mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.Unramified.Pi

Goal state before:
pp: R : Type u_1
I : Type u_2
inst✝⁵ : Finite I
f : I → Type u_3
inst✝⁴ : CommRing R
inst✝³ : (i : I) → CommRing (f i)
inst✝² : (i : I) → Algebra R (f i)
val✝ : Fintype I
H : ∀ (i : I), FormallyUnramified R (f i)
B : Type (max u_2 u_3)
inst✝¹ : CommRing B
inst✝ : Algebra R B
J : Ideal B
hJ : J ^ 2 = ⊥
f₁ f₂ : ((i : I) → f i) →ₐ[R] B
e✝ : (Ideal.Quotient.mkₐ R J).comp f₁ = (Ideal.Quotient.mkₐ R J).comp f₂
g : (i : I) → f i
x : I
hf : ∀ (x : (i : I) → f i), f₁ x - f₂ x ∈ J
e : (i : I) → f i := Pi.single x 1
he : IsIdempotentElem e
h₁ : f₁ e * (1 - f₂ e) = 0
h₂ : f₂ e * (1 - f₁ e) = 0
⊢ f₁ e - f₁ e * f₂ e = f₁ e * (1 - f₂ e)

Tactic:
mul_sub,

Goal state after:
R : Type u_1
I : Type u_2
inst✝⁵ : Finite I
f : I → Type u_3
inst✝⁴ : CommRing R
inst✝³ : (i : I) → CommRing (f i)
inst✝² : (i : I) → Algebra R (f i)
val✝ : Fintype I
H : ∀ (i : I), FormallyUnramified R (f i)
B : Type (max u_2 u_3)
inst✝¹ : CommRing B
inst✝ : Algebra R B
J : Ideal B
hJ : J ^ 2 = ⊥
f₁ f₂ : ((i : I) → f i) →ₐ[R] B
e✝ : (Ideal.Quotient.mkₐ R J).comp f₁ = (Ideal.Quotient.mkₐ R J).comp f₂
g : (i : I) → f i
x : I
hf : ∀ (x : (i : I) → f i), f₁ x - f₂ x ∈ J
e : (i : I) → f i := Pi.single x 1
he : IsIdempotentElem e
h₁ : f₁ e * (1 - f₂ e) = 0
h₂ : f₂ e * (1 - f₁ e) = 0
⊢ f₁ e - f₁ e * f₂ e = f₁ e * 1 - f₁ e * f₂ e
Module: Mathlib.RingTheory.Unramified.Pi

Goal state before:
pp: R : Type u_1
I : Type u_2
inst✝⁵ : Finite I
f : I → Type u_3
inst✝⁴ : CommRing R
inst✝³ : (i : I) → CommRing (f i)
inst✝² : (i : I) → Algebra R (f i)
val✝ : Fintype I
H : ∀ (i : I), FormallyUnramified R (f i)
B : Type (max u_2 u_3)
inst✝¹ : CommRing B
inst✝ : Algebra R B
J : Ideal B
hJ : J ^ 2 = ⊥
f₁ f₂ : ((i : I) → f i) →ₐ[R] B
e✝ : (Ideal.Quotient.mkₐ R J).comp f₁ = (Ideal.Quotient.mkₐ R J).comp f₂
g : (i : I) → f i
x : I
hf : ∀ (x : (i : I) → f i), f₁ x - f₂ x ∈ J
e : (i : I) → f i := Pi.single x 1
he : IsIdempotentElem e
h₁ : f₁ e * (1 - f₂ e) = 0
h₂ : f₂ e * (1 - f₁ e) = 0
⊢ f₂ e - f₁ e * f₂ e = f₂ e * (1 - f₁ e)

Tactic:
mul_sub,

Goal state after:
R : Type u_1
I : Type u_2
inst✝⁵ : Finite I
f : I → Type u_3
inst✝⁴ : CommRing R
inst✝³ : (i : I) → CommRing (f i)
inst✝² : (i : I) → Algebra R (f i)
val✝ : Fintype I
H : ∀ (i : I), FormallyUnramified R (f i)
B : Type (max u_2 u_3)
inst✝¹ : CommRing B
inst✝ : Algebra R B
J : Ideal B
hJ : J ^ 2 = ⊥
f₁ f₂ : ((i : I) → f i) →ₐ[R] B
e✝ : (Ideal.Quotient.mkₐ R J).comp f₁ = (Ideal.Quotient.mkₐ R J).comp f₂
g : (i : I) → f i
x : I
hf : ∀ (x : (i : I) → f i), f₁ x - f₂ x ∈ J
e : (i : I) → f i := Pi.single x 1
he : IsIdempotentElem e
h₁ : f₁ e * (1 - f₂ e) = 0
h₂ : f₂ e * (1 - f₁ e) = 0
⊢ f₂ e - f₁ e * f₂ e = f₂ e * 1 - f₂ e * f₁ e
Module: Mathlib.RingTheory.Unramified.Pi

Goal state before:
pp: R : Type u_1
I : Type u_2
inst✝⁵ : Finite I
f : I → Type u_3
inst✝⁴ : CommRing R
inst✝³ : (i : I) → CommRing (f i)
inst✝² : (i : I) → Algebra R (f i)
val✝ : Fintype I
H✝ : ∀ (i : I), FormallyUnramified R (f i)
B : Type (max u_2 u_3)
inst✝¹ : CommRing B
inst✝ : Algebra R B
J : Ideal B
hJ : J ^ 2 = ⊥
f₁ f₂ : ((i : I) → f i) →ₐ[R] B
e✝ : (Ideal.Quotient.mkₐ R J).comp f₁ = (Ideal.Quotient.mkₐ R J).comp f₂
g : (i : I) → f i
x : I
hf : ∀ (x : (i : I) → f i), f₁ x - f₂ x ∈ J
e : (i : I) → f i := Pi.single x 1
he : IsIdempotentElem e
h₁ : f₁ e * (1 - f₂ e) = 0
h₂ : f₂ e * (1 - f₁ e) = 0
H : f₁ e = f₂ e
J' : Ideal B := Ideal.span {1 - f₁ e}
f₁' : f x →ₐ[R] B ⧸ J' :=
  AlgHom.ofLinearMap (((Ideal.Quotient.mkₐ R J').comp f₁).toLinearMap ∘ₗ LinearMap.single R f x) ⋯ ⋯
f₂' : f x →ₐ[R] B ⧸ J' :=
  AlgHom.ofLinearMap (((Ideal.Quotient.mkₐ R J').comp f₂).toLinearMap ∘ₗ LinearMap.single R f x) ⋯ ⋯
this : f₁' = f₂'
c : B
hc : f₁ e * (f₁ (Pi.single x (g x)) - f₂ (Pi.single x (g x))) = f₁ e * (1 - f₁ e) * c
⊢ f₁ (Pi.single x (g x)) = f₂ (Pi.single x (g x))

Tactic:
mul_sub,

Goal state after:
R : Type u_1
I : Type u_2
inst✝⁵ : Finite I
f : I → Type u_3
inst✝⁴ : CommRing R
inst✝³ : (i : I) → CommRing (f i)
inst✝² : (i : I) → Algebra R (f i)
val✝ : Fintype I
H✝ : ∀ (i : I), FormallyUnramified R (f i)
B : Type (max u_2 u_3)
inst✝¹ : CommRing B
inst✝ : Algebra R B
J : Ideal B
hJ : J ^ 2 = ⊥
f₁ f₂ : ((i : I) → f i) →ₐ[R] B
e✝ : (Ideal.Quotient.mkₐ R J).comp f₁ = (Ideal.Quotient.mkₐ R J).comp f₂
g : (i : I) → f i
x : I
hf : ∀ (x : (i : I) → f i), f₁ x - f₂ x ∈ J
e : (i : I) → f i := Pi.single x 1
he : IsIdempotentElem e
h₁ : f₁ e * (1 - f₂ e) = 0
h₂ : f₂ e * (1 - f₁ e) = 0
H : f₁ e = f₂ e
J' : Ideal B := Ideal.span {1 - f₁ e}
f₁' : f x →ₐ[R] B ⧸ J' :=
  AlgHom.ofLinearMap (((Ideal.Quotient.mkₐ R J').comp f₁).toLinearMap ∘ₗ LinearMap.single R f x) ⋯ ⋯
f₂' : f x →ₐ[R] B ⧸ J' :=
  AlgHom.ofLinearMap (((Ideal.Quotient.mkₐ R J').comp f₂).toLinearMap ∘ₗ LinearMap.single R f x) ⋯ ⋯
this : f₁' = f₂'
c : B
hc : f₁ e * f₁ (Pi.single x (g x)) - f₁ e * f₂ (Pi.single x (g x)) = f₁ e * (1 - f₁ e) * c
⊢ f₁ (Pi.single x (g x)) = f₂ (Pi.single x (g x))
Module: Mathlib.RingTheory.Unramified.Pi

Goal state before:
pp: R : Type u_1
I : Type u_2
inst✝⁵ : Finite I
f : I → Type u_3
inst✝⁴ : CommRing R
inst✝³ : (i : I) → CommRing (f i)
inst✝² : (i : I) → Algebra R (f i)
val✝ : Fintype I
H✝ : ∀ (i : I), FormallyUnramified R (f i)
B : Type (max u_2 u_3)
inst✝¹ : CommRing B
inst✝ : Algebra R B
J : Ideal B
hJ : J ^ 2 = ⊥
f₁ f₂ : ((i : I) → f i) →ₐ[R] B
e✝ : (Ideal.Quotient.mkₐ R J).comp f₁ = (Ideal.Quotient.mkₐ R J).comp f₂
g : (i : I) → f i
x : I
hf : ∀ (x : (i : I) → f i), f₁ x - f₂ x ∈ J
e : (i : I) → f i := Pi.single x 1
he : IsIdempotentElem e
h₁ : f₁ e * (1 - f₂ e) = 0
h₂ : f₂ e * (1 - f₁ e) = 0
H : f₁ e = f₂ e
J' : Ideal B := Ideal.span {1 - f₁ e}
f₁' : f x →ₐ[R] B ⧸ J' :=
  AlgHom.ofLinearMap (((Ideal.Quotient.mkₐ R J').comp f₁).toLinearMap ∘ₗ LinearMap.single R f x) ⋯ ⋯
f₂' : f x →ₐ[R] B ⧸ J' :=
  AlgHom.ofLinearMap (((Ideal.Quotient.mkₐ R J').comp f₂).toLinearMap ∘ₗ LinearMap.single R f x) ⋯ ⋯
this : f₁' = f₂'
c : B
hc : f₁ e * f₁ (Pi.single x (g x)) - f₁ e * f₂ (Pi.single x (g x)) = f₁ e * (1 - f₁ e) * c
⊢ f₁ (Pi.single x (g x)) = f₂ (Pi.single x (g x))

Tactic:
mul_sub,

Goal state after:
R : Type u_1
I : Type u_2
inst✝⁵ : Finite I
f : I → Type u_3
inst✝⁴ : CommRing R
inst✝³ : (i : I) → CommRing (f i)
inst✝² : (i : I) → Algebra R (f i)
val✝ : Fintype I
H✝ : ∀ (i : I), FormallyUnramified R (f i)
B : Type (max u_2 u_3)
inst✝¹ : CommRing B
inst✝ : Algebra R B
J : Ideal B
hJ : J ^ 2 = ⊥
f₁ f₂ : ((i : I) → f i) →ₐ[R] B
e✝ : (Ideal.Quotient.mkₐ R J).comp f₁ = (Ideal.Quotient.mkₐ R J).comp f₂
g : (i : I) → f i
x : I
hf : ∀ (x : (i : I) → f i), f₁ x - f₂ x ∈ J
e : (i : I) → f i := Pi.single x 1
he : IsIdempotentElem e
h₁ : f₁ e * (1 - f₂ e) = 0
h₂ : f₂ e * (1 - f₁ e) = 0
H : f₁ e = f₂ e
J' : Ideal B := Ideal.span {1 - f₁ e}
f₁' : f x →ₐ[R] B ⧸ J' :=
  AlgHom.ofLinearMap (((Ideal.Quotient.mkₐ R J').comp f₁).toLinearMap ∘ₗ LinearMap.single R f x) ⋯ ⋯
f₂' : f x →ₐ[R] B ⧸ J' :=
  AlgHom.ofLinearMap (((Ideal.Quotient.mkₐ R J').comp f₂).toLinearMap ∘ₗ LinearMap.single R f x) ⋯ ⋯
this : f₁' = f₂'
c : B
hc : f₁ e * f₁ (Pi.single x (g x)) - f₁ e * f₂ (Pi.single x (g x)) = (f₁ e * 1 - f₁ e * f₁ e) * c
⊢ f₁ (Pi.single x (g x)) = f₂ (Pi.single x (g x))
Module: Mathlib.RingTheory.Kaehler.TensorProduct

Goal state before:
pp: R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝¹¹ : CommRing R
inst✝¹⁰ : CommRing S
inst✝⁹ : Algebra R S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra R B
inst✝⁴ : Algebra A B
inst✝³ : Algebra S B
inst✝² : IsScalarTower R A B
inst✝¹ : IsScalarTower R S B
inst✝ : Algebra.IsPushout R S A B
⊢ ↑S (derivationTensorProduct R S A B).liftKaehlerDifferential ∘ₗ LinearMap.liftBaseChange S (↑R (map R S A B)) =
    LinearMap.id

Tactic:
refine LinearMap.restrictScalars_injective R ?_
  apply TensorProduct.ext'
  intro x y
  obtain ⟨y, rfl⟩ := tensorProductTo_surjective _ _ y
  induction y
  · simp only [map_zero, TensorProduct.tmul_zero]
  · simp only [LinearMap.restrictScalars_comp, Derivation.tensorProductTo_tmul, LinearMap.coe_comp,
      LinearMap.coe_restrictScalars, Function.comp_apply, LinearMap.liftBaseChange_tmul, map_smul,
      map_D, LinearMap.map_smul_of_tower, Derivation.liftKaehlerDifferential_comp_D,
      LinearMap.id_coe, id_eq, derivationTensorProduct_algebraMap]
    rw [smul_comm, TensorProduct.smul_tmul', smul_eq_mul, mul_one]
    rfl
  · simp only [map_add, TensorProduct.tmul_add, *]

Goal state after:
Module: Mathlib.RingTheory.Kaehler.TensorProduct

Goal state before:
pp: R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : Algebra R S
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : Algebra S B
inst✝¹ : IsScalarTower R A B
inst✝ : IsScalarTower R S B
h : Algebra.IsPushout R S A B
⊢ IsBaseChange S (↑R (map R S A B))

Tactic:
by
  convert!
    (TensorProduct.isBaseChange R Ω[A⁄R] S).comp
      (IsBaseChange.ofEquiv (tensorKaehlerEquivBase R S A B))
  refine LinearMap.ext fun x ↦ ?_
  simp only [LinearMap.coe_restrictScalars, LinearMap.coe_comp, LinearEquiv.coe_coe,
    Function.comp_apply, mk_apply, tensorKaehlerEquivBase_tmul, one_smul]

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

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
I : Ideal R
M : Type u_3
inst✝¹ : AddCommGroup M
inst✝ : Module R M
r : R ⧸ I • ⊤
x : M ⧸ I • ⊤
b₁ b₂ : R
h : b₁ ≈ b₂
⊢ b₁ • x = b₂ • x

Tactic:
induction x using Quotient.inductionOn'

Goal state after:
case h
R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
I : Ideal R
M : Type u_3
inst✝¹ : AddCommGroup M
inst✝ : Module R M
r : R ⧸ I • ⊤
b₁ b₂ : R
h : b₁ ≈ b₂
a✝ : M
⊢ b₁ • Quotient.mk'' a✝ = b₂ • Quotient.mk'' a✝
Module: Mathlib.RingTheory.AdicCompletion.Algebra

Goal state before:
pp: R✝ : Type u_1
S✝ : Type u_2
inst✝⁵ : CommRing R✝
inst✝⁴ : CommRing S✝
I✝ : Ideal R✝
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module R✝ M
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
⊢ ⟨fun n ↦ (factor ⋯) ((f n) 0), ⋯⟩ = 0

Tactic:
simp only [map_zero]
    ext; simp

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✝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
ι : Type u_5
κ : Type u_6
v : ι → M
a : κ → ι
hainj : Function.Injective a
hsa : ∀ (i : κ), s (v (a i)) = 0
hli : LinearIndependent R v
⊢ Disjoint (Submodule.span R (Set.range (v ∘ a))) g.ker

Tactic:
Submodule.disjoint_def,

Goal state after:
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
ι : Type u_5
κ : Type u_6
v : ι → M
a : κ → ι
hainj : Function.Injective a
hsa : ∀ (i : κ), s (v (a i)) = 0
hli : LinearIndependent R v
⊢ ∀ x ∈ Submodule.span R (Set.range (v ∘ a)), x ∈ g.ker → x = 0
Module: Mathlib.RingTheory.Extension.Presentation.Submersive

Goal state before:
pp: case intro.intro
R : Type u
S : Type v
ι : Type w
σ : Type t
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
P : PreSubmersivePresentation R S ι σ
ι' : Type u_1
σ' : Type u_2
e : ι' ≃ ι
f : σ' ≃ σ
inst✝¹ : Finite σ
inst✝ : Finite σ'
val✝¹ : Fintype σ
val✝ : Fintype σ'
⊢ (P.reindex e f).jacobian = P.jacobian

Tactic:
simp_rw [PreSubmersivePresentation.jacobian_eq_jacobiMatrix_det]

Goal state after:
case intro.intro
R : Type u
S : Type v
ι : Type w
σ : Type t
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
P : PreSubmersivePresentation R S ι σ
ι' : Type u_1
σ' : Type u_2
e : ι' ≃ ι
f : σ' ≃ σ
inst✝¹ : Finite σ
inst✝ : Finite σ'
val✝¹ : Fintype σ
val✝ : Fintype σ'
⊢ (algebraMap (P.reindex e f).Ring S) (P.reindex e f).jacobiMatrix.det = (algebraMap P.Ring S) P.jacobiMatrix.det
Module: Mathlib.RingTheory.Extension.Presentation.Core

Goal state before:
pp: R : Type u_1
S : Type u_2
ι : Type u_3
σ : Type u_4
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : Finite σ
P : SubmersivePresentation R S ι σ
inst✝¹ : DecidableEq σ
inst✝ : Fintype σ
⊢ ∃ v, ∑ i, v i * P.relation i = P.jacobiMatrix.det * P.σ ↑⋯.unit⁻¹ - 1

Tactic:
have H : P.jacobiMatrix.det * P.σ ↑(P.jacobian_isUnit.unit⁻¹) - 1 ∈ P.ker := by
    simp [PreSubmersivePresentation.jacobian_eq_jacobiMatrix_det]

Goal state after:
R : Type u_1
S : Type u_2
ι : Type u_3
σ : Type u_4
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : Finite σ
P : SubmersivePresentation R S ι σ
inst✝¹ : DecidableEq σ
inst✝ : Fintype σ
H : P.jacobiMatrix.det * P.σ ↑⋯.unit⁻¹ - 1 ∈ P.ker
⊢ ∃ v, ∑ i, v i * P.relation i = P.jacobiMatrix.det * P.σ ↑⋯.unit⁻¹ - 1
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula

Goal state before:
pp: F : Type u
inst✝ : Field F
W : Projective F
P : Fin 3 → F
hPz : P z ≠ 0
n d : F
hd : d ≠ 0
⊢ -n / P z / d * (W.toAffine.addX (P x / P z) (P x / P z) (-n / P z / d) - P x / P z) + P y / P z =
    (-n * (n ^ 2 - W.a₁ * n * P z * d - W.a₂ * P z ^ 2 * d ^ 2 - 2 * P x * P z * d ^ 2 - P x * P z * d ^ 2) +
          P y * P z ^ 2 * d ^ 3) /
        P z ^ 2 /
      (P z * d ^ 3)

Tactic:
toAffine_addX_of_eq hPz hd

Goal state after:
F : Type u
inst✝ : Field F
W : Projective F
P : Fin 3 → F
hPz : P z ≠ 0
n d : F
hd : d ≠ 0
⊢ -n / P z / d *
        ((n ^ 2 - W.a₁ * n * P z * d - W.a₂ * P z ^ 2 * d ^ 2 - 2 * P x * P z * d ^ 2) * d / P z / (P z * d ^ 3) -
          P x / P z) +
      P y / P z =
    (-n * (n ^ 2 - W.a₁ * n * P z * d - W.a₂ * P z ^ 2 * d ^ 2 - 2 * P x * P z * d ^ 2 - P x * P z * d ^ 2) +
          P y * P z ^ 2 * d ^ 3) /
        P z ^ 2 /
      (P z * d ^ 3)
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula

Goal state before:
pp: case a
R : Type r
inst✝ : CommRing R
W' : Projective R
P Q : Fin 3 → R
hP : W'.Equation P
hQ : W'.Equation Q
⊢ W'.addZ P Q * (P z * Q z) +
        (Q z ^ 3 * 0 +
          P z ^ 3 *
            (Q y ^ 2 * Q z + W'.a₁ * Q x * Q y * Q z + W'.a₃ * Q y * Q z ^ 2 -
              (Q x ^ 3 + W'.a₂ * Q x ^ 2 * Q z + W'.a₄ * Q x * Q z ^ 2 + W'.a₆ * Q z ^ 3))) -
      ((P x * Q z - Q x * P z) ^ 3 +
        (Q z ^ 3 *
            (P y ^ 2 * P z + W'.a₁ * P x * P y * P z + W'.a₃ * P y * P z ^ 2 -
              (P x ^ 3 + W'.a₂ * P x ^ 2 * P z + W'.a₄ * P x * P z ^ 2 + W'.a₆ * P z ^ 3)) +
          P z ^ 3 * 0)) =
    0

Tactic:
rw [addZ]; ring1

Goal state after:
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] := Finsupp.linearCombination S fun i ↦ (D R S) (a i)
l₁ : (κ →₀ S) →ₗ[S] κ →₀ Localization.AtPrime p :=
  Finsupp.mapRange.linearMap (Algebra.linearMap S (Localization.AtPrime p))
l₂ : Ω[S⁄R] →ₗ[S] Ω[Localization.AtPrime p⁄R] := map R R S (Localization.AtPrime p)
eₚ : (κ →₀ Localization.AtPrime p) →ₗ[Localization.AtPrime p] Ω[Localization.AtPrime p⁄R] :=
  (IsLocalizedModule.mapExtendScalars p.primeCompl l₁ l₂ (Localization.AtPrime p)) e
this : eₚ = ↑b.repr.symm
heₚ : Function.Bijective ⇑eₚ
⊢ ∃ f ∉ p, IsStandardSmooth R (Localization.Away f)

Tactic:
have : Finite κ := Module.Finite.finite_basis b

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] := Finsupp.linearCombination S fun i ↦ (D R S) (a i)
l₁ : (κ →₀ S) →ₗ[S] κ →₀ Localization.AtPrime p :=
  Finsupp.mapRange.linearMap (Algebra.linearMap S (Localization.AtPrime p))
l₂ : Ω[S⁄R] →ₗ[S] Ω[Localization.AtPrime p⁄R] := map R R S (Localization.AtPrime p)
eₚ : (κ →₀ Localization.AtPrime p) →ₗ[Localization.AtPrime p] Ω[Localization.AtPrime p⁄R] :=
  (IsLocalizedModule.mapExtendScalars p.primeCompl l₁ l₂ (Localization.AtPrime p)) e
this✝ : eₚ = ↑b.repr.symm
heₚ : Function.Bijective ⇑eₚ
this : Finite κ
⊢ ∃ f ∉ p, IsStandardSmooth R (Localization.Away f)
Module: Mathlib.RingTheory.Smooth.NoetherianDescent

Goal state before:
pp: case refine_1
R : Type u_1
inst✝⁴ : CommRing R
A : Type u
B : Type u_2
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra A B
D : DescentAux A B
p : D.vars → MvPolynomial D.vars ↥(subalgebra R D)
hp : ∀ (i : D.vars), (MvPolynomial.map (algebraMap (↥(subalgebra R D)) A)) (p i) = D.h i
i : D.rels
hinj : Function.Injective ⇑(MvPolynomial.map (algebraMap (↥(subalgebra R D)) A))
⊢ (aeval p) (Presentation.relationOfHasCoeffs (↥(subalgebra R D)) i) ∈
    Ideal.span (Set.range (Presentation.relationOfHasCoeffs ↥(subalgebra R D))) ^ 2

Tactic:
rw [Ideal.mem_span_pow_iff_exists_isHomogeneous]

Goal state after:
case refine_1
R : Type u_1
inst✝⁴ : CommRing R
A : Type u
B : Type u_2
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : CommRing B
inst✝ : Algebra A B
D : DescentAux A B
p : D.vars → MvPolynomial D.vars ↥(subalgebra R D)
hp : ∀ (i : D.vars), (MvPolynomial.map (algebraMap (↥(subalgebra R D)) A)) (p i) = D.h i
i : D.rels
hinj : Function.Injective ⇑(MvPolynomial.map (algebraMap (↥(subalgebra R D)) A))
⊢ ∃ p_1,
    p_1.IsHomogeneous 2 ∧
      (eval (Presentation.relationOfHasCoeffs ↥(subalgebra R D))) p_1 =
        (aeval p) (Presentation.relationOfHasCoeffs (↥(subalgebra R D)) i)
Module: Mathlib.AlgebraicGeometry.Morphisms.Smooth

Goal state before:
pp: n m : ℕ
X Y : Scheme
f : X ⟶ Y
⊢ HasRingHomProperty (@SmoothOfRelativeDimension n) fun {R S} [CommRing R] [CommRing S] ↦
    Locally fun {R S} [CommRing R] [CommRing S] ↦ IsStandardSmoothOfRelativeDimension n

Tactic:
apply HasRingHomProperty.locally_of_iff
  · exact (isStandardSmoothOfRelativeDimension_localizationPreserves n).away
  · exact isStandardSmoothOfRelativeDimension_stableUnderCompositionWithLocalizationAway n
  · intro X Y f
    rw [smoothOfRelativeDimension_iff]
    congr!
    simp [Subtype.exists]
    grind [Scheme.affineOpens]

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
hQ : W.Equation Q
hPz : P z ≠ 0
hQz : Q z ≠ 0
⊢ W.negAddY P Q =
    ((P y * Q z - Q y * P z) *
          ((P y * Q z - Q y * P z) ^ 2 * P z * Q z +
                    W.a₁ * (P y * Q z - Q y * P z) * P z * Q z * (P x * Q z - Q x * P z) -
                  W.a₂ * P z * Q z * (P x * Q z - Q x * P z) ^ 2 -
                P x * Q z * (P x * Q z - Q x * P z) ^ 2 -
              Q x * P z * (P x * Q z - Q x * P z) ^ 2 -
            P x * Q z * (P x * Q z - Q x * P z) ^ 2) +
        P y * Q z * (P x * Q z - Q x * P z) ^ 3) /
      (P z * Q z) ^ 2

Tactic:
by
  rw [← negAddY_eq' hP hQ, mul_div_cancel_right₀ _ <| pow_ne_zero 2 <| mul_ne_zero hPz hQz]

Goal state after:
Module: Mathlib.FieldTheory.SeparablyGenerated

Goal state before:
pp: k : Type u_1
K : Type u_2
ι : Type u_3
inst✝² : Field k
inst✝¹ : Field K
inst✝ : Algebra k K
a : ι → K
F : MvPolynomial ι k
HF : ∀ (F' : MvPolynomial ι k), F' ≠ 0 → (aeval a) F' = 0 → F.totalDegree ≤ F'.totalDegree
hF0 : F ≠ 0
hFa : (aeval a) F = 0
⊢ Irreducible F

Tactic:
refine ⟨fun h' ↦ (h'.map (aeval a)).ne_zero hFa, fun q₁ q₂ e ↦ ?_⟩
  wlog h₁ : aeval a q₁ = 0 generalizing q₁ q₂
  · exact .symm (this q₂ q₁ (e.trans (mul_comm ..)) <| by
      simpa [h₁, hFa] using Eq.symm <| congr(aeval a $e))
  have ne := mul_ne_zero_iff.mp (e ▸ hF0)
  have := HF q₁ ne.1 h₁
  rw [e, totalDegree_mul_of_isDomain ne.1 ne.2, add_le_iff_nonpos_right, nonpos_iff_eq_zero] at this
  refine .inr (isUnit_iff_totalDegree_of_isReduced.mpr ⟨?_, this⟩)
  rw [totalDegree_eq_zero_iff_eq_C.mp this] at ne
  simpa using ne.2

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

Goal state before:
pp: W P Q : MorphismProperty Scheme
X Y : Scheme
inst✝³ : W.IsStableUnderBaseChange
inst✝² : Scheme.IsJointlySurjectivePreserving W
inst✝¹ : P.RespectsIso
inst✝ : P.RespectsLeft IsOpenImmersion
⊢ IsZariskiLocalAtSource (sourceLocalClosure IsOpenImmersion P)

Tactic:
refine .mk_of_iff_of_zeroHypercover fun f 𝒰 ↦ ?_

Goal state after:
W P Q : MorphismProperty Scheme
X Y : Scheme
inst✝³ : W.IsStableUnderBaseChange
inst✝² : Scheme.IsJointlySurjectivePreserving W
inst✝¹ : P.RespectsIso
inst✝ : P.RespectsLeft IsOpenImmersion
X✝ Y✝ : Scheme
f : X✝ ⟶ Y✝
𝒰 : Scheme.zariskiPrecoverage.ZeroHypercover X✝
⊢ sourceLocalClosure IsOpenImmersion P f ↔ ∀ (i : 𝒰.I₀), sourceLocalClosure IsOpenImmersion P (𝒰.f i ≫ f)
Module: Mathlib.AlgebraicGeometry.Morphisms.LocalClosure

Goal state before:
pp: W P Q : MorphismProperty Scheme
X✝ Y✝ : Scheme
inst✝⁴ : W.IsStableUnderBaseChange
inst✝³ : Scheme.IsJointlySurjectivePreserving W
inst✝² : W.IsStableUnderComposition
inst✝¹ : P.IsStableUnderBaseChange
inst✝ : P.IsStableUnderComposition
X Y Z : Scheme
f : X ⟶ Y
g : Y ⟶ Z
x✝² : sourceLocalClosure W P f
x✝¹ : sourceLocalClosure W P g
𝒰 : Scheme.Cover (Scheme.precoverage W) X
hf : ∀ (i : 𝒰.I₀), P (𝒰.f i ≫ f)
𝒱 : Scheme.Cover (Scheme.precoverage W) Y
hg : ∀ (i : 𝒱.I₀), P (𝒱.f i ≫ g)
x✝ : (Precoverage.ZeroHypercover.bind 𝒰 fun i ↦ Precoverage.ZeroHypercover.pullback₁ (𝒰.f i ≫ f) 𝒱).I₀
l : 𝒰.I₀
r : ((fun i ↦ ((fun i ↦ Precoverage.ZeroHypercover.pullback₁ (𝒰.f i ≫ f) 𝒱) i).toPreZeroHypercover) l).I₀
⊢ P ((Precoverage.ZeroHypercover.bind 𝒰 fun i ↦ Precoverage.ZeroHypercover.pullback₁ (𝒰.f i ≫ f) 𝒱).f ⟨l, r⟩ ≫ f ≫ g)

Tactic:
simpa [← pullbackRightPullbackFstIso_inv_snd_fst_assoc, pullback.condition_assoc] using
    P.comp_mem _ _ (P.pullback_snd _ _ (hf _)) (hg r)

Goal state after:
Module: Mathlib.FieldTheory.SeparablyGenerated

Goal state before:
pp: k : Type u_1
K : Type u_2
ι : Type u_3
inst✝² : Field k
inst✝¹ : Field K
inst✝ : Algebra k K
p : ℕ
hp : Nat.Prime p
a : ι → K
F : MvPolynomial ι k
HF : ∀ (F' : MvPolynomial ι k), F' ≠ 0 → (aeval a) F' = 0 → F.totalDegree ≤ F'.totalDegree
hF0 : F ≠ 0
hFa : (aeval a) F = 0
this : ∀ (i : ι), ∀ σ ∈ F.support, p ∣ σ i
σ' : (ι →₀ ℕ) → ι →₀ ℕ
hσ' : ∀ σ ∈ F.support, σ = p • σ' σ
hσ'' : ∀ (σ : ↥F.support), ↑σ = p • σ' ↑σ
H : ∀ (ι : Type u_3) (x : Fintype ι) (v : ι → K), LinearIndependent k v → LinearIndependent k fun x ↦ v x ^ p
F' : ↥F.support →₀ k
hF' : (Finsupp.linearCombination k fun s ↦ (aeval a) ((monomial (σ' ↑s)) 1)) F' = 0
hF'0 : ¬F' = 0
F'' : MvPolynomial ι k := AddMonoidAlgebra.ofCoeff (Finsupp.mapDomain (fun s ↦ σ' ↑s) F')
hF''0 : F'' ≠ 0
hF'' : (aeval a) F'' = 0
σ : ↥F.support
hσ₂ : σ ∈ F'.support
hσ : σ' ↑σ ∈ F''.support
⊢ (p * (σ' ↑σ).sum fun x e ↦ e) ≤ (↑σ).sum fun x e ↦ e

Tactic:
conv_rhs => rw [hσ' _ σ.2, Finsupp.sum_smul_index (fun _ ↦ rfl), ← Finsupp.mul_sum]

Goal state after:
Module: Mathlib.RingTheory.Flat.FaithfullyFlat.Descent

Goal state before:
pp: ⊢ CodescendsAlong (fun {R S} [CommRing R] [CommRing S] f ↦ Function.Surjective ⇑f) fun {R S} [CommRing R] [CommRing S] ↦
    FaithfullyFlat

Tactic:
apply CodescendsAlong.mk _ surjective_respectsIso

Goal state after:
⊢ ∀ ⦃R S T : Type u_1⦄ [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S]
    [inst_4 : Algebra R T],
    (algebraMap R S).FaithfullyFlat →
      Function.Surjective ⇑(algebraMap S (S ⊗[R] T)) → Function.Surjective ⇑(algebraMap R T)