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

Goal state before:
pp: F : Type u
inst✝ : Field F
W : Projective F
P : Fin 3 → F
⊢ W.dblU P = (W.a₁ * P y * P z - (3 * P x ^ 2 + 2 * W.a₂ * P x * P z + W.a₄ * P z ^ 2)) ^ 3 / P z ^ 2

Tactic:
dblU,

Goal state after:
F : Type u
inst✝ : Field F
W : Projective F
P : Fin 3 → F
⊢ (eval P) W.polynomialX ^ 3 / P z ^ 2 =
    (W.a₁ * P y * P z - (3 * P x ^ 2 + 2 * W.a₂ * P x * P z + W.a₄ * P z ^ 2)) ^ 3 / P z ^ 2
Module: Mathlib.RingTheory.Etale.Field

Goal state before:
pp: K : Type u_1
A : Type u
inst✝⁹ : Field K
inst✝⁸ : CommRing A
inst✝⁷ : Algebra K A
B : Type u_3
inst✝⁶ : CommRing B
inst✝⁵ : EssFiniteType K A
inst✝⁴ : FormallyEtale K A
inst✝³ : Algebra K B
inst✝² : EssFiniteType K B
inst✝¹ : FormallyEtale K B
inst✝ : IsSepClosed K
f : A →ₐ[K] B
x : A
p : PrimeSpectrum B
heq : ofId K (B ⧸ p.asIdeal) = (Ideal.quotientMapₐ p.asIdeal f ⋯).comp (ofId K (A ⧸ Ideal.comap f p.asIdeal))
h :
  (Ideal.quotientMapₐ p.asIdeal f ⋯) ((Ideal.Quotient.mk (Ideal.comap f p.asIdeal)) x) =
    (Ideal.Quotient.mk p.asIdeal) (f x)
⊢ (ofId K (B ⧸ p.asIdeal))
      ((AlgEquiv.ofBijective (ofId K (A ⧸ Ideal.comap (↑f) p.asIdeal)) ⋯).symm
        ((Ideal.Quotient.mk (Ideal.comap (↑f) p.asIdeal)) x)) =
    (algebraMap K (B ⧸ p.asIdeal))
      ((AlgEquiv.ofBijective (ofId K (B ⧸ p.asIdeal)) ⋯).symm ((Ideal.Quotient.mk p.asIdeal) (f x)))

Tactic:
← ofId_apply

Goal state after:
K : Type u_1
A : Type u
inst✝⁹ : Field K
inst✝⁸ : CommRing A
inst✝⁷ : Algebra K A
B : Type u_3
inst✝⁶ : CommRing B
inst✝⁵ : EssFiniteType K A
inst✝⁴ : FormallyEtale K A
inst✝³ : Algebra K B
inst✝² : EssFiniteType K B
inst✝¹ : FormallyEtale K B
inst✝ : IsSepClosed K
f : A →ₐ[K] B
x : A
p : PrimeSpectrum B
heq : ofId K (B ⧸ p.asIdeal) = (Ideal.quotientMapₐ p.asIdeal f ⋯).comp (ofId K (A ⧸ Ideal.comap f p.asIdeal))
h :
  (Ideal.quotientMapₐ p.asIdeal f ⋯) ((Ideal.Quotient.mk (Ideal.comap f p.asIdeal)) x) =
    (Ideal.Quotient.mk p.asIdeal) (f x)
⊢ (ofId K (B ⧸ p.asIdeal))
      ((AlgEquiv.ofBijective (ofId K (A ⧸ Ideal.comap (↑f) p.asIdeal)) ⋯).symm
        ((Ideal.Quotient.mk (Ideal.comap (↑f) p.asIdeal)) x)) =
    (ofId K (B ⧸ p.asIdeal))
      ((AlgEquiv.ofBijective (ofId K (B ⧸ p.asIdeal)) ⋯).symm ((Ideal.Quotient.mk p.asIdeal) (f x)))
Module: Mathlib.RingTheory.Smooth.Fiber

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
inst✝⁵ : Module.Flat R S
inst✝⁴ : FinitePresentation R S
p : Ideal R
q : Ideal S
inst✝³ : p.IsPrime
inst✝² : q.IsPrime
inst✝¹ : q.LiesOver p
inst✝ : FormallySmooth p.ResidueField (p.Fiber S)
Rp : Type u_1 := Localization.AtPrime p
Sp : Type u_2 := Localization (algebraMapSubmonoid S p.primeCompl)
Sq : Type u_2 := Localization.AtPrime q
this : Algebra (Localization.AtPrime p) (Localization.AtPrime q) := Localization.AtPrime.algebraOfLiesOver p q
⊢ ∀ (y : ↥(algebraMapSubmonoid S p.primeCompl)), IsUnit ((ofId S Sq) ↑y)

Tactic:
rintro ⟨_, x, hx, rfl⟩
      simpa using! IsLocalization.map_units (M := q.primeCompl) Sq ⟨algebraMap _ _ x,
        by simp_all [q.over_def p]⟩

Goal state after:
Module: Mathlib.RingTheory.AdicCompletion.Algebra

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
I : Ideal R
m n : ℕ
hmn : m ≤ n
r : AdicCauchySequence I R
⊢ (Ideal.Quotient.mk (I ^ m)) (↑r n) = (Ideal.Quotient.mk (I ^ m)) (↑r m)

Tactic:
by
  have h : I ^ m = I ^ m • ⊤ := by simp
  rw [← Ideal.Quotient.mk_eq_mk, ← Ideal.Quotient.mk_eq_mk, h]
  exact (r.property hmn).symm

Goal state after:
Module: Mathlib.RingTheory.AdicCompletion.Algebra

Goal state before:
pp: R : Type u_4
S : Type u_5
inst✝¹ : NonAssocSemiring R
inst✝ : CommRing S
I : Ideal S
f : (n : ℕ) → R →+* S ⧸ I ^ n
hf : ∀ {m n : ℕ} (hle : m ≤ n), (Ideal.Quotient.factorPow I hle).comp (f n) = f m
n : ℕ
x : R
h : I ^ n • ⊤ ≤ I ^ n
⊢ (factor h) ((eval I S n) ((liftRingHom I f ⋯) x)) = (f n) x

Tactic:
by
  simp [liftRingHom, eval]

Goal state after:
Module: Mathlib.LinearAlgebra.Basis.Exact

Goal state before:
pp: R✝ : Type u_1
M✝ : Type u_2
K✝ : Type u_3
P✝ : Type u_4
inst✝¹³ : Ring R✝
inst✝¹² : AddCommGroup M✝
inst✝¹¹ : AddCommGroup K✝
inst✝¹⁰ : AddCommGroup P✝
inst✝⁹ : Module R✝ M✝
inst✝⁸ : Module R✝ K✝
inst✝⁷ : Module R✝ P✝
g✝ : M✝ →ₗ[R✝] P✝
s✝ : M✝ →ₗ[R✝] K✝
κ✝ : Type u_6
σ✝ : Type u_7
R : Type u_1
M : Type u_2
K : Type u_3
P : Type u_4
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup K
inst✝³ : AddCommGroup P
inst✝² : Module R M
inst✝¹ : Module R K
inst✝ : Module R P
f : K →ₗ[R] M
g : M →ₗ[R] P
s : M →ₗ[R] K
hs : s ∘ₗ f = LinearMap.id
hfg : Function.Exact ⇑f ⇑g
κ : Type u_6
σ : Type u_7
v : κ ⊕ σ → M
hg : Function.Surjective ⇑g
hslzero : ∀ (i : κ), s (v (Sum.inl i)) = 0
hsp : ⊤ ≤ Submodule.span R (Set.range v)
c : κ ⊕ σ →₀ R
this : (c.sum fun i a ↦ a • v i) ∈ Submodule.span R (Set.range v)
h : ∑ x ∈ c.support.toRight, c (Sum.inr x) • s (v (Sum.inr x)) = 0
hli : ∀ (i : σ), c (Sum.inr i) = 0
⊢ g (c.sum fun i a ↦ a • v i) ∈ Submodule.span R (Set.range (⇑g ∘ v ∘ Sum.inl))

Tactic:
simp only [Finsupp.sum, Finset.sum_sum_eq_sum_toLeft_add_sum_toRight, hli, zero_smul,
    Finset.sum_const_zero, add_zero, map_sum, map_smul]

Goal state after:
R✝ : Type u_1
M✝ : Type u_2
K✝ : Type u_3
P✝ : Type u_4
inst✝¹³ : Ring R✝
inst✝¹² : AddCommGroup M✝
inst✝¹¹ : AddCommGroup K✝
inst✝¹⁰ : AddCommGroup P✝
inst✝⁹ : Module R✝ M✝
inst✝⁸ : Module R✝ K✝
inst✝⁷ : Module R✝ P✝
g✝ : M✝ →ₗ[R✝] P✝
s✝ : M✝ →ₗ[R✝] K✝
κ✝ : Type u_6
σ✝ : Type u_7
R : Type u_1
M : Type u_2
K : Type u_3
P : Type u_4
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup K
inst✝³ : AddCommGroup P
inst✝² : Module R M
inst✝¹ : Module R K
inst✝ : Module R P
f : K →ₗ[R] M
g : M →ₗ[R] P
s : M →ₗ[R] K
hs : s ∘ₗ f = LinearMap.id
hfg : Function.Exact ⇑f ⇑g
κ : Type u_6
σ : Type u_7
v : κ ⊕ σ → M
hg : Function.Surjective ⇑g
hslzero : ∀ (i : κ), s (v (Sum.inl i)) = 0
hsp : ⊤ ≤ Submodule.span R (Set.range v)
c : κ ⊕ σ →₀ R
this : (c.sum fun i a ↦ a • v i) ∈ Submodule.span R (Set.range v)
h : ∑ x ∈ c.support.toRight, c (Sum.inr x) • s (v (Sum.inr x)) = 0
hli : ∀ (i : σ), c (Sum.inr i) = 0
⊢ ∑ x ∈ c.support.toLeft, c (Sum.inl x) • g (v (Sum.inl x)) ∈ Submodule.span R (Set.range (⇑g ∘ v ∘ Sum.inl))
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.negDblY P = -W.dblU P

Tactic:
dblU,

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
⊢ W.negDblY P = -((eval P) W.polynomialX ^ 3 / P z ^ 2)
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.dblU P

Tactic:
dblU,

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
⊢ W.dblY P = (eval P) W.polynomialX ^ 3 / P z ^ 2
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
⊢ Submodule.span S (⇑(Finsupp.linearEquivFunOnFinite S S σ) '' ⇑(P.cotangentRestrict ⋯) '' Set.range ⇑b) = ⊤

Tactic:
Submodule.span_image_linearEquiv,

Goal state after:
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
⊢ Submodule.map (↑(Finsupp.linearEquivFunOnFinite S S σ))
      (Submodule.span S (⇑(P.cotangentRestrict ⋯) '' Set.range ⇑b)) =
    ⊤
Module: Mathlib.RingTheory.Smooth.StandardSmoothOfFree

Goal state before:
pp: R✝ : Type u_1
S✝ : Type u_2
inst✝⁸ : CommRing R✝
inst✝⁷ : CommRing S✝
inst✝⁶ : Algebra R✝ S✝
R : Type u_1
S : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : FinitePresentation R S
p : Ideal S
inst✝¹ : p.IsPrime
inst✝ : IsSmoothAt R p
h : Smooth R S
κ : Type u_2
a : κ → S
b : Module.Basis κ (Localization.AtPrime p) Ω[Localization.AtPrime p⁄R]
hb : ∀ (i : κ), b i = (⇑(map R R S (Localization.AtPrime p)) ∘ ⇑(D R S)) (a i)
e : (κ →₀ S) →ₗ[S] Ω[S⁄R] := ⋯
l₁ : (κ →₀ S) →ₗ[S] κ →₀ Localization.AtPrime p := ⋯
l₂ : Ω[S⁄R] →ₗ[S] Ω[Localization.AtPrime p⁄R] := ⋯
eₚ : (κ →₀ Localization.AtPrime p) →ₗ[Localization.AtPrime p] Ω[Localization.AtPrime p⁄R] := ⋯
i : κ
⊢ (eₚ ∘ₗ Finsupp.lsingle i) 1 = (↑b.repr.symm ∘ₗ Finsupp.lsingle i) 1

Tactic:
trans IsLocalizedModule.map p.primeCompl l₁ l₂ e <| l₁ <| Finsupp.single i 1

Goal state after:
R✝ : Type u_1
S✝ : Type u_2
inst✝⁸ : CommRing R✝
inst✝⁷ : CommRing S✝
inst✝⁶ : Algebra R✝ S✝
R : Type u_1
S : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : FinitePresentation R S
p : Ideal S
inst✝¹ : p.IsPrime
inst✝ : IsSmoothAt R p
h : Smooth R S
κ : Type u_2
a : κ → S
b : Module.Basis κ (Localization.AtPrime p) Ω[Localization.AtPrime p⁄R]
hb : ∀ (i : κ), b i = (⇑(map R R S (Localization.AtPrime p)) ∘ ⇑(D R S)) (a i)
e : (κ →₀ S) →ₗ[S] Ω[S⁄R] := ⋯
l₁ : (κ →₀ S) →ₗ[S] κ →₀ Localization.AtPrime p := ⋯
l₂ : Ω[S⁄R] →ₗ[S] Ω[Localization.AtPrime p⁄R] := ⋯
eₚ : (κ →₀ Localization.AtPrime p) →ₗ[Localization.AtPrime p] Ω[Localization.AtPrime p⁄R] := ⋯
i : κ
⊢ (eₚ ∘ₗ Finsupp.lsingle i) 1 = ((IsLocalizedModule.map p.primeCompl l₁ l₂) e) (l₁ (Finsupp.single i 1))

R✝ : Type u_1
S✝ : Type u_2
inst✝⁸ : CommRing R✝
inst✝⁷ : CommRing S✝
inst✝⁶ : Algebra R✝ S✝
R : Type u_1
S : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : FinitePresentation R S
p : Ideal S
inst✝¹ : p.IsPrime
inst✝ : IsSmoothAt R p
h : Smooth R S
κ : Type u_2
a : κ → S
b : Module.Basis κ (Localization.AtPrime p) Ω[Localization.AtPrime p⁄R]
hb : ∀ (i : κ), b i = (⇑(map R R S (Localization.AtPrime p)) ∘ ⇑(D R S)) (a i)
e : (κ →₀ S) →ₗ[S] Ω[S⁄R] := ⋯
l₁ : (κ →₀ S) →ₗ[S] κ →₀ Localization.AtPrime p := ⋯
l₂ : Ω[S⁄R] →ₗ[S] Ω[Localization.AtPrime p⁄R] := ⋯
eₚ : (κ →₀ Localization.AtPrime p) →ₗ[Localization.AtPrime p] Ω[Localization.AtPrime p⁄R] := ⋯
i : κ
⊢ ((IsLocalizedModule.map p.primeCompl l₁ l₂) e) (l₁ (Finsupp.single i 1)) = (↑b.repr.symm ∘ₗ Finsupp.lsingle i) 1
Module: Mathlib.RingTheory.Extension.Cotangent.Basis

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
σ✝ : Type u_3
ι : Type u_4
P : Generators R S ι
σ : Type u_5
b : Module.Basis σ S P.toExtension.Cotangent
D : Aux P b
h : (algebraMap P.Ring S) D.g = 1
⊢ IsUnit ((algebraMap P.Ring S) D.g)

Tactic:
by simp [h]

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

Goal state before:
pp: case inr
R : Type u_1
S : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : FinitePresentation R S
α : Type u_4
P : Generators R S α
inst✝ : Finite α
σ : Type u_5
b₀ : Module.Basis σ S P.toExtension.Cotangent
h✝ : Nontrivial S
f : P.toExtension.Cotangent → ↥P.toExtension.ker
hf : ∀ (b : P.toExtension.Cotangent), Extension.Cotangent.mk (f b) = b
v : σ → ↥P.ker := fun i ↦ f (b₀ i)
J : Ideal P.Ring := Ideal.span (Set.range (Subtype.val ∘ v))
hJfg : P.ker.FG
hJ : J ≤ P.ker
⊢ Submodule.restrictScalars P.Ring (Submodule.span S (Set.range fun x ↦ Extension.Cotangent.mk (v x))) = ⊤

Tactic:
Submodule.restrictScalars_eq_top_iff

Goal state after:
case inr
R : Type u_1
S : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : FinitePresentation R S
α : Type u_4
P : Generators R S α
inst✝ : Finite α
σ : Type u_5
b₀ : Module.Basis σ S P.toExtension.Cotangent
h✝ : Nontrivial S
f : P.toExtension.Cotangent → ↥P.toExtension.ker
hf : ∀ (b : P.toExtension.Cotangent), Extension.Cotangent.mk (f b) = b
v : σ → ↥P.ker := fun i ↦ f (b₀ i)
J : Ideal P.Ring := Ideal.span (Set.range (Subtype.val ∘ v))
hJfg : P.ker.FG
hJ : J ≤ P.ker
⊢ Submodule.span S (Set.range fun x ↦ Extension.Cotangent.mk (v x)) = ⊤
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula

Goal state before:
pp: 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
⊢ -3 * 0 ^ 2 * Q x * Q y + 3 * 0 * Q x ^ 2 * P y - P y ^ 2 * Q y * Q z + P y * Q y ^ 2 * 0 + W'.a₁ * 0 * Q y ^ 2 * 0 -
                              W'.a₁ * Q x * P y ^ 2 * Q z -
                            W'.a₂ * 0 ^ 2 * Q y * Q z +
                          W'.a₂ * Q x ^ 2 * P y * 0 +
                        2 * W'.a₂ * 0 * Q x * P y * Q z -
                      2 * W'.a₂ * 0 * Q x * Q y * 0 -
                    W'.a₃ * P y ^ 2 * Q z ^ 2 +
                  W'.a₃ * Q y ^ 2 * 0 ^ 2 +
                W'.a₄ * 0 * P y * Q z ^ 2 -
              2 * W'.a₄ * 0 * Q y * 0 * Q z +
            2 * W'.a₄ * Q x * P y * 0 * Q z -
          W'.a₄ * Q x * Q y * 0 ^ 2 +
        3 * W'.a₆ * P y * 0 * Q z ^ 2 -
      3 * W'.a₆ * Q y * 0 ^ 2 * Q z =
    P y ^ 2 * Q z * W'.negY Q

Tactic:
negY

Goal state after:
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
⊢ -3 * 0 ^ 2 * Q x * Q y + 3 * 0 * Q x ^ 2 * P y - P y ^ 2 * Q y * Q z + P y * Q y ^ 2 * 0 + W'.a₁ * 0 * Q y ^ 2 * 0 -
                              W'.a₁ * Q x * P y ^ 2 * Q z -
                            W'.a₂ * 0 ^ 2 * Q y * Q z +
                          W'.a₂ * Q x ^ 2 * P y * 0 +
                        2 * W'.a₂ * 0 * Q x * P y * Q z -
                      2 * W'.a₂ * 0 * Q x * Q y * 0 -
                    W'.a₃ * P y ^ 2 * Q z ^ 2 +
                  W'.a₃ * Q y ^ 2 * 0 ^ 2 +
                W'.a₄ * 0 * P y * Q z ^ 2 -
              2 * W'.a₄ * 0 * Q y * 0 * Q z +
            2 * W'.a₄ * Q x * P y * 0 * Q z -
          W'.a₄ * Q x * Q y * 0 ^ 2 +
        3 * W'.a₆ * P y * 0 * Q z ^ 2 -
      3 * W'.a₆ * Q y * 0 ^ 2 * Q z =
    P y ^ 2 * Q z * (-Q y - W'.a₁ * Q x - W'.a₃ * Q 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 Q : Fin 3 → R
hQ : W'.Equation Q
hQz : Q z = 0
⊢ -3 * P x ^ 2 * 0 * Q y + 3 * P x * 0 ^ 2 * P y - P y ^ 2 * Q y * 0 + P y * Q y ^ 2 * P z +
                                W'.a₁ * P x * Q y ^ 2 * P z -
                              W'.a₁ * 0 * P y ^ 2 * 0 -
                            W'.a₂ * P x ^ 2 * Q y * 0 +
                          W'.a₂ * 0 ^ 2 * P y * P z +
                        2 * W'.a₂ * P x * 0 * P y * 0 -
                      2 * W'.a₂ * P x * 0 * Q y * P z -
                    W'.a₃ * P y ^ 2 * 0 ^ 2 +
                  W'.a₃ * Q y ^ 2 * P z ^ 2 +
                W'.a₄ * P x * P y * 0 ^ 2 -
              2 * W'.a₄ * P x * Q y * P z * 0 +
            2 * W'.a₄ * 0 * P y * P z * 0 -
          W'.a₄ * 0 * Q y * P z ^ 2 +
        3 * W'.a₆ * P y * P z * 0 ^ 2 -
      3 * W'.a₆ * Q y * P z ^ 2 * 0 =
    -(Q y ^ 2 * P z) * W'.negY P

Tactic:
negY

Goal state after:
R : Type r
inst✝¹ : CommRing R
W' : Projective R
inst✝ : NoZeroDivisors R
P Q : Fin 3 → R
hQ : W'.Equation Q
hQz : Q z = 0
⊢ -3 * P x ^ 2 * 0 * Q y + 3 * P x * 0 ^ 2 * P y - P y ^ 2 * Q y * 0 + P y * Q y ^ 2 * P z +
                                W'.a₁ * P x * Q y ^ 2 * P z -
                              W'.a₁ * 0 * P y ^ 2 * 0 -
                            W'.a₂ * P x ^ 2 * Q y * 0 +
                          W'.a₂ * 0 ^ 2 * P y * P z +
                        2 * W'.a₂ * P x * 0 * P y * 0 -
                      2 * W'.a₂ * P x * 0 * Q y * P z -
                    W'.a₃ * P y ^ 2 * 0 ^ 2 +
                  W'.a₃ * Q y ^ 2 * P z ^ 2 +
                W'.a₄ * P x * P y * 0 ^ 2 -
              2 * W'.a₄ * P x * Q y * P z * 0 +
            2 * W'.a₄ * 0 * P y * P z * 0 -
          W'.a₄ * 0 * Q y * P z ^ 2 +
        3 * W'.a₆ * P y * P z * 0 ^ 2 -
      3 * W'.a₆ * Q y * P z ^ 2 * 0 =
    -(Q y ^ 2 * P z) * (-P y - W'.a₁ * P x - W'.a₃ * P z)
Module: Mathlib.AlgebraicGeometry.Morphisms.Proper

Goal state before:
pp: this : @IsAffineHom ⊓ @IsSeparated = @IsAffineHom
⊢ @IsFinite = @IsProper ⊓ @IsAffineHom

Tactic:
rw [inf_comm, isProper_eq, inf_assoc, ← inf_assoc, this, eq_inf,
    IsIntegralHom.eq_universallyClosed_inf_isAffineHom, inf_assoc, inf_left_comm]

Goal state after:
Module: Mathlib.AlgebraicGeometry.Morphisms.FormallyUnramified

Goal state before:
pp: X Y Z' Z : Scheme
i : Z' ⟶ Z
hi : IsNilpotent (Scheme.Hom.ker i)
inst✝¹ : IsClosedImmersion i
f : X ⟶ Y
inst✝ : FormallyUnramified f
g₁ g₂ : Z ⟶ X
hig : i ≫ g₁ = i ≫ g₂
hgf : g₁ ≫ f = g₂ ≫ f
⊢ g₁ = g₂

Tactic:
have : IsDominant i := by
    obtain ⟨n, hn⟩ := hi
    rw [isDominant_iff, denseRange_iff_closure_range, ← i.support_ker,
      ← i.ker.support_pow (n + 1) (by simp), pow_succ, hn]
    simp

Goal state after:
X Y Z' Z : Scheme
i : Z' ⟶ Z
hi : IsNilpotent (Scheme.Hom.ker i)
inst✝¹ : IsClosedImmersion i
f : X ⟶ Y
inst✝ : FormallyUnramified f
g₁ g₂ : Z ⟶ X
hig : i ≫ g₁ = i ≫ g₂
hgf : g₁ ≫ f = g₂ ≫ f
this : IsDominant i
⊢ g₁ = g₂
Module: Mathlib.FieldTheory.SeparablyGenerated

Goal state before:
pp: case inl
k : Type u_1
K : Type u_2
inst✝⁴ : Field k
inst✝³ : Field K
inst✝² : Algebra k K
inst✝¹ : PerfectField k
inst✝ : Algebra.EssFiniteType k K
h✝ : CharZero k
s : Finset K
hs : adjoin k ↑s = ⊤
this : Algebra.IsAlgebraic (↥(Algebra.adjoin k ↑s)) K
⊢ ∃ s, IsTranscendenceBasis k Subtype.val ∧ Algebra.IsSeparable (↥(adjoin k ↑s)) K

Tactic:
obtain ⟨t, hts, ht⟩ := exists_isTranscendenceBasis_subset (R := k) (s : Set K)

Goal state after:
case inl
k : Type u_1
K : Type u_2
inst✝⁴ : Field k
inst✝³ : Field K
inst✝² : Algebra k K
inst✝¹ : PerfectField k
inst✝ : Algebra.EssFiniteType k K
h✝ : CharZero k
s : Finset K
hs : adjoin k ↑s = ⊤
this : Algebra.IsAlgebraic (↥(Algebra.adjoin k ↑s)) K
t : Set K
hts : t ⊆ ↑s
ht : IsTranscendenceBasis k Subtype.val
⊢ ∃ s, IsTranscendenceBasis k Subtype.val ∧ Algebra.IsSeparable (↥(adjoin k ↑s)) K
Module: Mathlib.AlgebraicGeometry.Morphisms.Descent

Goal state before:
pp: P : MorphismProperty Scheme
X : Scheme
inst✝² : CompactSpace ↥X
inst✝¹ : IsZariskiLocalAtSource P
inst✝ : P.ContainsIdentities
𝒰 : X.OpenCover := X.affineCover.finiteSubcover
⊢ ∃ Y p, Surjective p ∧ P p ∧ IsAffine Y

Tactic:
let p : ∐ (fun i : 𝒰.I₀ ↦ 𝒰.X i) ⟶ X := Sigma.desc (fun i ↦ 𝒰.f i)

Goal state after:
P : MorphismProperty Scheme
X : Scheme
inst✝² : CompactSpace ↥X
inst✝¹ : IsZariskiLocalAtSource P
inst✝ : P.ContainsIdentities
𝒰 : X.OpenCover := X.affineCover.finiteSubcover
p : (∐ fun i ↦ 𝒰.X i) ⟶ X := Sigma.desc fun i ↦ 𝒰.f i
⊢ ∃ Y p, Surjective p ∧ P p ∧ IsAffine Y
Module: Mathlib.AlgebraicGeometry.Morphisms.Descent

Goal state before:
pp: P P' : MorphismProperty Scheme
inst✝¹ : IsZariskiLocalAtTarget P
inst✝ : P'.IsStableUnderBaseChange
H : ∀ {R : CommRingCat} {X Y : Scheme} (f : X ⟶ Spec R) (g : Y ⟶ Spec R), P' f → P (pullback.fst f g) → P g
⊢ ∀ {X Y Z : Scheme} {f : X ⟶ Z} {g : Y ⟶ Z} [inst : HasPullback f g], P' f → P (pullback.fst f g) → P g

Tactic:
introv h hf

Goal state after:
P P' : MorphismProperty Scheme
inst✝² : IsZariskiLocalAtTarget P
inst✝¹ : P'.IsStableUnderBaseChange
H : ∀ {R : CommRingCat} {X Y : Scheme} (f : X ⟶ Spec R) (g : Y ⟶ Spec R), P' f → P (pullback.fst f g) → P g
X Y Z : Scheme
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : HasPullback f g
h : P' f
hf : P (pullback.fst f g)
⊢ P g
Module: Mathlib.AlgebraicGeometry.Morphisms.Descent

Goal state before:
pp: case inr.refine_2
P P' : MorphismProperty Scheme
inst✝³ : P'.IsStableUnderBaseChange
inst✝² : P'.IsStableUnderComposition
inst✝¹ : P.IsStableUnderBaseChange
H₁ : @IsLocalIso ⊓ @Surjective ≤ P'
inst✝ : IsZariskiLocalAtTarget P
H :
  ∀ {R S : CommRingCat} {Y : Scheme} (φ : R ⟶ S) (g : Y ⟶ Spec R),
    P' (Spec.map φ) → P (pullback.fst (Spec.map φ) g) → P g
R : CommRingCat
X Y✝ : Scheme
f : X ⟶ Spec R
g : Y✝ ⟶ Spec R
hf : (P' ⊓ @QuasiCompact) f
h : P (pullback.fst f g)
this : ∀ {X : Scheme} (f : X ⟶ Spec R), (P' ⊓ @QuasiCompact) f → P (pullback.fst f g) → (∃ T, X = Spec T) → P g
hX : ¬∃ T, X = Spec T
x✝ : CompactSpace ↥X
Y : Scheme
p : Y ⟶ X
hsurj : Surjective p
hP' : IsLocalIso p
hY : IsAffine Y
⊢ P (pullback.fst ((Y.isoSpec.inv ≫ p) ≫ f) g)

Tactic:
rw [← pullbackRightPullbackFstIso_inv_fst f g _, P.cancel_left_of_respectsIso]

Goal state after:
case inr.refine_2
P P' : MorphismProperty Scheme
inst✝³ : P'.IsStableUnderBaseChange
inst✝² : P'.IsStableUnderComposition
inst✝¹ : P.IsStableUnderBaseChange
H₁ : @IsLocalIso ⊓ @Surjective ≤ P'
inst✝ : IsZariskiLocalAtTarget P
H :
  ∀ {R S : CommRingCat} {Y : Scheme} (φ : R ⟶ S) (g : Y ⟶ Spec R),
    P' (Spec.map φ) → P (pullback.fst (Spec.map φ) g) → P g
R : CommRingCat
X Y✝ : Scheme
f : X ⟶ Spec R
g : Y✝ ⟶ Spec R
hf : (P' ⊓ @QuasiCompact) f
h : P (pullback.fst f g)
this : ∀ {X : Scheme} (f : X ⟶ Spec R), (P' ⊓ @QuasiCompact) f → P (pullback.fst f g) → (∃ T, X = Spec T) → P g
hX : ¬∃ T, X = Spec T
x✝ : CompactSpace ↥X
Y : Scheme
p : Y ⟶ X
hsurj : Surjective p
hP' : IsLocalIso p
hY : IsAffine Y
⊢ P (pullback.fst (Y.isoSpec.inv ≫ p) (pullback.fst f g))
Module: Mathlib.AlgebraicGeometry.Morphisms.Descent

Goal state before:
pp: P P' : MorphismProperty Scheme
Q Q' : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
inst✝³ : P'.IsStableUnderBaseChange
inst✝² : P'.IsStableUnderComposition
inst✝¹ : P.IsStableUnderBaseChange
H₁ : @IsLocalIso ⊓ @Surjective ≤ P'
H₂ : ∀ {R S : CommRingCat} {f : R ⟶ S}, P' (Spec.map f) → Q' (CommRingCat.Hom.hom f)
inst✝ : HasRingHomProperty P fun {R S} [CommRing R] [CommRing S] ↦ Q
hQQ' : RingHom.CodescendsAlong (fun {R S} [CommRing R] [CommRing S] ↦ Q) fun {R S} [CommRing R] [CommRing S] ↦ Q'
⊢ ∀ {R S : CommRingCat} {Y : Scheme} (φ : R ⟶ S) (g : Y ⟶ Spec R),
    P' (Spec.map φ) → P (pullback.fst (Spec.map φ) g) → P g

Tactic:
introv h hf

Goal state after:
P P' : MorphismProperty Scheme
Q Q' : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
inst✝³ : P'.IsStableUnderBaseChange
inst✝² : P'.IsStableUnderComposition
inst✝¹ : P.IsStableUnderBaseChange
H₁ : @IsLocalIso ⊓ @Surjective ≤ P'
H₂ : ∀ {R S : CommRingCat} {f : R ⟶ S}, P' (Spec.map f) → Q' (CommRingCat.Hom.hom f)
inst✝ : HasRingHomProperty P fun {R S} [CommRing R] [CommRing S] ↦ Q
hQQ' : RingHom.CodescendsAlong (fun {R S} [CommRing R] [CommRing S] ↦ Q) fun {R S} [CommRing R] [CommRing S] ↦ Q'
R S : CommRingCat
Y : Scheme
φ : R ⟶ S
g : Y ⟶ Spec R
h : P' (Spec.map φ)
hf : P (pullback.fst (Spec.map φ) g)
⊢ P g