mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula

Goal state before:
pp: F : Type u
inst✝¹ : Field F
W : Projective F
inst✝ : DecidableEq F
P Q : Fin 3 → F
hP : W.Equation P
hQ : W.Equation Q
hPz : P z ≠ 0
hQz : Q z ≠ 0
hx : P x * Q z = Q x * P z
hy : P y * Q z ≠ W.negY Q * P z
⊢ (-(eval P) W.polynomialX *
            ((eval P) W.polynomialX ^ 2 - W.a₁ * (eval P) W.polynomialX * P z * (P y - W.negY P) -
                  W.a₂ * P z ^ 2 * (P y - W.negY P) ^ 2 -
                2 * P x * P z * (P y - W.negY P) ^ 2 -
              P x * P z * (P y - W.negY P) ^ 2) +
          P y * P z ^ 2 * (P y - W.negY P) ^ 3) /
        P z ^ 2 /
      (P z * (P y - W.negY P) ^ 3) =
    (-(eval P) W.polynomialX *
            ((eval P) W.polynomialX ^ 2 - W.a₁ * (eval P) W.polynomialX * P z * (P y - W.negY P) -
                  W.a₂ * P z ^ 2 * (P y - W.negY P) ^ 2 -
                2 * P x * P z * (P y - W.negY P) ^ 2 -
              P x * P z * (P y - W.negY P) ^ 2) +
          P y * P z ^ 2 * (P y - W.negY P) ^ 3) /
        P z ^ 2 /
      (P z * (P y - W.negY P) ^ 3)

Tactic:
]

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula

Goal state before:
pp: R : Type r
inst✝¹ : CommRing R
W' : Projective R
inst✝ : NoZeroDivisors R
P : Fin 3 → R
hP : W'.Equation P
hPz : P z = 0
⊢ - -P y ^ 4 - W'.a₁ * 0 - W'.a₃ * 0 = P y ^ 4

Tactic:
]

Goal state after:
R : Type r
inst✝¹ : CommRing R
W' : Projective R
inst✝ : NoZeroDivisors R
P : Fin 3 → R
hP : W'.Equation P
hPz : P z = 0
⊢ - -P y ^ 4 - W'.a₁ * 0 - W'.a₃ * 0 = P y ^ 4
Module: Mathlib.RingTheory.Smooth.StandardSmoothCotangent

Goal state before:
pp: R : Type u_1
S : Type u_2
ι : Type u_3
σ : Type u_4
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : Finite σ
P : SubmersivePresentation R S ι σ
⊢ IsCompl {x | x ∈ (Set.range P.map)ᶜ} (Set.range P.map)

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_2
ι : Type u_3
σ : Type u_4
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : Finite σ
P : SubmersivePresentation R S ι σ
⊢ IsCompl {x | x ∈ (Set.range P.map)ᶜ} (Set.range P.map)
Module: Mathlib.RingTheory.Extension.Cotangent.LocalizationAway

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
ι : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
g : S
inst✝ : IsLocalization.Away g T
P : Generators R S ι
f : P.Ring
h : (algebraMap P.Ring S) f = g
⊢ Ideal.span {C g * X () - 1} = Ideal.span {C g * X () - 1}

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Smooth.StandardSmoothCotangent

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : Nontrivial S
n : ℕ
inst✝ : IsStandardSmoothOfRelativeDimension n R S
w✝³ w✝² : Type
w✝¹ : Finite w✝²
w✝ : Finite w✝³
P : SubmersivePresentation R S w✝³ w✝²
hP : P.dimension = n
⊢ ↑n = ↑n

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Smooth.StandardSmoothCotangent

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : Nontrivial S
inst✝ : IsStandardSmooth R S
n : ℕ
h : Module.rank S Ω[S⁄R] = ↑n
w✝² w✝¹ : Type
w✝ : Finite w✝¹
left✝ : Finite w✝²
P : SubmersivePresentation R S w✝² w✝¹
⊢ Module.rank S Ω[S⁄R] = ↑n

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : Nontrivial S
inst✝ : IsStandardSmooth R S
n : ℕ
h : Module.rank S Ω[S⁄R] = ↑n
w✝² w✝¹ : Type
w✝ : Finite w✝¹
left✝ : Finite w✝²
P : SubmersivePresentation R S w✝² w✝¹
⊢ Module.rank S Ω[S⁄R] = ↑n
Module: Mathlib.RingTheory.Smooth.StandardSmoothCotangent

Goal state before:
pp: R : Type u_1
S : Type u_2
ι : Type u_3
σ : Type u_4
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : IsStandardSmooth R S
⊢ Module.Projective S Ω[S⁄R] ∧ Subsingleton (H1Cotangent R S)

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_2
ι : Type u_3
σ : Type u_4
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : IsStandardSmooth R S
⊢ Module.Projective S Ω[S⁄R] ∧ Subsingleton (H1Cotangent R S)
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula

Goal state before:
pp: case a
R : Type r
inst✝¹ : CommRing R
W' : Projective R
inst✝ : NoZeroDivisors R
P Q : Fin 3 → R
hP : W'.Equation P
hPz : P z ≠ 0
hQz : Q z ≠ 0
hx : P x * Q z = Q x * P z
hy : P y * Q z = Q y * P z
hy' : P y * Q z = W'.negY Q * P z
⊢ (-W'.negDblY P - W'.a₁ * 0 - W'.a₃ * 0) * P z ^ 2 + W'.negDblY P * P z ^ 2 -
      ((eval P) W'.polynomialX ^ 3 + -(eval P) W'.polynomialX ^ 3) =
    0

Tactic:
]

Goal state after:
case a
R : Type r
inst✝¹ : CommRing R
W' : Projective R
inst✝ : NoZeroDivisors R
P Q : Fin 3 → R
hP : W'.Equation P
hPz : P z ≠ 0
hQz : Q z ≠ 0
hx : P x * Q z = Q x * P z
hy : P y * Q z = Q y * P z
hy' : P y * Q z = W'.negY Q * P z
⊢ (-W'.negDblY P - W'.a₁ * 0 - W'.a₃ * 0) * P z ^ 2 + W'.negDblY P * P z ^ 2 -
      ((eval P) W'.polynomialX ^ 3 + -(eval P) W'.polynomialX ^ 3) =
    0
Module: Mathlib.RingTheory.Extension.Cotangent.Free

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
ι : Type u_3
σ : Type u_4
κ : Type u_5
P : Generators R S ι
u : σ → ι
hu : Function.Injective u
v : κ → ι
huv : IsCompl (Set.range v) (Set.range u)
hm : Submodule.span S (Set.range fun i ↦ (D R S) (P.val (v i))) = ⊤
hk : Disjoint P.toExtension.toKaehler.ker (Submodule.span S (Set.range fun x ↦ P.cotangentSpaceBasis (v x)))
inst✝ : Subsingleton (H1Cotangent R S)
⊢ Function.Bijective
    ⇑(LinearMap.linearProjOfIsCompl (Submodule.span S (Set.range fun x ↦ Finsupp.single (v x) 1))
          (Finsupp.lmapDomain S S u) ⋯ ⋯ ∘ₗ
        ↑P.cotangentSpaceBasis.repr ∘ₗ P.toExtension.cotangentComplex)

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
ι : Type u_3
σ : Type u_4
κ : Type u_5
P : Generators R S ι
u : σ → ι
hu : Function.Injective u
v : κ → ι
huv : IsCompl (Set.range v) (Set.range u)
hm : Submodule.span S (Set.range fun i ↦ (D R S) (P.val (v i))) = ⊤
hk : Disjoint P.toExtension.toKaehler.ker (Submodule.span S (Set.range fun x ↦ P.cotangentSpaceBasis (v x)))
inst✝ : Subsingleton (H1Cotangent R S)
⊢ Function.Bijective
    ⇑(LinearMap.linearProjOfIsCompl (Submodule.span S (Set.range fun x ↦ Finsupp.single (v x) 1))
          (Finsupp.lmapDomain S S u) ⋯ ⋯ ∘ₗ
        ↑P.cotangentSpaceBasis.repr ∘ₗ P.toExtension.cotangentComplex)
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula

Goal state before:
pp: F : Type u
inst✝ : Field F
W : Projective F
P Q : Fin 3 → F
hP : W.Equation P
hPz : P z ≠ 0
hQz : Q z ≠ 0
hx : P x * Q z = Q x * P z
hy : P y * Q z = Q y * P z
hy' : P y * Q z = W.negY Q * P z
⊢ W.dblY P = W.dblY P

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Extension.Cotangent.LocalizationAway

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
ι : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
g : S
inst✝ : IsLocalization.Away g T
P : Generators R S ι
f : P.Ring
h : (algebraMap P.Ring S) f = g
this :
  (localizationAway T g).ker =
    Ideal.map ((localizationAway T g).ofComp P).toAlgHom (Ideal.span {(rename Sum.inr) f * X (Sum.inl ()) - 1})
⊢ RingHom.ker ((localizationAway T g).ofComp P).toAlgHom ⊔ Ideal.span {(rename Sum.inr) f * X (Sum.inl ()) - 1} ⊔
      RingHom.ker ((localizationAway T g).ofComp P).toAlgHom =
    RingHom.ker ((localizationAway T g).ofComp P).toAlgHom ⊔ Ideal.span {(rename Sum.inr) f * X (Sum.inl ()) - 1}

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_2
T : Type u_3
ι : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
g : S
inst✝ : IsLocalization.Away g T
P : Generators R S ι
f : P.Ring
h : (algebraMap P.Ring S) f = g
this :
  (localizationAway T g).ker =
    Ideal.map ((localizationAway T g).ofComp P).toAlgHom (Ideal.span {(rename Sum.inr) f * X (Sum.inl ()) - 1})
⊢ RingHom.ker ((localizationAway T g).ofComp P).toAlgHom ⊔ Ideal.span {(rename Sum.inr) f * X (Sum.inl ()) - 1} ⊔
      RingHom.ker ((localizationAway T g).ofComp P).toAlgHom =
    RingHom.ker ((localizationAway T g).ofComp P).toAlgHom ⊔ Ideal.span {(rename Sum.inr) f * X (Sum.inl ()) - 1}
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula

Goal state before:
pp: F : Type u
inst✝¹ : Field F
W : Projective F
inst✝ : DecidableEq F
P Q : Fin 3 → F
hP : W.Equation P
hQ : W.Equation Q
hPz : P z ≠ 0
hQz : Q z ≠ 0
hx : P x * Q z = Q x * P z
hy : P y * Q z ≠ W.negY Q * P z
⊢ W.toAffine.negY
      (W.toAffine.addX (P x / P z) (Q x / Q z) (W.toAffine.slope (P x / P z) (Q x / Q z) (P y / P z) (Q y / Q z)))
      (W.toAffine.negAddY (P x / P z) (Q x / Q z) (P y / P z)
        (W.toAffine.slope (P x / P z) (Q x / Q z) (P y / P z) (Q y / Q z))) =
    W.toAffine.negY
      (W.toAffine.addX (P x / P z) (Q x / Q z) (W.toAffine.slope (P x / P z) (Q x / Q z) (P y / P z) (Q y / Q z)))
      (W.toAffine.negAddY (P x / P z) (Q x / Q z) (P y / P z)
        (W.toAffine.slope (P x / P z) (Q x / Q z) (P y / P z) (Q y / Q z)))

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Extension.Cotangent.LocalizationAway

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
ι : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
g : S
inst✝ : IsLocalization.Away g T
P : Generators R S ι
⊢ (compLocalizationAwayAlgHom T g P) (((localizationAway T g).toComp P).toAlgHom (P.σ g)) *
        (compLocalizationAwayAlgHom T g P) (X (Sum.inl ())) -
      1 =
    0

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_2
T : Type u_3
ι : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
g : S
inst✝ : IsLocalization.Away g T
P : Generators R S ι
⊢ (compLocalizationAwayAlgHom T g P) (((localizationAway T g).toComp P).toAlgHom (P.σ g)) *
        (compLocalizationAwayAlgHom T g P) (X (Sum.inl ())) -
      1 =
    0
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W' : Projective R
P : Fin 3 → R
u : R
⊢ ![u ^ 4 * W'.dblX P, u ^ 4 * W'.dblY P, u ^ 4 * W'.dblZ P] = u ^ 4 • W'.dblXYZ P

Tactic:
smul_fin3,

Goal state after:
R : Type r
inst✝ : CommRing R
W' : Projective R
P : Fin 3 → R
u : R
⊢ ![u ^ 4 * W'.dblX P, u ^ 4 * W'.dblY P, u ^ 4 * W'.dblZ P] =
    ![u ^ 4 * W'.dblXYZ P x, u ^ 4 * W'.dblXYZ P y, u ^ 4 * W'.dblXYZ P z]
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W' : Projective R
P : Fin 3 → R
u : R
⊢ ![u ^ 4 * W'.dblX P, u ^ 4 * W'.dblY P, u ^ 4 * W'.dblZ P] =
    ![u ^ 4 * W'.dblX P, u ^ 4 * W'.dblY P, u ^ 4 * W'.dblZ P]

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Extension.Cotangent.LocalizationAway

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
ι : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
g : S
inst✝ : IsLocalization.Away g T
P : Generators R S ι
⊢ (algebraMap (P.Ring ⧸ P.ker ^ 2) (Localization.Away ((Ideal.Quotient.mk (P.ker ^ 2)) (P.σ g))))
          ((algebraMap P.Ring (P.Ring ⧸ P.ker ^ 2)) (P.σ g)) *
        IsLocalization.Away.invSelf ((Ideal.Quotient.mk (P.ker ^ 2)) (P.σ g)) -
      1 =
    0

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_2
T : Type u_3
ι : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
g : S
inst✝ : IsLocalization.Away g T
P : Generators R S ι
⊢ (algebraMap (P.Ring ⧸ P.ker ^ 2) (Localization.Away ((Ideal.Quotient.mk (P.ker ^ 2)) (P.σ g))))
          ((algebraMap P.Ring (P.Ring ⧸ P.ker ^ 2)) (P.σ g)) *
        IsLocalization.Away.invSelf ((Ideal.Quotient.mk (P.ker ^ 2)) (P.σ g)) -
      1 =
    0
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula

Goal state before:
pp: R : Type r
inst✝¹ : CommRing R
W' : Projective R
inst✝ : NoZeroDivisors R
P : Fin 3 → R
hP : W'.Equation P
hPz : P z = 0
⊢ ![0, P y ^ 4, 0] = P y ^ 4 • ![0, 1, 0]

Tactic:
smul_fin3,

Goal state after:
R : Type r
inst✝¹ : CommRing R
W' : Projective R
inst✝ : NoZeroDivisors R
P : Fin 3 → R
hP : W'.Equation P
hPz : P z = 0
⊢ ![0, P y ^ 4, 0] = ![P y ^ 4 * ![0, 1, 0] x, P y ^ 4 * ![0, 1, 0] y, P y ^ 4 * ![0, 1, 0] z]
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula

Goal state before:
pp: R : Type r
inst✝¹ : CommRing R
W' : Projective R
inst✝ : NoZeroDivisors R
P : Fin 3 → R
hP : W'.Equation P
hPz : P z = 0
⊢ ![0, P y ^ 4, 0] = ![0, P y ^ 4, 0]

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Extension.Cotangent.Free

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
ι : Type u_3
κ : Type u_5
P : Generators R S ι
v : κ → ι
h : LinearIndependent S fun k ↦ (D R S) (P.val (v k))
⊢ ∀ x ∈ P.toExtension.toKaehler.ker ⊓ Submodule.span S (Set.range fun x ↦ P.cotangentSpaceBasis (v x)), x = 0

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
ι : Type u_3
κ : Type u_5
P : Generators R S ι
v : κ → ι
h : LinearIndependent S fun k ↦ (D R S) (P.val (v k))
⊢ ∀ x ∈ P.toExtension.toKaehler.ker ⊓ Submodule.span S (Set.range fun x ↦ P.cotangentSpaceBasis (v x)), x = 0
Module: Mathlib.RingTheory.Extension.Cotangent.Free

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
ι : Type u_3
κ : Type u_5
P : Generators R S ι
v : κ → ι
h : LinearIndependent S fun k ↦ (D R S) (P.val (v k))
x : P.toExtension.CotangentSpace
hx : x ∈ ↑P.toExtension.toKaehler.ker
hxs : ∃ c, (c.sum fun i a ↦ a • P.cotangentSpaceBasis (v i)) = x
⊢ x = 0

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
ι : Type u_3
κ : Type u_5
P : Generators R S ι
v : κ → ι
h : LinearIndependent S fun k ↦ (D R S) (P.val (v k))
x : P.toExtension.CotangentSpace
hx : x ∈ ↑P.toExtension.toKaehler.ker
hxs : ∃ c, (c.sum fun i a ↦ a • P.cotangentSpaceBasis (v i)) = x
⊢ x = 0
Module: Mathlib.RingTheory.Extension.Cotangent.LocalizationAway

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
ι : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
g : S
inst✝ : IsLocalization.Away g T
P : Generators R S ι
a : ((localizationAway T g).comp P).Ring
hx : ((rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1) * a ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1}
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula

Goal state before:
pp: F : Type u
inst✝ : Field F
W : Projective F
P Q : Fin 3 → F
hP : W.Equation P
hPz : P z ≠ 0
hQz : Q z ≠ 0
hx : P x * Q z = Q x * P z
hy : P y * Q z = Q y * P z
hy' : P y * Q z = W.negY Q * P z
⊢ ![0, W.dblU P, 0] = W.dblU P • ![0, 1, 0]

Tactic:
smul_fin3,

Goal state after:
F : Type u
inst✝ : Field F
W : Projective F
P Q : Fin 3 → F
hP : W.Equation P
hPz : P z ≠ 0
hQz : Q z ≠ 0
hx : P x * Q z = Q x * P z
hy : P y * Q z = Q y * P z
hy' : P y * Q z = W.negY Q * P z
⊢ ![0, W.dblU P, 0] = ![W.dblU P * ![0, 1, 0] x, W.dblU P * ![0, 1, 0] y, W.dblU P * ![0, 1, 0] z]
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula

Goal state before:
pp: F : Type u
inst✝ : Field F
W : Projective F
P Q : Fin 3 → F
hP : W.Equation P
hPz : P z ≠ 0
hQz : Q z ≠ 0
hx : P x * Q z = Q x * P z
hy : P y * Q z = Q y * P z
hy' : P y * Q z = W.negY Q * P z
⊢ ![0, W.dblU P, 0] = ![0, W.dblU P, 0]

Tactic:
]

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula

Goal state before:
pp: F : Type u
inst✝¹ : Field F
W : Projective F
inst✝ : DecidableEq F
P Q : Fin 3 → F
hP : W.Equation P
hQ : W.Equation Q
hPz : P z ≠ 0
hQz : Q z ≠ 0
hx : P x * Q z = Q x * P z
hy : P y * Q z ≠ W.negY Q * P z
hZ : IsUnit (W.dblZ P)
⊢ ![W.dblX P, W.dblY P, W.dblZ P] =
    W.dblZ P •
      ![W.toAffine.addX (P x / P z) (Q x / Q z) (W.toAffine.slope (P x / P z) (Q x / Q z) (P y / P z) (Q y / Q z)),
        W.toAffine.addY (P x / P z) (Q x / Q z) (P y / P z)
          (W.toAffine.slope (P x / P z) (Q x / Q z) (P y / P z) (Q y / Q z)),
        1]

Tactic:
smul_fin3,

Goal state after:
F : Type u
inst✝¹ : Field F
W : Projective F
inst✝ : DecidableEq F
P Q : Fin 3 → F
hP : W.Equation P
hQ : W.Equation Q
hPz : P z ≠ 0
hQz : Q z ≠ 0
hx : P x * Q z = Q x * P z
hy : P y * Q z ≠ W.negY Q * P z
hZ : IsUnit (W.dblZ P)
⊢ ![W.dblX P, W.dblY P, W.dblZ P] =
    ![W.dblZ P *
        ![W.toAffine.addX (P x / P z) (Q x / Q z) (W.toAffine.slope (P x / P z) (Q x / Q z) (P y / P z) (Q y / Q z)),
            W.toAffine.addY (P x / P z) (Q x / Q z) (P y / P z)
              (W.toAffine.slope (P x / P z) (Q x / Q z) (P y / P z) (Q y / Q z)),
            1]
          x,
      W.dblZ P *
        ![W.toAffine.addX (P x / P z) (Q x / Q z) (W.toAffine.slope (P x / P z) (Q x / Q z) (P y / P z) (Q y / Q z)),
            W.toAffine.addY (P x / P z) (Q x / Q z) (P y / P z)
              (W.toAffine.slope (P x / P z) (Q x / Q z) (P y / P z) (Q y / Q z)),
            1]
          y,
      W.dblZ P *
        ![W.toAffine.addX (P x / P z) (Q x / Q z) (W.toAffine.slope (P x / P z) (Q x / Q z) (P y / P z) (Q y / Q z)),
            W.toAffine.addY (P x / P z) (Q x / Q z) (P y / P z)
              (W.toAffine.slope (P x / P z) (Q x / Q z) (P y / P z) (Q y / Q z)),
            1]
          z]
Module: Mathlib.RingTheory.Extension.Cotangent.LocalizationAway

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
ι : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
g : S
inst✝ : IsLocalization.Away g T
P : Generators R S ι
hsple :
  ∀ {x : ((localizationAway T g).comp P).Ring},
    x ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} → (compLocalizationAwayAlgHom T g P) x = 0
⊢ Ideal.map ((localizationAway T g).toComp P).toAlgHom P.ker *
          Ideal.map ((localizationAway T g).toComp P).toAlgHom P.ker ⊔
        Ideal.map ((localizationAway T g).toComp P).toAlgHom P.ker *
          Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} ⊔
      (Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} *
          Ideal.map ((localizationAway T g).toComp P).toAlgHom P.ker ⊔
        Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} *
          Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1}) ≤
    RingHom.ker (compLocalizationAwayAlgHom T g P)

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_2
T : Type u_3
ι : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
g : S
inst✝ : IsLocalization.Away g T
P : Generators R S ι
hsple :
  ∀ {x : ((localizationAway T g).comp P).Ring},
    x ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} → (compLocalizationAwayAlgHom T g P) x = 0
⊢ Ideal.map ((localizationAway T g).toComp P).toAlgHom P.ker *
          Ideal.map ((localizationAway T g).toComp P).toAlgHom P.ker ⊔
        Ideal.map ((localizationAway T g).toComp P).toAlgHom P.ker *
          Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} ⊔
      (Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} *
          Ideal.map ((localizationAway T g).toComp P).toAlgHom P.ker ⊔
        Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} *
          Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1}) ≤
    RingHom.ker (compLocalizationAwayAlgHom T g P)
Module: Mathlib.RingTheory.Extension.Cotangent.Free

Goal state before:
pp: case hli
R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
ι : Type u_3
σ : Type u_4
P : PreSubmersivePresentation R S ι σ
inst✝ : Finite σ
b : Module.Basis σ S P.toExtension.Cotangent
hb : ∀ (r : σ), b r = Extension.Cotangent.mk ⟨P.relation r, ⋯⟩
h : Function.Bijective ⇑(P.cotangentRestrict ⋯)
heq :
  (fun j i ↦ (aeval P.val) ((pderiv (P.map i)) (P.relation j))) =
    ⇑(Finsupp.linearEquivFunOnFinite S S σ) ∘ ⇑(P.cotangentRestrict ⋯) ∘ ⇑b
⊢ LinearIndependent S (⇑(Finsupp.linearEquivFunOnFinite S S σ) ∘ ⇑(P.cotangentRestrict ⋯) ∘ ⇑b)

Tactic:
]

Goal state after:
case hli
R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
ι : Type u_3
σ : Type u_4
P : PreSubmersivePresentation R S ι σ
inst✝ : Finite σ
b : Module.Basis σ S P.toExtension.Cotangent
hb : ∀ (r : σ), b r = Extension.Cotangent.mk ⟨P.relation r, ⋯⟩
h : Function.Bijective ⇑(P.cotangentRestrict ⋯)
heq :
  (fun j i ↦ (aeval P.val) ((pderiv (P.map i)) (P.relation j))) =
    ⇑(Finsupp.linearEquivFunOnFinite S S σ) ∘ ⇑(P.cotangentRestrict ⋯) ∘ ⇑b
⊢ LinearIndependent S (⇑(Finsupp.linearEquivFunOnFinite S S σ) ∘ ⇑(P.cotangentRestrict ⋯) ∘ ⇑b)
Module: Mathlib.RingTheory.Extension.Cotangent.Free

Goal state before:
pp: case hsp
R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
ι : Type u_3
σ : Type u_4
P : PreSubmersivePresentation R S ι σ
inst✝ : Finite σ
b : Module.Basis σ S P.toExtension.Cotangent
hb : ∀ (r : σ), b r = Extension.Cotangent.mk ⟨P.relation r, ⋯⟩
h : Function.Bijective ⇑(P.cotangentRestrict ⋯)
heq :
  (fun j i ↦ (aeval P.val) ((pderiv (P.map i)) (P.relation j))) =
    ⇑(Finsupp.linearEquivFunOnFinite S S σ) ∘ ⇑(P.cotangentRestrict ⋯) ∘ ⇑b
⊢ ⊤ = ⊤

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Extension.Cotangent.LocalizationAway

Goal state before:
pp: case a.a
R : Type u_1
S : Type u_2
T : Type u_3
ι : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
g : S
inst✝ : IsLocalization.Away g T
P : Generators R S ι
hsple :
  ∀ {x : ((localizationAway T g).comp P).Ring},
    x ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} → (compLocalizationAwayAlgHom T g P) x = 0
⊢ P.ker ^ 2 ≤ Ideal.comap ((localizationAway T g).toComp P).toAlgHom (RingHom.ker (compLocalizationAwayAlgHom T g P))

Tactic:
]

Goal state after:
case a.a
R : Type u_1
S : Type u_2
T : Type u_3
ι : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
g : S
inst✝ : IsLocalization.Away g T
P : Generators R S ι
hsple :
  ∀ {x : ((localizationAway T g).comp P).Ring},
    x ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} → (compLocalizationAwayAlgHom T g P) x = 0
⊢ P.ker ^ 2 ≤ Ideal.comap ((localizationAway T g).toComp P).toAlgHom (RingHom.ker (compLocalizationAwayAlgHom T g P))
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula

Goal state before:
pp: F : Type u
inst✝¹ : Field F
W : Projective F
inst✝ : DecidableEq F
P Q : Fin 3 → F
hP : W.Equation P
hQ : W.Equation Q
hPz : P z ≠ 0
hQz : Q z ≠ 0
hx : P x * Q z = Q x * P z
hy : P y * Q z ≠ W.negY Q * P z
hZ : IsUnit (W.dblZ P)
⊢ ![W.dblX P, W.dblY P, W.dblZ P] = ![W.dblX P, W.dblY P, W.dblZ P]

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Extension.Cotangent.LocalizationAway

Goal state before:
pp: case a.a
R : Type u_1
S : Type u_2
T : Type u_3
ι : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
g : S
inst✝ : IsLocalization.Away g T
P : Generators R S ι
hsple :
  ∀ {x : ((localizationAway T g).comp P).Ring},
    x ∈ Ideal.span {(rename Sum.inr) (P.σ g) * X (Sum.inl ()) - 1} → (compLocalizationAwayAlgHom T g P) x = 0
x : P.Ring
hx : x ∈ P.ker ^ 2
⊢ 0 = 0

Tactic:
]

Goal state after: