l3lab/ntp-mathlib huggingface dataset:
theorem isSheaf_iff_preservesFiniteProducts
    (F : Stonean.{u}ᵒᵖ ⥤ Type (u+1)) :
    IsSheaf (coherentTopology Stonean) F ↔ Nonempty (PreservesFiniteProducts F) := by
  rw [isSheaf_coherent_iff_regular_and_extensive, and_iff_left ?_]
  · exact CategoryTheory.isSheaf_iff_preservesFiniteProducts F
  · 

Proof state:
F : Stoneanᵒᵖ ⥤ Type (u + 1)
⊢ IsSheaf (Coverage.toGrothendieck Stonean (regularCoverage Stonean)) F

Next tactic:
rw [Presieve.isSheaf_coverage]
theorem isSheaf_iff_preservesFiniteProducts
    (F : Stonean.{u}ᵒᵖ ⥤ Type (u+1)) :
    IsSheaf (coherentTopology Stonean) F ↔ Nonempty (PreservesFiniteProducts F) := by
  rw [isSheaf_coherent_iff_regular_and_extensive, and_iff_left ?_]
  · exact CategoryTheory.isSheaf_iff_preservesFiniteProducts F
  · rw [Presieve.isSheaf_coverage]
    

Proof state:
F : Stoneanᵒᵖ ⥤ Type (u + 1)
⊢ ∀ {X : Stonean}, ∀ R ∈ Coverage.covering (regularCoverage Stonean) X, IsSheafFor F R

Next tactic:
intro X R ⟨Y, hR⟩
theorem isSheaf_iff_preservesFiniteProducts
    (F : Stonean.{u}ᵒᵖ ⥤ Type (u+1)) :
    IsSheaf (coherentTopology Stonean) F ↔ Nonempty (PreservesFiniteProducts F) := by
  rw [isSheaf_coherent_iff_regular_and_extensive, and_iff_left ?_]
  · exact CategoryTheory.isSheaf_iff_preservesFiniteProducts F
  · rw [Presieve.isSheaf_coverage]
    intro X R ⟨Y, hR⟩
    

Proof state:
F : Stoneanᵒᵖ ⥤ Type (u + 1)
X : Stonean
R : Presieve X
Y : Stonean
hR : ∃ f, (R = ofArrows (fun x => Y) fun x => f) ∧ EffectiveEpi f
⊢ IsSheafFor F R

Next tactic:
have _ : R.regular := ⟨Y, hR⟩
theorem isSheaf_iff_preservesFiniteProducts
    (F : Stonean.{u}ᵒᵖ ⥤ Type (u+1)) :
    IsSheaf (coherentTopology Stonean) F ↔ Nonempty (PreservesFiniteProducts F) := by
  rw [isSheaf_coherent_iff_regular_and_extensive, and_iff_left ?_]
  · exact CategoryTheory.isSheaf_iff_preservesFiniteProducts F
  · rw [Presieve.isSheaf_coverage]
    intro X R ⟨Y, hR⟩
    have _ : R.regular := ⟨Y, hR⟩
    

Proof state:
F : Stoneanᵒᵖ ⥤ Type (u + 1)
X : Stonean
R : Presieve X
Y : Stonean
hR : ∃ f, (R = ofArrows (fun x => Y) fun x => f) ∧ EffectiveEpi f
x✝ : regular R
⊢ IsSheafFor F R

Next tactic:
exact isSheafFor_regular_of_projective R F
theorem isSheaf_iff_preservesFiniteProducts'
    {A : Type (u+2)} [Category.{u+1} A] (G : A ⥤ Type (u+1))
    [HasLimits A] [PreservesLimits G] [ReflectsIsomorphisms G] (F : Stonean.{u}ᵒᵖ ⥤ A) :
    Presheaf.IsSheaf (coherentTopology Stonean) F ↔
    Nonempty (PreservesFiniteProducts (F ⋙ G)) := by
  

Proof state:
A : Type (u + 2)
inst✝³ : Category.{u + 1, u + 2} A
G : A ⥤ Type (u + 1)
inst✝² : HasLimits A
inst✝¹ : PreservesLimits G
inst✝ : ReflectsIsomorphisms G
F : Stoneanᵒᵖ ⥤ A
⊢ Presheaf.IsSheaf (coherentTopology Stonean) F ↔ Nonempty (PreservesFiniteProducts (F ⋙ G))

Next tactic:
rw [Presheaf.isSheaf_iff_isSheaf_forget (coherentTopology Stonean) F G,
    isSheaf_iff_isSheaf_of_type, isSheaf_iff_preservesFiniteProducts]
/-- The condensed set associated to a finite-product-preserving presheaf on `Stonean`. -/
noncomputable def ofSheafStonean (F : Stonean.{u}ᵒᵖ ⥤ A) [PreservesFiniteProducts F] :
    Condensed A :=
  StoneanCompHaus.equivalence A |>.functor.obj {
    val := F
    cond := by
      

Proof state:
A : Type (u + 2)
inst✝⁴ : Category.{u + 1, u + 2} A
G : A ⥤ Type (u + 1)
inst✝³ : HasLimits A
inst✝² : PreservesLimits G
inst✝¹ : ReflectsIsomorphisms G
F : Stoneanᵒᵖ ⥤ A
inst✝ : PreservesFiniteProducts F
⊢ Presheaf.IsSheaf (coherentTopology Stonean) F

Next tactic:
rw [Stonean.isSheaf_iff_preservesFiniteProducts' G F]
/-- The condensed set associated to a finite-product-preserving presheaf on `Stonean`. -/
noncomputable def ofSheafStonean (F : Stonean.{u}ᵒᵖ ⥤ A) [PreservesFiniteProducts F] :
    Condensed A :=
  StoneanCompHaus.equivalence A |>.functor.obj {
    val := F
    cond := by
      rw [Stonean.isSheaf_iff_preservesFiniteProducts' G F]
      

Proof state:
A : Type (u + 2)
inst✝⁴ : Category.{u + 1, u + 2} A
G : A ⥤ Type (u + 1)
inst✝³ : HasLimits A
inst✝² : PreservesLimits G
inst✝¹ : ReflectsIsomorphisms G
F : Stoneanᵒᵖ ⥤ A
inst✝ : PreservesFiniteProducts F
⊢ Nonempty (PreservesFiniteProducts (F ⋙ G))

Next tactic:
exact ⟨⟨fun _ _ ↦ inferInstance⟩⟩
/--
The condensed set associated to a presheaf on `Profinite` which preserves finite products and
satisfies the equalizer condition.
-/
noncomputable def ofSheafProfinite (F : Profinite.{u}ᵒᵖ ⥤ A) [PreservesFiniteProducts F]
    (hF : EqualizerCondition (F ⋙ G)) : Condensed A :=
  ProfiniteCompHaus.equivalence A |>.functor.obj {
    val := F
    cond := by
      

Proof state:
A : Type (u + 2)
inst✝⁴ : Category.{u + 1, u + 2} A
G : A ⥤ Type (u + 1)
inst✝³ : HasLimits A
inst✝² : PreservesLimits G
inst✝¹ : ReflectsIsomorphisms G
F : Profiniteᵒᵖ ⥤ A
inst✝ : PreservesFiniteProducts F
hF : EqualizerCondition (F ⋙ G)
⊢ Presheaf.IsSheaf (coherentTopology Profinite) F

Next tactic:
rw [Profinite.isSheaf_iff_preservesFiniteProducts_and_equalizerCondition' G F]
/--
The condensed set associated to a presheaf on `Profinite` which preserves finite products and
satisfies the equalizer condition.
-/
noncomputable def ofSheafProfinite (F : Profinite.{u}ᵒᵖ ⥤ A) [PreservesFiniteProducts F]
    (hF : EqualizerCondition (F ⋙ G)) : Condensed A :=
  ProfiniteCompHaus.equivalence A |>.functor.obj {
    val := F
    cond := by
      rw [Profinite.isSheaf_iff_preservesFiniteProducts_and_equalizerCondition' G F]
      

Proof state:
A : Type (u + 2)
inst✝⁴ : Category.{u + 1, u + 2} A
G : A ⥤ Type (u + 1)
inst✝³ : HasLimits A
inst✝² : PreservesLimits G
inst✝¹ : ReflectsIsomorphisms G
F : Profiniteᵒᵖ ⥤ A
inst✝ : PreservesFiniteProducts F
hF : EqualizerCondition (F ⋙ G)
⊢ Nonempty (PreservesFiniteProducts (F ⋙ G)) ∧ EqualizerCondition (F ⋙ G)

Next tactic:
exact ⟨⟨⟨fun _ _ ↦ inferInstance⟩⟩, hF⟩
/--
The condensed set associated to a presheaf on `CompHaus` which preserves finite products and
satisfies the equalizer condition.
-/
noncomputable def ofSheafCompHaus (F : CompHaus.{u}ᵒᵖ ⥤ A) [PreservesFiniteProducts F]
    (hF : EqualizerCondition (F ⋙ G)) : Condensed A where
  val := F
  cond := by
    

Proof state:
A : Type (u + 2)
inst✝⁴ : Category.{u + 1, u + 2} A
G : A ⥤ Type (u + 1)
inst✝³ : HasLimits A
inst✝² : PreservesLimits G
inst✝¹ : ReflectsIsomorphisms G
F : CompHausᵒᵖ ⥤ A
inst✝ : PreservesFiniteProducts F
hF : EqualizerCondition (F ⋙ G)
⊢ Presheaf.IsSheaf (coherentTopology CompHaus) F

Next tactic:
rw [CompHaus.isSheaf_iff_preservesFiniteProducts_and_equalizerCondition' G F]
/--
The condensed set associated to a presheaf on `CompHaus` which preserves finite products and
satisfies the equalizer condition.
-/
noncomputable def ofSheafCompHaus (F : CompHaus.{u}ᵒᵖ ⥤ A) [PreservesFiniteProducts F]
    (hF : EqualizerCondition (F ⋙ G)) : Condensed A where
  val := F
  cond := by
    rw [CompHaus.isSheaf_iff_preservesFiniteProducts_and_equalizerCondition' G F]
    

Proof state:
A : Type (u + 2)
inst✝⁴ : Category.{u + 1, u + 2} A
G : A ⥤ Type (u + 1)
inst✝³ : HasLimits A
inst✝² : PreservesLimits G
inst✝¹ : ReflectsIsomorphisms G
F : CompHausᵒᵖ ⥤ A
inst✝ : PreservesFiniteProducts F
hF : EqualizerCondition (F ⋙ G)
⊢ Nonempty (PreservesFiniteProducts (F ⋙ G)) ∧ EqualizerCondition (F ⋙ G)

Next tactic:
exact ⟨⟨⟨fun _ _ ↦ inferInstance⟩⟩, hF⟩
theorem mkDerivationₗ_X (f : σ → A) (i : σ) : mkDerivationₗ R f (X i) = f i :=
  (mkDerivationₗ_monomial f _ _).trans <| by 

Proof state:
σ : Type u_1
R : Type u_2
A : Type u_3
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid A
inst✝¹ : Module R A
inst✝ : Module (MvPolynomial σ R) A
f : σ → A
i : σ
⊢ (1 • Finsupp.sum (fun₀ | i => 1) fun i_1 k => (monomial ((fun₀ | i => 1) - fun₀ | i_1 => 1)) ↑k • f i_1) = f i

Next tactic:
simp
@[simp]
theorem derivation_C_mul (D : Derivation R (MvPolynomial σ R) A) (a : R) (f : MvPolynomial σ R) :
    C (σ := σ) a • D f = a • D f := by
  

Proof state:
σ : Type u_1
R : Type u_2
A : Type u_3
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid A
inst✝¹ : Module R A
inst✝ : Module (MvPolynomial σ R) A
D : Derivation R (MvPolynomial σ R) A
a : R
f : MvPolynomial σ R
⊢ C a • D f = a • D f

Next tactic:
have : C (σ := σ) a • D f = D (C a * f) := by simp
@[simp]
theorem derivation_C_mul (D : Derivation R (MvPolynomial σ R) A) (a : R) (f : MvPolynomial σ R) :
    C (σ := σ) a • D f = a • D f := by
  have : C (σ := σ) a • D f = D (C a * f) := by 

Proof state:
σ : Type u_1
R : Type u_2
A : Type u_3
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid A
inst✝¹ : Module R A
inst✝ : Module (MvPolynomial σ R) A
D : Derivation R (MvPolynomial σ R) A
a : R
f : MvPolynomial σ R
⊢ C a • D f = D (C a * f)

Next tactic:
simp
@[simp]
theorem derivation_C_mul (D : Derivation R (MvPolynomial σ R) A) (a : R) (f : MvPolynomial σ R) :
    C (σ := σ) a • D f = a • D f := by
  have : C (σ := σ) a • D f = D (C a * f) := by simp
  

Proof state:
σ : Type u_1
R : Type u_2
A : Type u_3
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid A
inst✝¹ : Module R A
inst✝ : Module (MvPolynomial σ R) A
D : Derivation R (MvPolynomial σ R) A
a : R
f : MvPolynomial σ R
this : C a • D f = D (C a * f)
⊢ C a • D f = a • D f

Next tactic:
rw [this, C_mul', D.map_smul]
theorem leibniz_iff_X (D : MvPolynomial σ R →ₗ[R] A) (h₁ : D 1 = 0) :
    (∀ p q, D (p * q) = p • D q + q • D p) ↔ ∀ s i, D (monomial s 1 * X i) =
    (monomial s 1 : MvPolynomial σ R) • D (X i) + (X i : MvPolynomial σ R) • D (monomial s 1) := by
  

Proof state:
σ : Type u_1
R : Type u_2
A : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid A
inst✝² : Module R A
inst✝¹ : Module (MvPolynomial σ R) A
inst✝ : IsScalarTower R (MvPolynomial σ R) A
D : MvPolynomial σ R →ₗ[R] A
h₁ : D 1 = 0
⊢ (∀ (p q : MvPolynomial σ R), D (p * q) = p • D q + q • D p) ↔
    ∀ (s : σ →₀ ℕ) (i : σ), D ((monomial s) 1 * X i) = (monomial s) 1 • D (X i) + X i • D ((monomial s) 1)

Next tactic:
refine' ⟨fun H p i => H _ _, fun H => _⟩
theorem leibniz_iff_X (D : MvPolynomial σ R →ₗ[R] A) (h₁ : D 1 = 0) :
    (∀ p q, D (p * q) = p • D q + q • D p) ↔ ∀ s i, D (monomial s 1 * X i) =
    (monomial s 1 : MvPolynomial σ R) • D (X i) + (X i : MvPolynomial σ R) • D (monomial s 1) := by
  refine' ⟨fun H p i => H _ _, fun H => _⟩
  

Proof state:
σ : Type u_1
R : Type u_2
A : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid A
inst✝² : Module R A
inst✝¹ : Module (MvPolynomial σ R) A
inst✝ : IsScalarTower R (MvPolynomial σ R) A
D : MvPolynomial σ R →ₗ[R] A
h₁ : D 1 = 0
H : ∀ (s : σ →₀ ℕ) (i : σ), D ((monomial s) 1 * X i) = (monomial s) 1 • D (X i) + X i • D ((monomial s) 1)
⊢ ∀ (p q : MvPolynomial σ R), D (p * q) = p • D q + q • D p

Next tactic:
have hC : ∀ r, D (C r) = 0 := by intro r; rw [C_eq_smul_one, D.map_smul, h₁, smul_zero]
theorem leibniz_iff_X (D : MvPolynomial σ R →ₗ[R] A) (h₁ : D 1 = 0) :
    (∀ p q, D (p * q) = p • D q + q • D p) ↔ ∀ s i, D (monomial s 1 * X i) =
    (monomial s 1 : MvPolynomial σ R) • D (X i) + (X i : MvPolynomial σ R) • D (monomial s 1) := by
  refine' ⟨fun H p i => H _ _, fun H => _⟩
  have hC : ∀ r, D (C r) = 0 := by 

Proof state:
σ : Type u_1
R : Type u_2
A : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid A
inst✝² : Module R A
inst✝¹ : Module (MvPolynomial σ R) A
inst✝ : IsScalarTower R (MvPolynomial σ R) A
D : MvPolynomial σ R →ₗ[R] A
h₁ : D 1 = 0
H : ∀ (s : σ →₀ ℕ) (i : σ), D ((monomial s) 1 * X i) = (monomial s) 1 • D (X i) + X i • D ((monomial s) 1)
⊢ ∀ (r : R), D (C r) = 0

Next tactic:
intro r
theorem leibniz_iff_X (D : MvPolynomial σ R →ₗ[R] A) (h₁ : D 1 = 0) :
    (∀ p q, D (p * q) = p • D q + q • D p) ↔ ∀ s i, D (monomial s 1 * X i) =
    (monomial s 1 : MvPolynomial σ R) • D (X i) + (X i : MvPolynomial σ R) • D (monomial s 1) := by
  refine' ⟨fun H p i => H _ _, fun H => _⟩
  have hC : ∀ r, D (C r) = 0 := by intro r; 

Proof state:
σ : Type u_1
R : Type u_2
A : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid A
inst✝² : Module R A
inst✝¹ : Module (MvPolynomial σ R) A
inst✝ : IsScalarTower R (MvPolynomial σ R) A
D : MvPolynomial σ R →ₗ[R] A
h₁ : D 1 = 0
H : ∀ (s : σ →₀ ℕ) (i : σ), D ((monomial s) 1 * X i) = (monomial s) 1 • D (X i) + X i • D ((monomial s) 1)
r : R
⊢ D (C r) = 0

Next tactic:
rw [C_eq_smul_one, D.map_smul, h₁, smul_zero]
theorem leibniz_iff_X (D : MvPolynomial σ R →ₗ[R] A) (h₁ : D 1 = 0) :
    (∀ p q, D (p * q) = p • D q + q • D p) ↔ ∀ s i, D (monomial s 1 * X i) =
    (monomial s 1 : MvPolynomial σ R) • D (X i) + (X i : MvPolynomial σ R) • D (monomial s 1) := by
  refine' ⟨fun H p i => H _ _, fun H => _⟩
  have hC : ∀ r, D (C r) = 0 := by intro r; rw [C_eq_smul_one, D.map_smul, h₁, smul_zero]
  

Proof state:
σ : Type u_1
R : Type u_2
A : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid A
inst✝² : Module R A
inst✝¹ : Module (MvPolynomial σ R) A
inst✝ : IsScalarTower R (MvPolynomial σ R) A
D : MvPolynomial σ R →ₗ[R] A
h₁ : D 1 = 0
H : ∀ (s : σ →₀ ℕ) (i : σ), D ((monomial s) 1 * X i) = (monomial s) 1 • D (X i) + X i • D ((monomial s) 1)
hC : ∀ (r : R), D (C r) = 0
⊢ ∀ (p q : MvPolynomial σ R), D (p * q) = p • D q + q • D p

Next tactic:
have : ∀ p i, D (p * X i) = p • D (X i) + (X i : MvPolynomial σ R) • D p := by
    intro p i
    induction' p using MvPolynomial.induction_on' with s r p q hp hq
    · rw [← mul_one r, ← C_mul_monomial, mul_assoc, C_mul', D.map_smul, H, C_mul', smul_assoc,
        smul_add, D.map_smul, smul_comm r (X i)]
    · rw [add_mul, map_add, map_add, hp, hq, add_smul, smul_add, add_add_add_comm]
theorem leibniz_iff_X (D : MvPolynomial σ R →ₗ[R] A) (h₁ : D 1 = 0) :
    (∀ p q, D (p * q) = p • D q + q • D p) ↔ ∀ s i, D (monomial s 1 * X i) =
    (monomial s 1 : MvPolynomial σ R) • D (X i) + (X i : MvPolynomial σ R) • D (monomial s 1) := by
  refine' ⟨fun H p i => H _ _, fun H => _⟩
  have hC : ∀ r, D (C r) = 0 := by intro r; rw [C_eq_smul_one, D.map_smul, h₁, smul_zero]
  have : ∀ p i, D (p * X i) = p • D (X i) + (X i : MvPolynomial σ R) • D p := by
    

Proof state:
σ : Type u_1
R : Type u_2
A : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid A
inst✝² : Module R A
inst✝¹ : Module (MvPolynomial σ R) A
inst✝ : IsScalarTower R (MvPolynomial σ R) A
D : MvPolynomial σ R →ₗ[R] A
h₁ : D 1 = 0
H : ∀ (s : σ →₀ ℕ) (i : σ), D ((monomial s) 1 * X i) = (monomial s) 1 • D (X i) + X i • D ((monomial s) 1)
hC : ∀ (r : R), D (C r) = 0
⊢ ∀ (p : MvPolynomial σ R) (i : σ), D (p * X i) = p • D (X i) + X i • D p

Next tactic:
intro p i
theorem leibniz_iff_X (D : MvPolynomial σ R →ₗ[R] A) (h₁ : D 1 = 0) :
    (∀ p q, D (p * q) = p • D q + q • D p) ↔ ∀ s i, D (monomial s 1 * X i) =
    (monomial s 1 : MvPolynomial σ R) • D (X i) + (X i : MvPolynomial σ R) • D (monomial s 1) := by
  refine' ⟨fun H p i => H _ _, fun H => _⟩
  have hC : ∀ r, D (C r) = 0 := by intro r; rw [C_eq_smul_one, D.map_smul, h₁, smul_zero]
  have : ∀ p i, D (p * X i) = p • D (X i) + (X i : MvPolynomial σ R) • D p := by
    intro p i
    

Proof state:
σ : Type u_1
R : Type u_2
A : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid A
inst✝² : Module R A
inst✝¹ : Module (MvPolynomial σ R) A
inst✝ : IsScalarTower R (MvPolynomial σ R) A
D : MvPolynomial σ R →ₗ[R] A
h₁ : D 1 = 0
H : ∀ (s : σ →₀ ℕ) (i : σ), D ((monomial s) 1 * X i) = (monomial s) 1 • D (X i) + X i • D ((monomial s) 1)
hC : ∀ (r : R), D (C r) = 0
p : MvPolynomial σ R
i : σ
⊢ D (p * X i) = p • D (X i) + X i • D p

Next tactic:
induction' p using MvPolynomial.induction_on' with s r p q hp hq
theorem leibniz_iff_X (D : MvPolynomial σ R →ₗ[R] A) (h₁ : D 1 = 0) :
    (∀ p q, D (p * q) = p • D q + q • D p) ↔ ∀ s i, D (monomial s 1 * X i) =
    (monomial s 1 : MvPolynomial σ R) • D (X i) + (X i : MvPolynomial σ R) • D (monomial s 1) := by
  refine' ⟨fun H p i => H _ _, fun H => _⟩
  have hC : ∀ r, D (C r) = 0 := by intro r; rw [C_eq_smul_one, D.map_smul, h₁, smul_zero]
  have : ∀ p i, D (p * X i) = p • D (X i) + (X i : MvPolynomial σ R) • D p := by
    intro p i
    induction' p using MvPolynomial.induction_on' with s r p q hp hq
    · 

Proof state:
case h1
σ : Type u_1
R : Type u_2
A : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid A
inst✝² : Module R A
inst✝¹ : Module (MvPolynomial σ R) A
inst✝ : IsScalarTower R (MvPolynomial σ R) A
D : MvPolynomial σ R →ₗ[R] A
h₁ : D 1 = 0
H : ∀ (s : σ →₀ ℕ) (i : σ), D ((monomial s) 1 * X i) = (monomial s) 1 • D (X i) + X i • D ((monomial s) 1)
hC : ∀ (r : R), D (C r) = 0
i : σ
s : σ →₀ ℕ
r : R
⊢ D ((monomial s) r * X i) = (monomial s) r • D (X i) + X i • D ((monomial s) r)

Next tactic:
rw [← mul_one r, ← C_mul_monomial, mul_assoc, C_mul', D.map_smul, H, C_mul', smul_assoc,
        smul_add, D.map_smul, smul_comm r (X i)]
theorem leibniz_iff_X (D : MvPolynomial σ R →ₗ[R] A) (h₁ : D 1 = 0) :
    (∀ p q, D (p * q) = p • D q + q • D p) ↔ ∀ s i, D (monomial s 1 * X i) =
    (monomial s 1 : MvPolynomial σ R) • D (X i) + (X i : MvPolynomial σ R) • D (monomial s 1) := by
  refine' ⟨fun H p i => H _ _, fun H => _⟩
  have hC : ∀ r, D (C r) = 0 := by intro r; rw [C_eq_smul_one, D.map_smul, h₁, smul_zero]
  have : ∀ p i, D (p * X i) = p • D (X i) + (X i : MvPolynomial σ R) • D p := by
    intro p i
    induction' p using MvPolynomial.induction_on' with s r p q hp hq
    · rw [← mul_one r, ← C_mul_monomial, mul_assoc, C_mul', D.map_smul, H, C_mul', smul_assoc,
        smul_add, D.map_smul, smul_comm r (X i)]
    · 

Proof state:
case h2
σ : Type u_1
R : Type u_2
A : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid A
inst✝² : Module R A
inst✝¹ : Module (MvPolynomial σ R) A
inst✝ : IsScalarTower R (MvPolynomial σ R) A
D : MvPolynomial σ R →ₗ[R] A
h₁ : D 1 = 0
H : ∀ (s : σ →₀ ℕ) (i : σ), D ((monomial s) 1 * X i) = (monomial s) 1 • D (X i) + X i • D ((monomial s) 1)
hC : ∀ (r : R), D (C r) = 0
i : σ
p q : MvPolynomial σ R
hp : D (p * X i) = p • D (X i) + X i • D p
hq : D (q * X i) = q • D (X i) + X i • D q
⊢ D ((p + q) * X i) = (p + q) • D (X i) + X i • D (p + q)

Next tactic:
rw [add_mul, map_add, map_add, hp, hq, add_smul, smul_add, add_add_add_comm]
theorem leibniz_iff_X (D : MvPolynomial σ R →ₗ[R] A) (h₁ : D 1 = 0) :
    (∀ p q, D (p * q) = p • D q + q • D p) ↔ ∀ s i, D (monomial s 1 * X i) =
    (monomial s 1 : MvPolynomial σ R) • D (X i) + (X i : MvPolynomial σ R) • D (monomial s 1) := by
  refine' ⟨fun H p i => H _ _, fun H => _⟩
  have hC : ∀ r, D (C r) = 0 := by intro r; rw [C_eq_smul_one, D.map_smul, h₁, smul_zero]
  have : ∀ p i, D (p * X i) = p • D (X i) + (X i : MvPolynomial σ R) • D p := by
    intro p i
    induction' p using MvPolynomial.induction_on' with s r p q hp hq
    · rw [← mul_one r, ← C_mul_monomial, mul_assoc, C_mul', D.map_smul, H, C_mul', smul_assoc,
        smul_add, D.map_smul, smul_comm r (X i)]
    · rw [add_mul, map_add, map_add, hp, hq, add_smul, smul_add, add_add_add_comm]
  

Proof state:
σ : Type u_1
R : Type u_2
A : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid A
inst✝² : Module R A
inst✝¹ : Module (MvPolynomial σ R) A
inst✝ : IsScalarTower R (MvPolynomial σ R) A
D : MvPolynomial σ R →ₗ[R] A
h₁ : D 1 = 0
H : ∀ (s : σ →₀ ℕ) (i : σ), D ((monomial s) 1 * X i) = (monomial s) 1 • D (X i) + X i • D ((monomial s) 1)
hC : ∀ (r : R), D (C r) = 0
this : ∀ (p : MvPolynomial σ R) (i : σ), D (p * X i) = p • D (X i) + X i • D p
⊢ ∀ (p q : MvPolynomial σ R), D (p * q) = p • D q + q • D p

Next tactic:
intro p q
theorem leibniz_iff_X (D : MvPolynomial σ R →ₗ[R] A) (h₁ : D 1 = 0) :
    (∀ p q, D (p * q) = p • D q + q • D p) ↔ ∀ s i, D (monomial s 1 * X i) =
    (monomial s 1 : MvPolynomial σ R) • D (X i) + (X i : MvPolynomial σ R) • D (monomial s 1) := by
  refine' ⟨fun H p i => H _ _, fun H => _⟩
  have hC : ∀ r, D (C r) = 0 := by intro r; rw [C_eq_smul_one, D.map_smul, h₁, smul_zero]
  have : ∀ p i, D (p * X i) = p • D (X i) + (X i : MvPolynomial σ R) • D p := by
    intro p i
    induction' p using MvPolynomial.induction_on' with s r p q hp hq
    · rw [← mul_one r, ← C_mul_monomial, mul_assoc, C_mul', D.map_smul, H, C_mul', smul_assoc,
        smul_add, D.map_smul, smul_comm r (X i)]
    · rw [add_mul, map_add, map_add, hp, hq, add_smul, smul_add, add_add_add_comm]
  intro p q
  

Proof state:
σ : Type u_1
R : Type u_2
A : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid A
inst✝² : Module R A
inst✝¹ : Module (MvPolynomial σ R) A
inst✝ : IsScalarTower R (MvPolynomial σ R) A
D : MvPolynomial σ R →ₗ[R] A
h₁ : D 1 = 0
H : ∀ (s : σ →₀ ℕ) (i : σ), D ((monomial s) 1 * X i) = (monomial s) 1 • D (X i) + X i • D ((monomial s) 1)
hC : ∀ (r : R), D (C r) = 0
this : ∀ (p : MvPolynomial σ R) (i : σ), D (p * X i) = p • D (X i) + X i • D p
p q : MvPolynomial σ R
⊢ D (p * q) = p • D q + q • D p

Next tactic:
induction q using MvPolynomial.induction_on with
  | h_C c =>
    rw [mul_comm, C_mul', hC, smul_zero, zero_add, D.map_smul, C_eq_smul_one, smul_one_smul]
  | h_add q₁ q₂ h₁ h₂ => simp only [mul_add, map_add, h₁, h₂, smul_add, add_smul]; abel
  | h_X q i hq =>
    simp only [this, ← mul_assoc, hq, mul_smul, smul_add, add_assoc]
    rw [smul_comm (X i), smul_comm (X i)]
theorem leibniz_iff_X (D : MvPolynomial σ R →ₗ[R] A) (h₁ : D 1 = 0) :
    (∀ p q, D (p * q) = p • D q + q • D p) ↔ ∀ s i, D (monomial s 1 * X i) =
    (monomial s 1 : MvPolynomial σ R) • D (X i) + (X i : MvPolynomial σ R) • D (monomial s 1) := by
  refine' ⟨fun H p i => H _ _, fun H => _⟩
  have hC : ∀ r, D (C r) = 0 := by intro r; rw [C_eq_smul_one, D.map_smul, h₁, smul_zero]
  have : ∀ p i, D (p * X i) = p • D (X i) + (X i : MvPolynomial σ R) • D p := by
    intro p i
    induction' p using MvPolynomial.induction_on' with s r p q hp hq
    · rw [← mul_one r, ← C_mul_monomial, mul_assoc, C_mul', D.map_smul, H, C_mul', smul_assoc,
        smul_add, D.map_smul, smul_comm r (X i)]
    · rw [add_mul, map_add, map_add, hp, hq, add_smul, smul_add, add_add_add_comm]
  intro p q
  induction q using MvPolynomial.induction_on with
  

Proof state:
case h_C
σ : Type u_1
R : Type u_2
A : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid A
inst✝² : Module R A
inst✝¹ : Module (MvPolynomial σ R) A
inst✝ : IsScalarTower R (MvPolynomial σ R) A
D : MvPolynomial σ R →ₗ[R] A
h₁ : D 1 = 0
H : ∀ (s : σ →₀ ℕ) (i : σ), D ((monomial s) 1 * X i) = (monomial s) 1 • D (X i) + X i • D ((monomial s) 1)
hC : ∀ (r : R), D (C r) = 0
this : ∀ (p : MvPolynomial σ R) (i : σ), D (p * X i) = p • D (X i) + X i • D p
p : MvPolynomial σ R
c : R
⊢ D (p * C c) = p • D (C c) + C c • D p

Next tactic:
| h_C c =>
    rw [mul_comm, C_mul', hC, smul_zero, zero_add, D.map_smul, C_eq_smul_one, smul_one_smul]
theorem leibniz_iff_X (D : MvPolynomial σ R →ₗ[R] A) (h₁ : D 1 = 0) :
    (∀ p q, D (p * q) = p • D q + q • D p) ↔ ∀ s i, D (monomial s 1 * X i) =
    (monomial s 1 : MvPolynomial σ R) • D (X i) + (X i : MvPolynomial σ R) • D (monomial s 1) := by
  refine' ⟨fun H p i => H _ _, fun H => _⟩
  have hC : ∀ r, D (C r) = 0 := by intro r; rw [C_eq_smul_one, D.map_smul, h₁, smul_zero]
  have : ∀ p i, D (p * X i) = p • D (X i) + (X i : MvPolynomial σ R) • D p := by
    intro p i
    induction' p using MvPolynomial.induction_on' with s r p q hp hq
    · rw [← mul_one r, ← C_mul_monomial, mul_assoc, C_mul', D.map_smul, H, C_mul', smul_assoc,
        smul_add, D.map_smul, smul_comm r (X i)]
    · rw [add_mul, map_add, map_add, hp, hq, add_smul, smul_add, add_add_add_comm]
  intro p q
  induction q using MvPolynomial.induction_on with
  | h_C c =>
    

Proof state:
case h_C
σ : Type u_1
R : Type u_2
A : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid A
inst✝² : Module R A
inst✝¹ : Module (MvPolynomial σ R) A
inst✝ : IsScalarTower R (MvPolynomial σ R) A
D : MvPolynomial σ R →ₗ[R] A
h₁ : D 1 = 0
H : ∀ (s : σ →₀ ℕ) (i : σ), D ((monomial s) 1 * X i) = (monomial s) 1 • D (X i) + X i • D ((monomial s) 1)
hC : ∀ (r : R), D (C r) = 0
this : ∀ (p : MvPolynomial σ R) (i : σ), D (p * X i) = p • D (X i) + X i • D p
p : MvPolynomial σ R
c : R
⊢ D (p * C c) = p • D (C c) + C c • D p

Next tactic:
rw [mul_comm, C_mul', hC, smul_zero, zero_add, D.map_smul, C_eq_smul_one, smul_one_smul]