l3lab/ntp-mathlib huggingface dataset:
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
  smul c m :=
    Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
      intro c₁ m₁ c₂ m₂ hc hm
      apply Ideal.Quotient.eq.2
      rw [Submodule.quotientRel_r_def] at hc hm
      intro i
      exact I.mul_sub_mul_mem hc (hm i)
  one_smul := by
    rintro ⟨a⟩
    convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
    congr with i; exact one_mul (a i)
  mul_smul := by
    rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
    convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
    congr 1; funext i; exact mul_assoc a b (c i)
  smul_add := by
    rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
    convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
    congr with i; exact mul_add a (b i) (c i)
  smul_zero := by
    rintro ⟨a⟩
    convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
    congr with _; exact mul_zero a
  add_smul := by
    rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
    convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
    congr with i; 

Proof state:
case mk.mk.mk.convert_3.h.e_6.h.h
R : Type u
inst✝ : CommRing R
I : Ideal R
a✝ b✝ : R
S ι : Type v
r✝ : R ⧸ I
a : R
s✝ : R ⧸ I
b : R
x✝ : (ι → R) ⧸ pi I ι
c : ι → R
i : ι
⊢ ((fun x x_1 => x + x_1) a b • c) i = (fun x x_1 => x + x_1) (a • c) (b • c) i

Next tactic:
exact add_mul a b (c i)
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
  smul c m :=
    Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
      intro c₁ m₁ c₂ m₂ hc hm
      apply Ideal.Quotient.eq.2
      rw [Submodule.quotientRel_r_def] at hc hm
      intro i
      exact I.mul_sub_mul_mem hc (hm i)
  one_smul := by
    rintro ⟨a⟩
    convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
    congr with i; exact one_mul (a i)
  mul_smul := by
    rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
    convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
    congr 1; funext i; exact mul_assoc a b (c i)
  smul_add := by
    rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
    convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
    congr with i; exact mul_add a (b i) (c i)
  smul_zero := by
    rintro ⟨a⟩
    convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
    congr with _; exact mul_zero a
  add_smul := by
    rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
    convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
    congr with i; exact add_mul a b (c i)
  zero_smul := by
    

Proof state:
R : Type u
inst✝ : CommRing R
I : Ideal R
a b : R
S ι : Type v
⊢ ∀ (x : (ι → R) ⧸ pi I ι), 0 • x = 0

Next tactic:
rintro ⟨a⟩
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
  smul c m :=
    Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
      intro c₁ m₁ c₂ m₂ hc hm
      apply Ideal.Quotient.eq.2
      rw [Submodule.quotientRel_r_def] at hc hm
      intro i
      exact I.mul_sub_mul_mem hc (hm i)
  one_smul := by
    rintro ⟨a⟩
    convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
    congr with i; exact one_mul (a i)
  mul_smul := by
    rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
    convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
    congr 1; funext i; exact mul_assoc a b (c i)
  smul_add := by
    rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
    convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
    congr with i; exact mul_add a (b i) (c i)
  smul_zero := by
    rintro ⟨a⟩
    convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
    congr with _; exact mul_zero a
  add_smul := by
    rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
    convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
    congr with i; exact add_mul a b (c i)
  zero_smul := by
    rintro ⟨a⟩
    

Proof state:
case mk
R : Type u
inst✝ : CommRing R
I : Ideal R
a✝ b : R
S ι : Type v
x✝ : (ι → R) ⧸ pi I ι
a : ι → R
⊢ 0 • Quot.mk Setoid.r a = 0

Next tactic:
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
  smul c m :=
    Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
      intro c₁ m₁ c₂ m₂ hc hm
      apply Ideal.Quotient.eq.2
      rw [Submodule.quotientRel_r_def] at hc hm
      intro i
      exact I.mul_sub_mul_mem hc (hm i)
  one_smul := by
    rintro ⟨a⟩
    convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
    congr with i; exact one_mul (a i)
  mul_smul := by
    rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
    convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
    congr 1; funext i; exact mul_assoc a b (c i)
  smul_add := by
    rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
    convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
    congr with i; exact mul_add a (b i) (c i)
  smul_zero := by
    rintro ⟨a⟩
    convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
    congr with _; exact mul_zero a
  add_smul := by
    rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
    convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
    congr with i; exact add_mul a b (c i)
  zero_smul := by
    rintro ⟨a⟩
    convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
    

Proof state:
case mk.convert_3
R : Type u
inst✝ : CommRing R
I : Ideal R
a✝ b : R
S ι : Type v
x✝ : (ι → R) ⧸ pi I ι
a : ι → R
⊢ (Quotient.mk (pi I ι)) (0 • a) = (Quotient.mk (pi I ι)) 0

Next tactic:
congr with i
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
  smul c m :=
    Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
      intro c₁ m₁ c₂ m₂ hc hm
      apply Ideal.Quotient.eq.2
      rw [Submodule.quotientRel_r_def] at hc hm
      intro i
      exact I.mul_sub_mul_mem hc (hm i)
  one_smul := by
    rintro ⟨a⟩
    convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
    congr with i; exact one_mul (a i)
  mul_smul := by
    rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
    convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
    congr 1; funext i; exact mul_assoc a b (c i)
  smul_add := by
    rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
    convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
    congr with i; exact mul_add a (b i) (c i)
  smul_zero := by
    rintro ⟨a⟩
    convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
    congr with _; exact mul_zero a
  add_smul := by
    rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
    convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
    congr with i; exact add_mul a b (c i)
  zero_smul := by
    rintro ⟨a⟩
    convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
    congr with i; 

Proof state:
case mk.convert_3.h.e_6.h.h
R : Type u
inst✝ : CommRing R
I : Ideal R
a✝ b : R
S ι : Type v
x✝ : (ι → R) ⧸ pi I ι
a : ι → R
i : ι
⊢ (0 • a) i = OfNat.ofNat 0 i

Next tactic:
exact zero_mul (a i)
/-- `R^n/I^n` is isomorphic to `(R/I)^n` as an `R/I`-module. -/
noncomputable def piQuotEquiv : ((ι → R) ⧸ I.pi ι) ≃ₗ[R ⧸ I] ι → (R ⧸ I) where
  toFun := fun x ↦
      Quotient.liftOn' x (fun f i => Ideal.Quotient.mk I (f i)) fun a b hab =>
        funext fun i => (Submodule.Quotient.eq' _).2 (QuotientAddGroup.leftRel_apply.mp hab i)
  map_add' := by 

Proof state:
R : Type u
inst✝ : CommRing R
I : Ideal R
a b : R
S ι : Type v
⊢ ∀ (x y : (ι → R) ⧸ pi I ι),
    (fun x =>
          Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
            (_ :
              ∀ (a b : ι → R),
                Setoid.r a b → (fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b))
        (x + y) =
      (fun x =>
            Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
              (_ :
                ∀ (a b : ι → R),
                  Setoid.r a b → (fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b))
          x +
        (fun x =>
            Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
              (_ :
                ∀ (a b : ι → R),
                  Setoid.r a b → (fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b))
          y

Next tactic:
rintro ⟨_⟩ ⟨_⟩
/-- `R^n/I^n` is isomorphic to `(R/I)^n` as an `R/I`-module. -/
noncomputable def piQuotEquiv : ((ι → R) ⧸ I.pi ι) ≃ₗ[R ⧸ I] ι → (R ⧸ I) where
  toFun := fun x ↦
      Quotient.liftOn' x (fun f i => Ideal.Quotient.mk I (f i)) fun a b hab =>
        funext fun i => (Submodule.Quotient.eq' _).2 (QuotientAddGroup.leftRel_apply.mp hab i)
  map_add' := by rintro ⟨_⟩ ⟨_⟩; 

Proof state:
case mk.mk
R : Type u
inst✝ : CommRing R
I : Ideal R
a b : R
S ι : Type v
x✝ : (ι → R) ⧸ pi I ι
a✝¹ : ι → R
y✝ : (ι → R) ⧸ pi I ι
a✝ : ι → R
⊢ (fun x =>
        Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
          (_ :
            ∀ (a b : ι → R),
              Setoid.r a b → (fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b))
      (Quot.mk Setoid.r a✝¹ + Quot.mk Setoid.r a✝) =
    (fun x =>
          Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
            (_ :
              ∀ (a b : ι → R),
                Setoid.r a b → (fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b))
        (Quot.mk Setoid.r a✝¹) +
      (fun x =>
          Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
            (_ :
              ∀ (a b : ι → R),
                Setoid.r a b → (fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b))
        (Quot.mk Setoid.r a✝)

Next tactic:
rfl
/-- If `f : R^n → R^m` is an `R`-linear map and `I ⊆ R` is an ideal, then the image of `I^n` is
    contained in `I^m`. -/
theorem map_pi {ι : Type*} [Finite ι] {ι' : Type w} (x : ι → R) (hi : ∀ i, x i ∈ I)
    (f : (ι → R) →ₗ[R] ι' → R) (i : ι') : f x i ∈ I := by
  

Proof state:
R : Type u
inst✝¹ : CommRing R
I : Ideal R
a b : R
S ι✝ : Type v
ι : Type u_1
inst✝ : Finite ι
ι' : Type w
x : ι → R
hi : ∀ (i : ι), x i ∈ I
f : (ι → R) →ₗ[R] ι' → R
i : ι'
⊢ f x i ∈ I

Next tactic:
classical
    cases nonempty_fintype ι
    rw [pi_eq_sum_univ x]
    simp only [Finset.sum_apply, smul_eq_mul, map_sum, Pi.smul_apply, map_smul]
    exact I.sum_mem fun j _ => I.mul_mem_right _ (hi j)
/-- If `f : R^n → R^m` is an `R`-linear map and `I ⊆ R` is an ideal, then the image of `I^n` is
    contained in `I^m`. -/
theorem map_pi {ι : Type*} [Finite ι] {ι' : Type w} (x : ι → R) (hi : ∀ i, x i ∈ I)
    (f : (ι → R) →ₗ[R] ι' → R) (i : ι') : f x i ∈ I := by
  classical
    

Proof state:
R : Type u
inst✝¹ : CommRing R
I : Ideal R
a b : R
S ι✝ : Type v
ι : Type u_1
inst✝ : Finite ι
ι' : Type w
x : ι → R
hi : ∀ (i : ι), x i ∈ I
f : (ι → R) →ₗ[R] ι' → R
i : ι'
⊢ f x i ∈ I

Next tactic:
cases nonempty_fintype ι
/-- If `f : R^n → R^m` is an `R`-linear map and `I ⊆ R` is an ideal, then the image of `I^n` is
    contained in `I^m`. -/
theorem map_pi {ι : Type*} [Finite ι] {ι' : Type w} (x : ι → R) (hi : ∀ i, x i ∈ I)
    (f : (ι → R) →ₗ[R] ι' → R) (i : ι') : f x i ∈ I := by
  classical
    cases nonempty_fintype ι
    

Proof state:
case intro
R : Type u
inst✝¹ : CommRing R
I : Ideal R
a b : R
S ι✝ : Type v
ι : Type u_1
inst✝ : Finite ι
ι' : Type w
x : ι → R
hi : ∀ (i : ι), x i ∈ I
f : (ι → R) →ₗ[R] ι' → R
i : ι'
val✝ : Fintype ι
⊢ f x i ∈ I

Next tactic:
rw [pi_eq_sum_univ x]
/-- If `f : R^n → R^m` is an `R`-linear map and `I ⊆ R` is an ideal, then the image of `I^n` is
    contained in `I^m`. -/
theorem map_pi {ι : Type*} [Finite ι] {ι' : Type w} (x : ι → R) (hi : ∀ i, x i ∈ I)
    (f : (ι → R) →ₗ[R] ι' → R) (i : ι') : f x i ∈ I := by
  classical
    cases nonempty_fintype ι
    rw [pi_eq_sum_univ x]
    

Proof state:
case intro
R : Type u
inst✝¹ : CommRing R
I : Ideal R
a b : R
S ι✝ : Type v
ι : Type u_1
inst✝ : Finite ι
ι' : Type w
x : ι → R
hi : ∀ (i : ι), x i ∈ I
f : (ι → R) →ₗ[R] ι' → R
i : ι'
val✝ : Fintype ι
⊢ f (∑ i : ι, x i • fun j => if i = j then 1 else 0) i ∈ I

Next tactic:
simp only [Finset.sum_apply, smul_eq_mul, map_sum, Pi.smul_apply, map_smul]
/-- If `f : R^n → R^m` is an `R`-linear map and `I ⊆ R` is an ideal, then the image of `I^n` is
    contained in `I^m`. -/
theorem map_pi {ι : Type*} [Finite ι] {ι' : Type w} (x : ι → R) (hi : ∀ i, x i ∈ I)
    (f : (ι → R) →ₗ[R] ι' → R) (i : ι') : f x i ∈ I := by
  classical
    cases nonempty_fintype ι
    rw [pi_eq_sum_univ x]
    simp only [Finset.sum_apply, smul_eq_mul, map_sum, Pi.smul_apply, map_smul]
    

Proof state:
case intro
R : Type u
inst✝¹ : CommRing R
I : Ideal R
a b : R
S ι✝ : Type v
ι : Type u_1
inst✝ : Finite ι
ι' : Type w
x : ι → R
hi : ∀ (i : ι), x i ∈ I
f : (ι → R) →ₗ[R] ι' → R
i : ι'
val✝ : Fintype ι
⊢ ∑ x_1 : ι, x x_1 * f (fun j => if x_1 = j then 1 else 0) i ∈ I

Next tactic:
exact I.sum_mem fun j _ => I.mul_mem_right _ (hi j)
theorem natDegree_cancelLeads_lt_of_natDegree_le_natDegree_of_comm
    (comm : p.leadingCoeff * q.leadingCoeff = q.leadingCoeff * p.leadingCoeff)
    (h : p.natDegree ≤ q.natDegree) (hq : 0 < q.natDegree) :
    (p.cancelLeads q).natDegree < q.natDegree := by
  

Proof state:
R : Type u_1
inst✝ : Ring R
p q : R[X]
comm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p
h : natDegree p ≤ natDegree q
hq : 0 < natDegree q
⊢ natDegree (cancelLeads p q) < natDegree q

Next tactic:
by_cases hp : p = 0
theorem natDegree_cancelLeads_lt_of_natDegree_le_natDegree_of_comm
    (comm : p.leadingCoeff * q.leadingCoeff = q.leadingCoeff * p.leadingCoeff)
    (h : p.natDegree ≤ q.natDegree) (hq : 0 < q.natDegree) :
    (p.cancelLeads q).natDegree < q.natDegree := by
  by_cases hp : p = 0
  · 

Proof state:
case pos
R : Type u_1
inst✝ : Ring R
p q : R[X]
comm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p
h : natDegree p ≤ natDegree q
hq : 0 < natDegree q
hp : p = 0
⊢ natDegree (cancelLeads p q) < natDegree q

Next tactic:
convert hq
theorem natDegree_cancelLeads_lt_of_natDegree_le_natDegree_of_comm
    (comm : p.leadingCoeff * q.leadingCoeff = q.leadingCoeff * p.leadingCoeff)
    (h : p.natDegree ≤ q.natDegree) (hq : 0 < q.natDegree) :
    (p.cancelLeads q).natDegree < q.natDegree := by
  by_cases hp : p = 0
  · convert hq
    

Proof state:
case h.e'_3
R : Type u_1
inst✝ : Ring R
p q : R[X]
comm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p
h : natDegree p ≤ natDegree q
hq : 0 < natDegree q
hp : p = 0
⊢ natDegree (cancelLeads p q) = 0

Next tactic:
simp [hp, cancelLeads]
theorem natDegree_cancelLeads_lt_of_natDegree_le_natDegree_of_comm
    (comm : p.leadingCoeff * q.leadingCoeff = q.leadingCoeff * p.leadingCoeff)
    (h : p.natDegree ≤ q.natDegree) (hq : 0 < q.natDegree) :
    (p.cancelLeads q).natDegree < q.natDegree := by
  by_cases hp : p = 0
  · convert hq
    simp [hp, cancelLeads]
  

Proof state:
case neg
R : Type u_1
inst✝ : Ring R
p q : R[X]
comm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p
h : natDegree p ≤ natDegree q
hq : 0 < natDegree q
hp : ¬p = 0
⊢ natDegree (cancelLeads p q) < natDegree q

Next tactic:
rw [cancelLeads, sub_eq_add_neg, tsub_eq_zero_iff_le.mpr h, pow_zero, mul_one]
theorem natDegree_cancelLeads_lt_of_natDegree_le_natDegree_of_comm
    (comm : p.leadingCoeff * q.leadingCoeff = q.leadingCoeff * p.leadingCoeff)
    (h : p.natDegree ≤ q.natDegree) (hq : 0 < q.natDegree) :
    (p.cancelLeads q).natDegree < q.natDegree := by
  by_cases hp : p = 0
  · convert hq
    simp [hp, cancelLeads]
  rw [cancelLeads, sub_eq_add_neg, tsub_eq_zero_iff_le.mpr h, pow_zero, mul_one]
  

Proof state:
case neg
R : Type u_1
inst✝ : Ring R
p q : R[X]
comm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p
h : natDegree p ≤ natDegree q
hq : 0 < natDegree q
hp : ¬p = 0
⊢ natDegree (C (leadingCoeff p) * q + -(C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p)) < natDegree q

Next tactic:
by_cases h0 :
    C p.leadingCoeff * q + -(C q.leadingCoeff * X ^ (q.natDegree - p.natDegree) * p) = 0
theorem natDegree_cancelLeads_lt_of_natDegree_le_natDegree_of_comm
    (comm : p.leadingCoeff * q.leadingCoeff = q.leadingCoeff * p.leadingCoeff)
    (h : p.natDegree ≤ q.natDegree) (hq : 0 < q.natDegree) :
    (p.cancelLeads q).natDegree < q.natDegree := by
  by_cases hp : p = 0
  · convert hq
    simp [hp, cancelLeads]
  rw [cancelLeads, sub_eq_add_neg, tsub_eq_zero_iff_le.mpr h, pow_zero, mul_one]
  by_cases h0 :
    C p.leadingCoeff * q + -(C q.leadingCoeff * X ^ (q.natDegree - p.natDegree) * p) = 0
  · 

Proof state:
case pos
R : Type u_1
inst✝ : Ring R
p q : R[X]
comm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p
h : natDegree p ≤ natDegree q
hq : 0 < natDegree q
hp : ¬p = 0
h0 : C (leadingCoeff p) * q + -(C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p) = 0
⊢ natDegree (C (leadingCoeff p) * q + -(C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p)) < natDegree q

Next tactic:
exact (le_of_eq (by simp only [h0, natDegree_zero])).trans_lt hq
theorem natDegree_cancelLeads_lt_of_natDegree_le_natDegree_of_comm
    (comm : p.leadingCoeff * q.leadingCoeff = q.leadingCoeff * p.leadingCoeff)
    (h : p.natDegree ≤ q.natDegree) (hq : 0 < q.natDegree) :
    (p.cancelLeads q).natDegree < q.natDegree := by
  by_cases hp : p = 0
  · convert hq
    simp [hp, cancelLeads]
  rw [cancelLeads, sub_eq_add_neg, tsub_eq_zero_iff_le.mpr h, pow_zero, mul_one]
  by_cases h0 :
    C p.leadingCoeff * q + -(C q.leadingCoeff * X ^ (q.natDegree - p.natDegree) * p) = 0
  · exact (le_of_eq (by 

Proof state:
R : Type u_1
inst✝ : Ring R
p q : R[X]
comm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p
h : natDegree p ≤ natDegree q
hq : 0 < natDegree q
hp : ¬p = 0
h0 : C (leadingCoeff p) * q + -(C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p) = 0
⊢ natDegree (C (leadingCoeff p) * q + -(C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p)) = 0

Next tactic:
simp only [h0, natDegree_zero]
theorem natDegree_cancelLeads_lt_of_natDegree_le_natDegree_of_comm
    (comm : p.leadingCoeff * q.leadingCoeff = q.leadingCoeff * p.leadingCoeff)
    (h : p.natDegree ≤ q.natDegree) (hq : 0 < q.natDegree) :
    (p.cancelLeads q).natDegree < q.natDegree := by
  by_cases hp : p = 0
  · convert hq
    simp [hp, cancelLeads]
  rw [cancelLeads, sub_eq_add_neg, tsub_eq_zero_iff_le.mpr h, pow_zero, mul_one]
  by_cases h0 :
    C p.leadingCoeff * q + -(C q.leadingCoeff * X ^ (q.natDegree - p.natDegree) * p) = 0
  · exact (le_of_eq (by simp only [h0, natDegree_zero])).trans_lt hq
  

Proof state:
case neg
R : Type u_1
inst✝ : Ring R
p q : R[X]
comm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p
h : natDegree p ≤ natDegree q
hq : 0 < natDegree q
hp : ¬p = 0
h0 : ¬C (leadingCoeff p) * q + -(C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p) = 0
⊢ natDegree (C (leadingCoeff p) * q + -(C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p)) < natDegree q

Next tactic:
apply lt_of_le_of_ne
theorem natDegree_cancelLeads_lt_of_natDegree_le_natDegree_of_comm
    (comm : p.leadingCoeff * q.leadingCoeff = q.leadingCoeff * p.leadingCoeff)
    (h : p.natDegree ≤ q.natDegree) (hq : 0 < q.natDegree) :
    (p.cancelLeads q).natDegree < q.natDegree := by
  by_cases hp : p = 0
  · convert hq
    simp [hp, cancelLeads]
  rw [cancelLeads, sub_eq_add_neg, tsub_eq_zero_iff_le.mpr h, pow_zero, mul_one]
  by_cases h0 :
    C p.leadingCoeff * q + -(C q.leadingCoeff * X ^ (q.natDegree - p.natDegree) * p) = 0
  · exact (le_of_eq (by simp only [h0, natDegree_zero])).trans_lt hq
  apply lt_of_le_of_ne
  · 

Proof state:
case neg.a
R : Type u_1
inst✝ : Ring R
p q : R[X]
comm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p
h : natDegree p ≤ natDegree q
hq : 0 < natDegree q
hp : ¬p = 0
h0 : ¬C (leadingCoeff p) * q + -(C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p) = 0
⊢ natDegree (C (leadingCoeff p) * q + -(C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p)) ≤ natDegree q

Next tactic:
compute_degree!
theorem natDegree_cancelLeads_lt_of_natDegree_le_natDegree_of_comm
    (comm : p.leadingCoeff * q.leadingCoeff = q.leadingCoeff * p.leadingCoeff)
    (h : p.natDegree ≤ q.natDegree) (hq : 0 < q.natDegree) :
    (p.cancelLeads q).natDegree < q.natDegree := by
  by_cases hp : p = 0
  · convert hq
    simp [hp, cancelLeads]
  rw [cancelLeads, sub_eq_add_neg, tsub_eq_zero_iff_le.mpr h, pow_zero, mul_one]
  by_cases h0 :
    C p.leadingCoeff * q + -(C q.leadingCoeff * X ^ (q.natDegree - p.natDegree) * p) = 0
  · exact (le_of_eq (by simp only [h0, natDegree_zero])).trans_lt hq
  apply lt_of_le_of_ne
  · compute_degree!
    

Proof state:
case neg.a.a
R : Type u_1
inst✝ : Ring R
p q : R[X]
comm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p
h : natDegree p ≤ natDegree q
hq : 0 < natDegree q
hp : ¬p = 0
h0 : ¬C (leadingCoeff p) * q + -(C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p) = 0
⊢ natDegree q - natDegree p + natDegree p ≤ natDegree q

Next tactic:
rwa [Nat.sub_add_cancel]
theorem natDegree_cancelLeads_lt_of_natDegree_le_natDegree_of_comm
    (comm : p.leadingCoeff * q.leadingCoeff = q.leadingCoeff * p.leadingCoeff)
    (h : p.natDegree ≤ q.natDegree) (hq : 0 < q.natDegree) :
    (p.cancelLeads q).natDegree < q.natDegree := by
  by_cases hp : p = 0
  · convert hq
    simp [hp, cancelLeads]
  rw [cancelLeads, sub_eq_add_neg, tsub_eq_zero_iff_le.mpr h, pow_zero, mul_one]
  by_cases h0 :
    C p.leadingCoeff * q + -(C q.leadingCoeff * X ^ (q.natDegree - p.natDegree) * p) = 0
  · exact (le_of_eq (by simp only [h0, natDegree_zero])).trans_lt hq
  apply lt_of_le_of_ne
  · compute_degree!
    rwa [Nat.sub_add_cancel]
  · 

Proof state:
case neg.a
R : Type u_1
inst✝ : Ring R
p q : R[X]
comm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p
h : natDegree p ≤ natDegree q
hq : 0 < natDegree q
hp : ¬p = 0
h0 : ¬C (leadingCoeff p) * q + -(C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p) = 0
⊢ natDegree (C (leadingCoeff p) * q + -(C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p)) ≠ natDegree q

Next tactic:
contrapose! h0
theorem natDegree_cancelLeads_lt_of_natDegree_le_natDegree_of_comm
    (comm : p.leadingCoeff * q.leadingCoeff = q.leadingCoeff * p.leadingCoeff)
    (h : p.natDegree ≤ q.natDegree) (hq : 0 < q.natDegree) :
    (p.cancelLeads q).natDegree < q.natDegree := by
  by_cases hp : p = 0
  · convert hq
    simp [hp, cancelLeads]
  rw [cancelLeads, sub_eq_add_neg, tsub_eq_zero_iff_le.mpr h, pow_zero, mul_one]
  by_cases h0 :
    C p.leadingCoeff * q + -(C q.leadingCoeff * X ^ (q.natDegree - p.natDegree) * p) = 0
  · exact (le_of_eq (by simp only [h0, natDegree_zero])).trans_lt hq
  apply lt_of_le_of_ne
  · compute_degree!
    rwa [Nat.sub_add_cancel]
  · contrapose! h0
    

Proof state:
case neg.a
R : Type u_1
inst✝ : Ring R
p q : R[X]
comm : leadingCoeff p * leadingCoeff q = leadingCoeff q * leadingCoeff p
h : natDegree p ≤ natDegree q
hq : 0 < natDegree q
hp : ¬p = 0
h0 : natDegree (C (leadingCoeff p) * q + -(C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p)) = natDegree q
⊢ C (leadingCoeff p) * q + -(C (leadingCoeff q) * X ^ (natDegree q - natDegree p) * p) = 0

Next tactic:
rw [← leadingCoeff_eq_zero, leadingCoeff, h0, mul_assoc, X_pow_mul, ← tsub_add_cancel_of_le h,
      add_comm _ p.natDegree]