[X]}, x ∈ I.degreeLE (↑N : WithBot ℕ) → Membership.mem (γ := Ideal R[X]) (Ideal.span (↑s : Set R[X])) x
this_1 : Eq (α := Submodule R[X] R[X]) (Submodule.span R[X] (↑s : Set R[X])) (Ideal.span (↑s : Set R[X]))
k : ℕ
ih :
  ∀ m < k,
    ∀ ⦃p : R[X]⦄ (hp : p ∈ I) (hn : p.natDegree = m),
      Membership.mem (γ := Submodule R[X] R[X]) (Ideal.span (↑s : Set R[X])) p
p : R[X]
hp : p ∈ I
hn : p.natDegree = k
h : N < k
hp0 : p ≠ 0
this_2 : Ne.{u + 1} (α := R) 0 1
this : Nontrivial R
q : R[X]
hq : q ∈ I
hdq : q.degree ≤ (↑N : WithBot ℕ)
hlqp : q.leadingCoeff = p.leadingCoeff
hq0 : q ≠ 0
⊢ q.natDegree ≤ k

Tactic:
refine le_trans (Polynomial.natDegree_le_of_degree_le hdq) (le_of_lt h)

Premises used: le_of_lt, Polynomial.natDegree_le_of_degree_le, le_trans
-- Module: Mathlib.RingTheory.Polynomial.Basic
Goal:
R : Type u
inst_1 : CommRing R
inst : IsNoetherianRing R
I : Ideal R[X]
M : Submodule R R := ⋯.min (r := fun (x1 x2 : Submodule R R) ↦ x1 > x2) (Set.range I.leadingCoeffNth) ⋯
hm : M ∈ Set.range I.leadingCoeffNth
N : ℕ
HN : I.leadingCoeffNth N = M
s : Finset R[X]
hs : Submodule.span R (↑s : Set R[X]) = I.degreeLE (↑N : WithBot ℕ)
hm2 : ∀ (k : ℕ), I.leadingCoeffNth k ≤ M
hs2 : ∀ {x : R[X]}, x ∈ I.degreeLE (↑N : WithBot ℕ) → Membership.mem (γ := Ideal R[X]) (Ideal.span (↑s : Set R[X])) x
this_1 : Eq (α := Submodule R[X] R[X]) (Submodule.span R[X] (↑s : Set R[X])) (Ideal.span (↑s : Set R[X]))
k : ℕ
ih :
  ∀ m < k,
    ∀ ⦃p : R[X]⦄ (hp : p ∈ I) (hn : p.natDegree = m),
      Membership.mem (γ := Submodule R[X] R[X]) (Ideal.span (↑s : Set R[X])) p
p : R[X]
hp : p ∈ I
hn : p.natDegree = k
h : N < k
hp0 : p ≠ 0
this_2 : Ne.{u + 1} (α := R) 0 1
this : Nontrivial R
q : R[X]
hq : q ∈ I
hdq : q.degree ≤ (↑N : WithBot ℕ)
hlqp : q.leadingCoeff = p.leadingCoeff
hq0 : q ≠ 0
rw : q.leadingCoeff ≠ 0
⊢ q.leadingCoeff * (HPow.hPow (α := R[X]) X (k - q.natDegree)).leadingCoeff ≠ 0

Tactic:
rw [Polynomial.leadingCoeff_X_pow, mul_one]

Premises used: Polynomial.leadingCoeff_X_pow, mul_one
-- Module: Mathlib.RingTheory.Polynomial.Basic
Goal:
R : Type u
inst_1 : CommRing R
inst : IsNoetherianRing R
I : Ideal R[X]
M : Submodule R R := ⋯.min (r := fun (x1 x2 : Submodule R R) ↦ x1 > x2) (Set.range I.leadingCoeffNth) ⋯
hm : M ∈ Set.range I.leadingCoeffNth
N : ℕ
HN : I.leadingCoeffNth N = M
s : Finset R[X]
hs : Submodule.span R (↑s : Set R[X]) = I.degreeLE (↑N : WithBot ℕ)
hm2 : ∀ (k : ℕ), I.leadingCoeffNth k ≤ M
hs2 : ∀ {x : R[X]}, x ∈ I.degreeLE (↑N : WithBot ℕ) → Membership.mem (γ := Ideal R[X]) (Ideal.span (↑s : Set R[X])) x
this_1 : Eq (α := Submodule R[X] R[X]) (Submodule.span R[X] (↑s : Set R[X])) (Ideal.span (↑s : Set R[X]))
k : ℕ
ih :
  ∀ m < k,
    ∀ ⦃p : R[X]⦄ (hp : p ∈ I) (hn : p.natDegree = m),
      Membership.mem (γ := Submodule R[X] R[X]) (Ideal.span (↑s : Set R[X])) p
p : R[X]
hp : p ∈ I
hn : p.natDegree = k
h : N < k
hp0 : p ≠ 0
this_2 : Ne.{u + 1} (α := R) 0 1
this : Nontrivial R
q : R[X]
hq : q ∈ I
hdq : q.degree ≤ (↑N : WithBot ℕ)
hlqp : q.leadingCoeff = p.leadingCoeff
hq0 : q ≠ 0
⊢ q.leadingCoeff ≠ 0

Tactic:
mt Polynomial.leadingCoeff_eq_zero.1 hq0

Premises used: mt, Iff.mp, Polynomial.leadingCoeff_eq_zero
-- Module: Mathlib.RingTheory.Polynomial.Basic
Goal:
R : Type u
inst_1 : CommRing R
inst : IsNoetherianRing R
I : Ideal R[X]
M : Submodule R R := ⋯.min (r := fun (x1 x2 : Submodule R R) ↦ x1 > x2) (Set.range I.leadingCoeffNth) ⋯
hm : M ∈ Set.range I.leadingCoeffNth
N : ℕ
HN : I.leadingCoeffNth N = M
s : Finset R[X]
hs : Submodule.span R (↑s : Set R[X]) = I.degreeLE (↑N : WithBot ℕ)
hm2 : ∀ (k : ℕ), I.leadingCoeffNth k ≤ M
hs2 : ∀ {x : R[X]}, x ∈ I.degreeLE (↑N : WithBot ℕ) → Membership.mem (γ := Ideal R[X]) (Ideal.span (↑s : Set R[X])) x
this_1 : Eq (α := Submodule R[X] R[X]) (Submodule.span R[X] (↑s : Set R[X])) (Ideal.span (↑s : Set R[X]))
k : ℕ
ih :
  ∀ m < k,
    ∀ ⦃p : R[X]⦄ (hp : p ∈ I) (hn : p.natDegree = m),
      Membership.mem (γ := Submodule R[X] R[X]) (Ideal.span (↑s : Set R[X])) p
p : R[X]
hp : p ∈ I
hn : p.natDegree = k
h : N < k
hp0 : p ≠ 0
this_2 : Ne.{u + 1} (α := R) 0 1
this : Nontrivial R
q : R[X]
hq : q ∈ I
hdq : q.degree ≤ (↑N : WithBot ℕ)
hlqp : q.leadingCoeff = p.leadingCoeff
hq0 : q ≠ 0
⊢ q.leadingCoeff ≠ 0

Tactic:
exact mt Polynomial.leadingCoeff_eq_zero.1 hq0

Premises used: mt, Iff.mp, Polynomial.leadingCoeff_eq_zero
-- Module: Mathlib.RingTheory.Polynomial.Basic
Goal:
R : Type u
inst_1 : CommRing R
inst : IsNoetherianRing R
I : Ideal R[X]
M : Submodule R R := ⋯.min (r := fun (x1 x2 : Submodule R R) ↦ x1 > x2) (Set.range I.leadingCoeffNth) ⋯
hm : M ∈ Set.range I.leadingCoeffNth
N : ℕ
HN : I.leadingCoeffNth N = M
s : Finset R[X]
hs : Submodule.span R (↑s : Set R[X]) = I.degreeLE (↑N : WithBot ℕ)
hm2 : ∀ (k : ℕ), I.leadingCoeffNth k ≤ M
hs2 : ∀ {x : R[X]}, x ∈ I.degreeLE (↑N : WithBot ℕ) → Membership.mem (γ := Ideal R[X]) (Ideal.span (↑s : Set R[X])) x
this_1 : Eq (α := Submodule R[X] R[X]) (Submodule.span R[X] (↑s : Set R[X])) (Ideal.span (↑s : Set R[X]))
k : ℕ
ih :
  ∀ m < k,
    ∀ ⦃p : R[X]⦄ (hp : p ∈ I) (hn : p.natDegree = m),
      Membership.mem (γ := Submodule R[X] R[X]) (Ideal.span (↑s : Set R[X])) p
p : R[X]
hp : p ∈ I
hn : p.natDegree = k
h : N < k
hp0 : p ≠ 0
this_2 : Ne.{u + 1} (α := R) 0 1
this : Nontrivial R
q : R[X]
hq : q ∈ I
hdq : q.degree ≤ (↑N : WithBot ℕ)
hlqp : q.leadingCoeff = p.leadingCoeff
hq0 : q ≠ 0
⊢ p.degree = (q * HPow.hPow (α := R[X]) X (k - q.natDegree)).degree

Tactic:
have h1 : p.degree = (q * Polynomial.X ^ (k - q.natDegree)).degree := by
            rw [Polynomial.degree_mul', Polynomial.degree_X_pow]
            · rw [Polynomial.degree_eq_natDegree hp0, Polynomial.degree_eq_natDegree hq0]
              rw [← Nat.cast_add, add_tsub_cancel_of_le, hn]
              · refine le_trans (Polynomial.natDegree_le_of_degree_le hdq) (le_of_lt h)
            rw [Polynomial.leadingCoeff_X_pow, mul_one]
            exact mt Polynomial.leadingCoeff_eq_zero.1 hq0
-- Module: Mathlib.RingTheory.Polynomial.Basic
Goal:
R : Type u
inst_1 : CommRing R
inst : IsNoetherianRing R
I : Ideal R[X]
M : Submodule R R := ⋯.min (r := fun (x1 x2 : Submodule R R) ↦ x1 > x2) (Set.range I.leadingCoeffNth) ⋯
hm : M ∈ Set.range I.leadingCoeffNth
N : ℕ
HN : I.leadingCoeffNth N = M
s : Finset R[X]
hs : Submodule.span R (↑s : Set R[X]) = I.degreeLE (↑N : WithBot ℕ)
hm2 : ∀ (k : ℕ), I.leadingCoeffNth k ≤ M
hs2 : ∀ {x : R[X]}, x ∈ I.degreeLE (↑N : WithBot ℕ) → Membership.mem (γ := Ideal R[X]) (Ideal.span (↑s : Set R[X])) x
this_1 : Eq (α := Submodule R[X] R[X]) (Submodule.span R[X] (↑s : Set R[X])) (Ideal.span (↑s : Set R[X]))
k : ℕ
ih :
  ∀ m < k,
    ∀ ⦃p : R[X]⦄ (hp : p ∈ I) (hn : p.natDegree = m),
      Membership.mem (γ := Submodule R[X] R[X]) (Ideal.span (↑s : Set R[X])) p
p : R[X]
hp : p ∈ I
hn : p.natDegree = k
h : N < k
hp0 : p ≠ 0
this_2 : Ne.{u + 1} (α := R) 0 1
this : Nontrivial R
q : R[X]
hq : q ∈ I
hdq : q.degree ≤ (↑N : WithBot ℕ)
hlqp : q.leadingCoeff = p.leadingCoeff
hq0 : q ≠ 0
h1 : p.degree = (q * HPow.hPow (α := R[X]) X (k - q.natDegree)).degree
rw : q.leadingCoeff = (q * HPow.hPow (α := R[X]) X (k - q.natDegree)).leadingCoeff
rw₁ rw₂ : q.leadingCoeff = q.leadingCoeff
⊢ p.leadingCoeff = (q * HPow.hPow (α := R[X]) X (k - q.natDegree)).leadingCoeff

Tactic:
rw [← hlqp, Polynomial.leadingCoeff_mul_X_pow]

Premises used: Polynomial.leadingCoeff_mul_X_pow