mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
n : σ →₀ ℕ
r : R
H : (monomial n) r ∈ nonZeroDivisorsRight (MvPowerSeries σ R)
s : R
hrs : s * r = 0
⊢ C s * (monomial n) r = 0

Tactic:
rw [← monomial_zero_eq_C, monomial_mul_monomial]; ext; simp [hrs]

Goal state after:
Module: Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝¹ : Semiring R
inst✝ : NoZeroDivisors R
w : σ → ℕ
f g : MvPowerSeries σ R
hf : weightedOrder w f < ⊤
hg : weightedOrder w g < ⊤
p : ℕ := (weightedOrder w f).toNat
hp : ↑p = weightedOrder w f
q : ℕ := (weightedOrder w g).toNat
hq : ↑q = weightedOrder w g
H : (weightedHomogeneousComponent w p) f = 0
⊢ weightedOrder w f < ?m.147

Tactic:
by simpa

Goal state after:
Module: Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝¹ : Semiring R
inst✝ : NoZeroDivisors R
w : σ → ℕ
f g : MvPowerSeries σ R
hf : weightedOrder w f < ⊤
hg : weightedOrder w g < ⊤
p : ℕ := (weightedOrder w f).toNat
hp : ↑p = weightedOrder w f
q : ℕ := (weightedOrder w g).toNat
hq : ↑q = weightedOrder w g
H : (weightedHomogeneousComponent w q) g = 0
⊢ weightedOrder w g < ?m.171

Tactic:
by simpa

Goal state after:
Module: Mathlib.RingTheory.PowerSeries.NoZeroDivisors

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
a : R
ha : a ≠ 0
p q : R⟦X⟧
n : ℕ
h : (coeff n) ((rescale a) p) = (coeff n) ((rescale a) q)
⊢ (coeff n) p = (coeff n) q

Tactic:
coeff_rescale,

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
a : R
ha : a ≠ 0
p q : R⟦X⟧
n : ℕ
h : a ^ n * (coeff n) p = (coeff n) ((rescale a) q)
⊢ (coeff n) p = (coeff n) q
Module: Mathlib.RingTheory.PowerSeries.NoZeroDivisors

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
a : R
ha : a ≠ 0
p q : R⟦X⟧
n : ℕ
h : a ^ n * (coeff n) p = (coeff n) ((rescale a) q)
⊢ (coeff n) p = (coeff n) q

Tactic:
coeff_rescale,

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
a : R
ha : a ≠ 0
p q : R⟦X⟧
n : ℕ
h : a ^ n * (coeff n) p = a ^ n * (coeff n) q
⊢ (coeff n) p = (coeff n) q
Module: Mathlib.AlgebraicGeometry.EllipticCurve.ModelsWithJ

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
⊢ (ofJ0 R).c₄ = 0

Tactic:
by
  rw [ofJ0, c₄, b₂, b₄]
  norm_num1

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.ModelsWithJ

Goal state before:
pp: case pos
R : Type u_1
inst✝³ : CommRing R
W : WeierstrassCurve R
j✝ : R
inst✝² : W.IsElliptic
F : Type u_2
inst✝¹ : Field F
j : F
inst✝ : DecidableEq F
h0 : j = 0
h3 : 3 = 0
this : Fact (IsUnit 2)
⊢ (ofJ 0).IsElliptic

Tactic:
ofJ_0_of_three_eq_zero h3

Goal state after:
case pos
R : Type u_1
inst✝³ : CommRing R
W : WeierstrassCurve R
j✝ : R
inst✝² : W.IsElliptic
F : Type u_2
inst✝¹ : Field F
j : F
inst✝ : DecidableEq F
h0 : j = 0
h3 : 3 = 0
this : Fact (IsUnit 2)
⊢ (ofJ1728 F).IsElliptic
Module: Mathlib.CategoryTheory.Sites.Hypercover.SheafOfTypes

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : C
E : PreOneHypercover X
F : Cᵒᵖ ⥤ Type u_2
h₁ : E.IsStronglySheafFor F
h₂ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), Presieve.IsSeparatedFor F (Sieve.pullback f E.sieve₀).arrows
S : Sieve X
H : ∀ (i : E.I₀), Presieve.IsSheafFor F (Sieve.pullback (E.f i) S).arrows
H' : ∀ ⦃i j : E.I₀⦄ (k : E.I₁ i j), Presieve.IsSeparatedFor F (Sieve.pullback (E.p₁ k ≫ E.f i) S).arrows
t : Presieve.FamilyOfElements F S.arrows
ht : t.Compatible
s : (i : E.I₀) → F.obj (op (E.X i))
hs : ∀ (i : E.I₀), (fun t_1 ↦ (Presieve.FamilyOfElements.pullback (E.f i) t).IsAmalgamation t_1) (s i)
huniq :
  ∀ (i : E.I₀) (y : F.obj (op (E.X i))),
    (fun t_1 ↦ (Presieve.FamilyOfElements.pullback (E.f i) t).IsAmalgamation t_1) y → y = s i
i j : E.I₀
Z : C
gi : Z ⟶ E.X i
gj : Z ⟶ E.X j
hgij : gi ≫ E.f i = gj ≫ E.f j
Y : C
f : Y ⟶ Z
x✝ : (E.sieve₁ gi gj).arrows f
k : E.I₁ i j
h : Y ⟶ E.Y k
hf₁ : f ≫ gi = h ≫ E.p₁ k
hf₂ : f ≫ gj = h ≫ E.p₂ k
W : C
p : W ⟶ E.Y k
hp : (Sieve.pullback (E.p₁ k ≫ E.f i) S).arrows p
⊢ (Sieve.pullback (E.f i) S).arrows (p ≫ E.p₁ k)

Tactic:
by simpa

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

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W' : Projective R
P : Fin 3 → R
⊢ (eval P) W'.polynomialZ =
    P y ^ 2 + W'.a₁ * P x * P y + 2 * W'.a₃ * P y * P z -
      (W'.a₂ * P x ^ 2 + 2 * W'.a₄ * P x * P z + 3 * W'.a₆ * P z ^ 2)

Tactic:
polynomialZ_eq

Goal state after:
R : Type r
inst✝ : CommRing R
W' : Projective R
P : Fin 3 → R
⊢ (eval P)
      (Y ^ 2 + C W'.a₁ * X * Y + C (2 * W'.a₃) * Y * Z -
        (C W'.a₂ * X ^ 2 + C (2 * W'.a₄) * X * Z + C (3 * W'.a₆) * Z ^ 2)) =
    P y ^ 2 + W'.a₁ * P x * P y + 2 * W'.a₃ * P y * P z -
      (W'.a₂ * P x ^ 2 + 2 * W'.a₄ * P x * P z + 3 * W'.a₆ * P z ^ 2)
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic

Goal state before:
pp: R : Type r
inst✝¹ : CommRing R
W' : Projective R
S : Type s
inst✝ : CommRing S
f : R →+* S
hf : Function.Injective ⇑f
P : Fin 3 → R
⊢ (W'.map f).Equation (⇑f ∘ P) ↔ W'.Equation P

Tactic:
by
  simp only [Equation, map_polynomial, eval_map, ← eval₂_comp, map_eq_zero_iff f hf]

Goal state after:
Module: Mathlib.RingTheory.Unramified.Pi

Goal state before:
pp: case intro.mpr
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
⊢ f₁ (Pi.single x (g x)) = f₂ (Pi.single x (g x))

Tactic:
let e : ∀ i, f i := Pi.single x 1

Goal state after:
case intro.mpr
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
⊢ f₁ (Pi.single x (g x)) = f₂ (Pi.single x (g x))
Module: Mathlib.RingTheory.Kaehler.TensorProduct

Goal state before:
pp: case smul
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
x : S ⊗[R] Ω[A⁄R]
s : S
b : B
e : (LinearMap.liftBaseChange S (↑R (map R S A B))) (b • x) = b • (LinearMap.liftBaseChange S (↑R (map R S A B))) x
⊢ (LinearMap.liftBaseChange S (↑R (map R S A B))) (s • b • x) =
    (s • b) • (LinearMap.liftBaseChange S (↑R (map R S A B))) x

Tactic:
map_smul,

Goal state after:
case smul
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
x : S ⊗[R] Ω[A⁄R]
s : S
b : B
e : (LinearMap.liftBaseChange S (↑R (map R S A B))) (b • x) = b • (LinearMap.liftBaseChange S (↑R (map R S A B))) x
⊢ s • (LinearMap.liftBaseChange S (↑R (map R S A B))) (b • x) =
    (s • b) • (LinearMap.liftBaseChange S (↑R (map R S A B))) x
Module: Mathlib.RingTheory.Kaehler.TensorProduct

Goal state before:
pp: case smul
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
b : B
x : S
y : B
e :
  (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) (y * b) =
    y • (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) b + b • (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) y
⊢ (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) (x • (y * b)) =
    (x • y) • (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) b + b • (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) (x • y)

Tactic:
map_smul,

Goal state after:
case smul
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
b : B
x : S
y : B
e :
  (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) (y * b) =
    y • (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) b + b • (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) y
⊢ x • (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) (y * b) =
    (x • y) • (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) b + b • (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) (x • y)
Module: Mathlib.RingTheory.Kaehler.TensorProduct

Goal state before:
pp: case smul
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
b : B
x : S
y : B
e :
  (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) (y * b) =
    y • (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) b + b • (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) y
⊢ x • (y • (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) b + b • (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) y) =
    (x • y) • (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) b + b • (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) (x • y)

Tactic:
map_smul,

Goal state after:
case smul
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
b : B
x : S
y : B
e :
  (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) (y * b) =
    y • (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) b + b • (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) y
⊢ x • (y • (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) b + b • (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) y) =
    (x • y) • (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) b + b • x • (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) y
Module: Mathlib.RingTheory.Kaehler.TensorProduct

Goal state before:
pp: case tmul.smul
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
z : A
s✝ : S
n✝ : B
e :
  (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) ((algebraMap A B) z * n✝) =
    (algebraMap A B) z • (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) n✝ +
      n✝ • (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) ((algebraMap A B) z)
⊢ (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) (s✝ • (n✝ * (algebraMap A B) z)) =
    (algebraMap A B) z • (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) (s✝ • n✝) +
      (s✝ • n✝) • (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) ((algebraMap A B) z)

Tactic:
map_smul,

Goal state after:
case tmul.smul
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
z : A
s✝ : S
n✝ : B
e :
  (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) ((algebraMap A B) z * n✝) =
    (algebraMap A B) z • (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) n✝ +
      n✝ • (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) ((algebraMap A B) z)
⊢ s✝ • (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) (n✝ * (algebraMap A B) z) =
    (algebraMap A B) z • (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) (s✝ • n✝) +
      (s✝ • n✝) • (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) ((algebraMap A B) z)
Module: Mathlib.RingTheory.Kaehler.TensorProduct

Goal state before:
pp: case tmul.smul
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
z : A
s✝ : S
n✝ : B
e :
  (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) ((algebraMap A B) z * n✝) =
    (algebraMap A B) z • (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) n✝ +
      n✝ • (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) ((algebraMap A B) z)
⊢ s✝ •
      ((algebraMap A B) z • (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) n✝ +
        n✝ • (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) ((algebraMap A B) z)) =
    (algebraMap A B) z • (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) (s✝ • n✝) +
      (s✝ • n✝) • (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) ((algebraMap A B) z)

Tactic:
map_smul,

Goal state after:
case tmul.smul
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
z : A
s✝ : S
n✝ : B
e :
  (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) ((algebraMap A B) z * n✝) =
    (algebraMap A B) z • (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) n✝ +
      n✝ • (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) ((algebraMap A B) z)
⊢ s✝ •
      ((algebraMap A B) z • (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) n✝ +
        n✝ • (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) ((algebraMap A B) z)) =
    (algebraMap A B) z • s✝ • (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) n✝ +
      (s✝ • n✝) • (⋯.lift ((mk R S Ω[A⁄R]) 1 ∘ₗ ↑(D R A))) ((algebraMap A B) z)
Module: Mathlib.RingTheory.Smooth.Fiber

Goal state before:
pp: R : Type u_1
S : Type u_2
P : Type u_3
inst✝¹³ : CommRing R
inst✝¹² : CommRing S
inst✝¹¹ : Algebra R S
inst✝¹⁰ : Module.Flat R S
inst✝⁹ : CommRing P
inst✝⁸ : Algebra R P
inst✝⁷ : Algebra P S
inst✝⁶ : IsScalarTower R P S
inst✝⁵ : IsLocalRing R
inst✝⁴ : IsLocalRing S
inst✝³ : IsLocalHom (algebraMap R S)
inst✝² : FormallySmooth 𝓀[R] (𝓀[R] ⊗[R] S)
M : Submonoid P
inst✝¹ : IsLocalization M S
inst✝ : FinitePresentation R P
n : ℕ
f₀ : MvPolynomial (Fin n) R →ₐ[R] P
hf₀ : Function.Surjective ⇑f₀
M' : Submonoid (MvPolynomial (Fin n) R) := Submonoid.comap f₀ M
P' : Type u_1 := Localization M'
fP : P' →ₐ[R] S := IsLocalization.liftAlgHom ⋯
hf₁ : Function.Surjective ⇑fP
this : (RingHom.ker f₀.toRingHom).FG
x : MvPolynomial (Fin n) R
s : ↥M'
hx : IsLocalization.mk' P' x s ∈ RingHom.ker fP
a : MvPolynomial (Fin n) R
ha : f₀ a ∈ M
e : f₀ a * f₀ x = 0
⊢ a * x ∈ RingHom.ker f₀.toRingHom

Tactic:
by simpa

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

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
S : Type u_3
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : FormallySmooth R A
f : S →ₐ[R] A
hf : Function.Surjective ⇑f
⊢ ∃ g, (AdicCompletion.kerProj hf).comp g = AlgHom.id R A

Tactic:
obtain ⟨g, hg⟩ := exists_adicCompletionEvalOneₐ_comp_eq
    (Ideal.quotientKerAlgEquivOfSurjective hf).symm.toAlgHom

Goal state after:
R : Type u_1
A : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
S : Type u_3
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : FormallySmooth R A
f : S →ₐ[R] A
hf : Function.Surjective ⇑f
g : A →ₐ[R] AdicCompletion (ker f) S
hg :
  (AlgHom.restrictScalars R (AdicCompletion.evalOneₐ (ker f))).comp g = ↑(Ideal.quotientKerAlgEquivOfSurjective hf).symm
⊢ ∃ g, (AdicCompletion.kerProj hf).comp g = AlgHom.id R A
Module: Mathlib.LinearAlgebra.Basis.Exact

Goal state before:
pp: case mem
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
y : K
x m : M
hm : m ∈ Set.range (v ∘ a)
⊢ s m = 0

Tactic:
obtain ⟨i, rfl⟩ := hm

Goal state after:
case mem
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
y : K
x : M
i : κ
⊢ s ((v ∘ a) i) = 0
Module: Mathlib.RingTheory.Smooth.StandardSmooth

Goal state before:
pp: n : ℕ
R : Type u
S : Type v
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
T : Type u_1
inst✝² : CommRing T
inst✝¹ : Algebra R T
e : S ≃ₐ[R] T
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
⊢ (P.ofAlgEquiv e).dimension = n

Tactic:
by simpa

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
hfg : Function.Exact ⇑f ⇑g
p q : Submodule R M
hpq : IsCompl p q
hmap : map g q = ⊤
x : ↥p
triv : x ∈ ⊤
a : M
haq : a ∈ ↑q
ha : g a = g ↑x
⊢ (p.projectionOnto q hpq) (↑x - a) = x

Tactic:
by simpa

Goal state after:
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✝ : IsStandardSmoothOfRelativeDimension 0 R S
⊢ Subsingleton Ω[S⁄R]

Tactic:
cases subsingleton_or_nontrivial S

Goal state after:
case inl
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✝ : IsStandardSmoothOfRelativeDimension 0 R S
h✝ : Subsingleton S
⊢ Subsingleton Ω[S⁄R]

case inr
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✝ : IsStandardSmoothOfRelativeDimension 0 R S
h✝ : Nontrivial S
⊢ Subsingleton Ω[S⁄R]
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)
f : P.toExtension.Cotangent →ₗ[S] ι →₀ S := ↑P.cotangentSpaceBasis.repr ∘ₗ P.toExtension.cotangentComplex
g : (ι →₀ S) →ₗ[S] Ω[S⁄R] := P.toExtension.toKaehler ∘ₗ ↑P.cotangentSpaceBasis.repr.symm
hfg : Function.Exact ⇑f ⇑g
⊢ Function.Bijective
    ⇑(LinearMap.linearProjOfIsCompl (Submodule.span S (Set.range fun x ↦ Finsupp.single (v x) 1))
          (Finsupp.lmapDomain S S u) ⋯ ⋯ ∘ₗ
        f)

Tactic:
apply LinearMap.linearProjOfIsCompl_comp_bijective_of_exact hfg

Goal state after:
case hf
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)
f : P.toExtension.Cotangent →ₗ[S] ι →₀ S := ⋯
g : (ι →₀ S) →ₗ[S] Ω[S⁄R] := ⋯
hfg : Function.Exact ⇑f ⇑g
⊢ Function.Injective ⇑f

case hker
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)
f : P.toExtension.Cotangent →ₗ[S] ι →₀ S := ⋯
g : (ι →₀ S) →ₗ[S] Ω[S⁄R] := ⋯
hfg : Function.Exact ⇑f ⇑g
⊢ Disjoint g.ker (Submodule.span S (Set.range fun x ↦ Finsupp.single (v x) 1))

case hmap
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)
f : P.toExtension.Cotangent →ₗ[S] ι →₀ S := ⋯
g : (ι →₀ S) →ₗ[S] Ω[S⁄R] := ⋯
hfg : Function.Exact ⇑f ⇑g
⊢ Submodule.map g (Submodule.span S (Set.range fun x ↦ Finsupp.single (v x) 1)) = ⊤
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})
⊢ Ideal.map ((localizationAway T g).toComp P).toAlgHom P.ker ⊔
      Ideal.comap ((localizationAway T g).ofComp P).toAlgHom (localizationAway T g).ker =
    Ideal.map ((localizationAway T g).toComp P).toAlgHom P.ker ⊔ Ideal.span {(rename Sum.inr) f * X (Sum.inl ()) - 1}

Tactic:
Algebra.Generators.map_toComp_ker,

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.comap ((localizationAway T g).ofComp P).toAlgHom (localizationAway T g).ker =
    RingHom.ker ((localizationAway T g).ofComp P).toAlgHom ⊔ Ideal.span {(rename Sum.inr) f * X (Sum.inl ()) - 1}