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

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Ring R
φ : MvPowerSeries σ R
u : Rˣ
h : constantCoeff φ = ↑u
n : σ →₀ ℕ
this : DecidableEq (σ →₀ ℕ) := Classical.decEq (σ →₀ ℕ)
H : ¬n = 0
⊢ (0, n) ∈ antidiagonal n

Tactic:
rw [mem_antidiagonal, zero_add]

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Ring R
φ : MvPowerSeries σ R
u : Rˣ
h : constantCoeff φ = ↑u
n : σ →₀ ℕ
this✝ : DecidableEq (σ →₀ ℕ) := Classical.decEq (σ →₀ ℕ)
H : ¬n = 0
this : (0, n) ∈ antidiagonal n
⊢ ∑ p ∈ insert (0, n) ((antidiagonal n).erase (0, n)), (coeff p.1) φ * (coeff p.2) (φ.invOfUnit u) = 0

Tactic:
Finset.sum_insert (Finset.notMem_erase _ _),

Goal state after:
σ : Type u_1
R : Type u_2
inst✝ : Ring R
φ : MvPowerSeries σ R
u : Rˣ
h : constantCoeff φ = ↑u
n : σ →₀ ℕ
this✝ : DecidableEq (σ →₀ ℕ) := Classical.decEq (σ →₀ ℕ)
H : ¬n = 0
this : (0, n) ∈ antidiagonal n
⊢ (coeff (0, n).1) φ * (coeff (0, n).2) (φ.invOfUnit u) +
      ∑ x ∈ (antidiagonal n).erase (0, n), (coeff x.1) φ * (coeff x.2) (φ.invOfUnit u) =
    0
Module: Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors

Goal state before:
pp: case mp
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
n : σ →₀ ℕ
r : R
⊢ (monomial n) r ∈ nonZeroDivisorsRight (MvPowerSeries σ R) → r ∈ nonZeroDivisorsRight R

Tactic:
intro H

Goal state after:
case mp
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)
⊢ r ∈ nonZeroDivisorsRight R
Module: Mathlib.RingTheory.MvPowerSeries.Inverse

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Ring R
φ : MvPowerSeries σ R
u : Rˣ
h : constantCoeff φ = ↑u
n : σ →₀ ℕ
this✝ : DecidableEq (σ →₀ ℕ) := Classical.decEq (σ →₀ ℕ)
H : ¬n = 0
this : (0, n) ∈ antidiagonal n
⊢ (-∑ x ∈ insert (0, n) ((antidiagonal n).erase (0, n)),
          if x.2 < (0, n).2 then (coeff x.1) φ * (coeff x.2) (φ.invOfUnit u) else 0) +
      ∑ x ∈ (insert (0, n) ((antidiagonal n).erase (0, n))).erase (0, n), (coeff x.1) φ * (coeff x.2) (φ.invOfUnit u) =
    0

Tactic:
Finset.sum_insert (Finset.notMem_erase _ _),

Goal state after:
σ : Type u_1
R : Type u_2
inst✝ : Ring R
φ : MvPowerSeries σ R
u : Rˣ
h : constantCoeff φ = ↑u
n : σ →₀ ℕ
this✝ : DecidableEq (σ →₀ ℕ) := Classical.decEq (σ →₀ ℕ)
H : ¬n = 0
this : (0, n) ∈ antidiagonal n
⊢ -((if (0, n).2 < (0, n).2 then (coeff (0, n).1) φ * (coeff (0, n).2) (φ.invOfUnit u) else 0) +
          ∑ x ∈ (antidiagonal n).erase (0, n),
            if x.2 < (0, n).2 then (coeff x.1) φ * (coeff x.2) (φ.invOfUnit u) else 0) +
      ∑ x ∈ (insert (0, n) ((antidiagonal n).erase (0, n))).erase (0, n), (coeff x.1) φ * (coeff x.2) (φ.invOfUnit u) =
    0
Module: Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors

Goal state before:
pp: case mpr
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
n : σ →₀ ℕ
r : R
⊢ r ∈ nonZeroDivisorsRight R → (monomial n) r ∈ nonZeroDivisorsRight (MvPowerSeries σ R)

Tactic:
intro H

Goal state after:
case mpr
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
n : σ →₀ ℕ
r : R
H : r ∈ nonZeroDivisorsRight R
⊢ (monomial n) r ∈ nonZeroDivisorsRight (MvPowerSeries σ R)
Module: Mathlib.RingTheory.MvPowerSeries.Inverse

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Ring R
φ : MvPowerSeries σ R
u : Rˣ
h : constantCoeff φ = ↑u
n : σ →₀ ℕ
this✝ : DecidableEq (σ →₀ ℕ) := Classical.decEq (σ →₀ ℕ)
H : ¬n = 0
this : (0, n) ∈ antidiagonal n
⊢ (-∑ x ∈ (antidiagonal n).erase (0, n), if x.2 < (0, n).2 then (coeff x.1) φ * (coeff x.2) (φ.invOfUnit u) else 0) +
      ∑ x ∈ (antidiagonal n).erase (0, n), (coeff x.1) φ * (coeff x.2) (φ.invOfUnit u) =
    0

Tactic:
add_comm,

Goal state after:
σ : Type u_1
R : Type u_2
inst✝ : Ring R
φ : MvPowerSeries σ R
u : Rˣ
h : constantCoeff φ = ↑u
n : σ →₀ ℕ
this✝ : DecidableEq (σ →₀ ℕ) := Classical.decEq (σ →₀ ℕ)
H : ¬n = 0
this : (0, n) ∈ antidiagonal n
⊢ (∑ x ∈ (antidiagonal n).erase (0, n), (coeff x.1) φ * (coeff x.2) (φ.invOfUnit u) +
      -∑ x ∈ (antidiagonal n).erase (0, n), if x.2 < (0, n).2 then (coeff x.1) φ * (coeff x.2) (φ.invOfUnit u) else 0) =
    0
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
⊢ ¬((weightedHomogeneousComponent w p) f = 0 ∨ (weightedHomogeneousComponent w q) g = 0)

Tactic:
intro H

Goal state after:
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 ∨ (weightedHomogeneousComponent w q) g = 0
⊢ False
Module: Mathlib.Topology.Algebra.Valued.NormedValued

Goal state before:
pp: L : Type u_1
inst✝¹ : Field L
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
val : Valued L Γ₀
hv : v.RankOne
x x' : L
⊢ ‖x‖ < ‖x'‖ ↔ v x < v x'

Tactic:
← v.restrict_lt_iff,

Goal state after:
L : Type u_1
inst✝¹ : Field L
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
val : Valued L Γ₀
hv : v.RankOne
x x' : L
⊢ ‖x‖ < ‖x'‖ ↔ v.restrict x < v.restrict x'
Module: Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors

Goal state before:
pp: case pos
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
this : (weightedHomogeneousComponent w (p + q)) (f * g) ≠ 0
⊢ weightedOrder w (f * g) ≤ ↑p + ↑q

Tactic:
← Nat.cast_add,

Goal state after:
case pos
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
this : (weightedHomogeneousComponent w (p + q)) (f * g) ≠ 0
⊢ weightedOrder w (f * g) ≤ ↑(p + q)
Module: Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors

Goal state before:
pp: case pos
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
this : (weightedHomogeneousComponent w (p + q)) (f * g) ≠ 0
⊢ ¬↑(p + q) < weightedOrder w (f * g)

Tactic:
intro H

Goal state after:
case pos
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
this : (weightedHomogeneousComponent w (p + q)) (f * g) ≠ 0
H : ↑(p + q) < weightedOrder w (f * g)
⊢ False
Module: Mathlib.RingTheory.PowerSeries.Inverse

Goal state before:
pp: k : Type u_2
inst✝ : Field k
⊢ (Ideal.span {X}).IsMaximal

Tactic:
Ideal.isMaximal_iff

Goal state after:
k : Type u_2
inst✝ : Field k
⊢ 1 ∉ Ideal.span {X} ∧ ∀ (J : Ideal k⟦X⟧) (x : k⟦X⟧), Ideal.span {X} ≤ J → x ∉ Ideal.span {X} → x ∈ J → 1 ∈ J
Module: Mathlib.NumberTheory.NumberField.Completion.FinitePlace

Goal state before:
pp: K : Type u_1
inst✝⁶ : Field K
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : Algebra R K
inst✝³ : IsDedekindDomain R
inst✝² : IsFractionRing R K
v : HeightOneSpectrum R
inst✝¹ : Module.Finite ℤ R
inst✝ : Module.Free ℤ R
x : R
h_x_nezero : x ≠ 0
⊢ ‖(embedding v) ((algebraMap R K) x)‖ *
      ↑(absNorm (v.asIdeal ^ (Associates.mk v.asIdeal).count (Associates.mk (span {x})).factors)) =
    1

Tactic:
map_pow,

Goal state after:
K : Type u_1
inst✝⁶ : Field K
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : Algebra R K
inst✝³ : IsDedekindDomain R
inst✝² : IsFractionRing R K
v : HeightOneSpectrum R
inst✝¹ : Module.Finite ℤ R
inst✝ : Module.Free ℤ R
x : R
h_x_nezero : x ≠ 0
⊢ ‖(embedding v) ((algebraMap R K) x)‖ *
      ↑(absNorm v.asIdeal ^ (Associates.mk v.asIdeal).count (Associates.mk (span {x})).factors) =
    1
Module: Mathlib.NumberTheory.NumberField.Completion.FinitePlace

Goal state before:
pp: K : Type u_1
inst✝⁶ : Field K
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : Algebra R K
inst✝³ : IsDedekindDomain R
inst✝² : IsFractionRing R K
v : HeightOneSpectrum R
inst✝¹ : Module.Finite ℤ R
inst✝ : Module.Free ℤ R
x : R
h_x_nezero : x ≠ 0
⊢ ↑(absNorm v.asIdeal) ^
      (Multiplicative.toAdd (unzero ⋯) + ↑((Associates.mk v.asIdeal).count (Associates.mk (span {x})).factors)) =
    1

Tactic:
zpow_eq_one_iff_right₀ (Nat.cast_nonneg' _) (mod_cast (one_lt_absNorm_nnreal v).ne')

Goal state after:
K : Type u_1
inst✝⁶ : Field K
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : Algebra R K
inst✝³ : IsDedekindDomain R
inst✝² : IsFractionRing R K
v : HeightOneSpectrum R
inst✝¹ : Module.Finite ℤ R
inst✝ : Module.Free ℤ R
x : R
h_x_nezero : x ≠ 0
⊢ Multiplicative.toAdd (unzero ⋯) + ↑((Associates.mk v.asIdeal).count (Associates.mk (span {x})).factors) = 0
Module: Mathlib.AlgebraicGeometry.Geometrically.Connected

Goal state before:
pp: X S : Scheme
f : X ⟶ S
inst✝ : GeometricallyConnected f
hf : IsOpenMap ⇑f
s : Set ↥S
hs : _root_.IsConnected s
hs' : IsClosed s
⊢ Topology.IsCoinducing ⇑f

Tactic:
exact (hf.isQuotientMap f.continuous f.surjective).isCoinducing

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Hypercover.SheafOfTypes

Goal state before:
pp: case trans
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Limits.HasPullbacks C
K : Precoverage C
inst✝ : K.IsStableUnderBaseChange
S : C
F : Cᵒᵖ ⥤ Type u_2
𝒰 : K.ZeroHypercover S
𝒱 : K.ZeroHypercover S
f✝ : Hom K 𝒰 𝒱
H₁ : Presieve.IsSheafFor F (Presieve.ofArrows 𝒰.X 𝒰.f)
H₂ : ∀ {X : C} (f : X ⟶ S), Presieve.IsSeparatedFor F (Presieve.ofArrows (pullback₂ f 𝒰).X (pullback₂ f 𝒰).f)
Y : C
f : Y ⟶ S
hf : (Sieve.generate (Presieve.ofArrows 𝒱.X 𝒱.f)).arrows f
⊢ Presieve.IsSeparatedFor F (Sieve.pullback f (Sieve.generate (Presieve.ofArrows 𝒰.X 𝒰.f))).arrows

Tactic:
rw [← Sieve.pullbackArrows_comm, ← Presieve.isSeparatedFor_iff_generate,
      ← Presieve.ofArrows_pullback]

Goal state after:
case trans
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Limits.HasPullbacks C
K : Precoverage C
inst✝ : K.IsStableUnderBaseChange
S : C
F : Cᵒᵖ ⥤ Type u_2
𝒰 : K.ZeroHypercover S
𝒱 : K.ZeroHypercover S
f✝ : Hom K 𝒰 𝒱
H₁ : Presieve.IsSheafFor F (Presieve.ofArrows 𝒰.X 𝒰.f)
H₂ : ∀ {X : C} (f : X ⟶ S), Presieve.IsSeparatedFor F (Presieve.ofArrows (pullback₂ f 𝒰).X (pullback₂ f 𝒰).f)
Y : C
f : Y ⟶ S
hf : (Sieve.generate (Presieve.ofArrows 𝒱.X 𝒱.f)).arrows f
⊢ Presieve.IsSeparatedFor F (Presieve.ofArrows (fun i ↦ pullback (𝒰.f i) f) fun x ↦ pullback.snd (𝒰.f x) f)
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W' : Projective R
⊢ W'.polynomialY = C 2 * Y * Z + C W'.a₁ * X * Z + C W'.a₃ * Z ^ 2

Tactic:
rw [polynomialY, polynomial]

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

Goal state before:
pp: F : Type u
inst✝ : Field F
W : Projective F
P : Fin 3 → F
hPz : P z = 0
⊢ toAffine W P = 0

Tactic:
by
  rw [toAffine, dif_neg <| not_and_not_right.mpr fun _ => hPz]

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Preserves

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
F : Cᵒᵖ ⥤ Type w
α : Type u_1
inst✝² : Small.{w, u_1} α
X : α → C
c : Cofan X
hc : IsColimit c
inst✝¹ : (ofArrows X c.inj).HasPairwisePullbacks
inst✝ : PreservesLimit (Discrete.functor fun x ↦ op (X x)) F
this : HasCoproduct X
hi :
  ∀ (b : (fun X ↦ X) (∏ᶜ fun b ↦ F.obj (op (X b)))),
    ∃! a,
      (ConcreteCategory.hom
            (F.map (opCoproductIsoProduct' hc (productIsProduct fun x ↦ op (X x))).inv ≫
              Equalizer.Presieve.Arrows.forkMap F X c.inj))
          a =
        b
b : Equalizer.Presieve.Arrows.FirstObj F X
a✝ :
  (ConcreteCategory.hom (Equalizer.Presieve.Arrows.firstMap F X c.inj)) b =
    (ConcreteCategory.hom (Equalizer.Presieve.Arrows.secondMap F X c.inj)) b
t : F.obj (∏ᶜ fun x ↦ op (X x))
ht₁ :
  (ConcreteCategory.hom
        (F.map (opCoproductIsoProduct' hc (productIsProduct fun x ↦ op (X x))).inv ≫
          Equalizer.Presieve.Arrows.forkMap F X c.inj))
      t =
    b
ht₂ :
  ∀ (y : (fun X ↦ X) (F.obj (∏ᶜ fun x ↦ op (X x)))),
    (fun a ↦
          (ConcreteCategory.hom
                (F.map (opCoproductIsoProduct' hc (productIsProduct fun x ↦ op (X x))).inv ≫
                  Equalizer.Presieve.Arrows.forkMap F X c.inj))
              a =
            b)
        y →
      y = t
y : F.obj (op c.pt)
hy : (fun x ↦ (ConcreteCategory.hom (Equalizer.Presieve.Arrows.forkMap F X c.inj)) x = b) y
⊢ (ConcreteCategory.hom
        (F.map (opCoproductIsoProduct' hc (productIsProduct fun x ↦ op (X x))).inv ≫
          Equalizer.Presieve.Arrows.forkMap F X c.inj))
      ((ConcreteCategory.hom (F.map (opCoproductIsoProduct' hc (productIsProduct fun x ↦ op (X x))).hom)) y) =
    b

Tactic:
simp [← hy, ← comp_apply]

Goal state after:
Module: Mathlib.Topology.Category.TopCat.ULift

Goal state before:
pp: case refine_1
J✝ : Type w
inst✝ : Category.{w', w} J✝
K : J✝ ⥤ TopCat
c : Cone K
hc : IsLimit c
j : J✝
t : Set ↑((K ⋙ uliftFunctor).obj j)
ht : IsOpen t
⊢ IsOpen (⇑Homeomorph.ulift.symm ⁻¹' ⇑(ConcreteCategory.hom (uliftFunctor.map (c.π.app j))) ⁻¹' t)

Tactic:
apply Homeomorph.ulift.continuous_invFun.isOpen_preimage

Goal state after:
case refine_1
J✝ : Type w
inst✝ : Category.{w', w} J✝
K : J✝ ⥤ TopCat
c : Cone K
hc : IsLimit c
j : J✝
t : Set ↑((K ⋙ uliftFunctor).obj j)
ht : IsOpen t
⊢ IsOpen (⇑(ConcreteCategory.hom (uliftFunctor.map (c.π.app j))) ⁻¹' t)
Module: Mathlib.RingTheory.RingHom.Unramified

Goal state before:
pp: ⊢ RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ FormallyUnramified

Tactic:
refine stableUnderComposition.respectsIso ?_
  intro R S _ _ e
  exact .of_surjective e.surjective

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

Goal state before:
pp: ⊢ OfLocalizationSpanTarget fun {R S} [CommRing R] [CommRing S] ↦ Smooth

Tactic:
by
  introv R hs hf
  have : f.FinitePresentation :=
    finitePresentation_ofLocalizationSpanTarget _ s hs fun r ↦ (hf r).finitePresentation
  algebraize [f]
  refine ⟨?_, ‹_›⟩
  rw [← Algebra.smoothLocus_eq_univ_iff, ← Set.univ_subset_iff, ← TopologicalSpace.Opens.coe_top,
    ← PrimeSpectrum.iSup_basicOpen_eq_top_iff'.mpr hs]
  simp only [TopologicalSpace.Opens.coe_iSup, Set.iUnion_subset_iff,
    Algebra.basicOpen_subset_smoothLocus_iff, ← formallySmooth_algebraMap]
  exact fun r hr ↦ (hf ⟨r, hr⟩).1

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

Goal state before:
pp: case intro.mpr
R : Type u_1
I : Type u_2
inst✝³ : Finite I
f : I → Type u_3
inst✝² : CommRing R
inst✝¹ : (i : I) → CommRing (f i)
inst✝ : (i : I) → Algebra R (f i)
val✝ : Fintype I
⊢ (∀ (i : I), FormallyUnramified R (f i)) → FormallyUnramified R ((i : I) → f i)

Tactic:
intro H

Goal state after:
case intro.mpr
R : Type u_1
I : Type u_2
inst✝³ : Finite I
f : I → Type u_3
inst✝² : CommRing R
inst✝¹ : (i : I) → CommRing (f i)
inst✝ : (i : I) → Algebra R (f i)
val✝ : Fintype I
H : ∀ (i : I), FormallyUnramified R (f i)
⊢ FormallyUnramified R ((i : I) → f i)
Module: Mathlib.RingTheory.Etale.Field

Goal state before:
pp: case h
K : Type u_1
L : Type u_2
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra.IsSeparable K L
inst✝ : EssFiniteType K L
this✝ : FormallyUnramified K L
this : Module.Finite K L
B : Type (max u_1 u_2)
x✝¹ : CommRing B
x✝ : Algebra K B
I : Ideal B
h : I ^ 2 = ⊥
f : L →ₐ[K] B ⧸ I
pb : PowerBasis K L := ⋯
x : B
hx : (Ideal.Quotient.mk I) x = f pb.gen
helper : ∀ (x : B), (IsScalarTower.toAlgHom K B (B ⧸ I)) x = (Ideal.Quotient.mk I) x
hx' : (Ideal.Quotient.mk I) ((aeval x) (minpoly K pb.gen)) = 0
u : B
hu : (aeval x) (derivative (minpoly K pb.gen)) * u + 1 ∈ I
⊢ (Ideal.Quotient.mkₐ K I).comp (pb.liftEquiv.symm ⟨x + u * (aeval x) (minpoly K pb.gen), ?w⟩) = f

pp: case w
K : Type u_1
L : Type u_2
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra.IsSeparable K L
inst✝ : EssFiniteType K L
this✝ : FormallyUnramified K L
this : Module.Finite K L
B : Type (max u_1 u_2)
x✝¹ : CommRing B
x✝ : Algebra K B
I : Ideal B
h : I ^ 2 = ⊥
f : L →ₐ[K] B ⧸ I
pb : PowerBasis K L := Field.powerBasisOfFiniteOfSeparable K L
x : B
hx : (Ideal.Quotient.mk I) x = f pb.gen
helper : ∀ (x : B), (IsScalarTower.toAlgHom K B (B ⧸ I)) x = (Ideal.Quotient.mk I) x
hx' : (Ideal.Quotient.mk I) ((aeval x) (minpoly K pb.gen)) = 0
u : B
hu : (aeval x) (derivative (minpoly K pb.gen)) * u + 1 ∈ I
⊢ (aeval (x + u * (aeval x) (minpoly K pb.gen))) (minpoly K pb.gen) = 0

Tactic:
· apply pb.algHom_ext
    simp [hx, hx']

Goal state after:
case w
K : Type u_1
L : Type u_2
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra.IsSeparable K L
inst✝ : EssFiniteType K L
this✝ : FormallyUnramified K L
this : Module.Finite K L
B : Type (max u_1 u_2)
x✝¹ : CommRing B
x✝ : Algebra K B
I : Ideal B
h : I ^ 2 = ⊥
f : L →ₐ[K] B ⧸ I
pb : PowerBasis K L := Field.powerBasisOfFiniteOfSeparable K L
x : B
hx : (Ideal.Quotient.mk I) x = f pb.gen
helper : ∀ (x : B), (IsScalarTower.toAlgHom K B (B ⧸ I)) x = (Ideal.Quotient.mk I) x
hx' : (Ideal.Quotient.mk I) ((aeval x) (minpoly K pb.gen)) = 0
u : B
hu : (aeval x) (derivative (minpoly K pb.gen)) * u + 1 ∈ I
⊢ (aeval (x + u * (aeval x) (minpoly K pb.gen))) (minpoly K pb.gen) = 0
Module: Mathlib.RingTheory.Etale.Field

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : Algebra.IsSeparable K L
B : Type (max u_1 u_2)
x✝¹ : CommRing B
x✝ : Algebra K B
I : Ideal B
h : I ^ 2 = ⊥
f : L →ₐ[K] B ⧸ I
g : (k : L) → ↥K⟮k⟯ →ₐ[K] B
hg₁ : ∀ (k : L), (fun g ↦ (Ideal.Quotient.mkₐ K I).comp g = f.comp (IsScalarTower.toAlgHom K (↥K⟮k⟯) L)) (g k)
hg₂ :
  ∀ (k : L) (y : ↥K⟮k⟯ →ₐ[K] B),
    (fun g ↦ (Ideal.Quotient.mkₐ K I).comp g = f.comp (IsScalarTower.toAlgHom K (↥K⟮k⟯) L)) y → y = g k
hg₃ : ∀ (x y : L) (h : x ∈ K⟮y⟯), (g y) ⟨x, h⟩ = (g x) (IntermediateField.AdjoinSimple.gen K x)
x y : L
⊢ ∀ x_1 ∈ {x, y}, IsIntegral K x_1

Tactic:
intro x _

Goal state after:
K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : Algebra.IsSeparable K L
B : Type (max u_1 u_2)
x✝² : CommRing B
x✝¹ : Algebra K B
I : Ideal B
h : I ^ 2 = ⊥
f : L →ₐ[K] B ⧸ I
g : (k : L) → ↥K⟮k⟯ →ₐ[K] B
hg₁ : ∀ (k : L), (fun g ↦ (Ideal.Quotient.mkₐ K I).comp g = f.comp (IsScalarTower.toAlgHom K (↥K⟮k⟯) L)) (g k)
hg₂ :
  ∀ (k : L) (y : ↥K⟮k⟯ →ₐ[K] B),
    (fun g ↦ (Ideal.Quotient.mkₐ K I).comp g = f.comp (IsScalarTower.toAlgHom K (↥K⟮k⟯) L)) y → y = g k
hg₃ : ∀ (x y : L) (h : x ∈ K⟮y⟯), (g y) ⟨x, h⟩ = (g x) (IntermediateField.AdjoinSimple.gen K x)
x✝ y x : L
a✝ : x ∈ {x✝, y}
⊢ IsIntegral K x
Module: Mathlib.RingTheory.Kaehler.TensorProduct

Goal state before:
pp: case tmul
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
s : S
a : A
x✝ : S
y✝ : Ω[A⁄R]
⊢ s • a • x✝ ⊗ₜ[R] y✝ = a • s • x✝ ⊗ₜ[R] y✝

Tactic:
rw [mulActionBaseChange_smul_tmul, smul_tmul', smul_tmul', mulActionBaseChange_smul_tmul]

Goal state after:
Module: Mathlib.RingTheory.Etale.Field

Goal state before:
pp: case mp
K : Type u_1
A : Type u
inst✝³ : Field K
inst✝² : CommRing A
inst✝¹ : Algebra K A
inst✝ : EssFiniteType K A
⊢ FormallyEtale K A → ∃ I, ∃ (_ : Finite I), ∃ Ai x x_1 x_2, ∀ (i : I), Algebra.IsSeparable K (Ai i)

Tactic:
intro H

Goal state after:
case mp
K : Type u_1
A : Type u
inst✝³ : Field K
inst✝² : CommRing A
inst✝¹ : Algebra K A
inst✝ : EssFiniteType K A
H : FormallyEtale K A
⊢ ∃ I, ∃ (_ : Finite I), ∃ Ai x x_1 x_2, ∀ (i : I), Algebra.IsSeparable K (Ai i)
Module: Mathlib.RingTheory.Etale.Field

Goal state before:
pp: case mp
K : Type u_1
A : Type u
inst✝² : Field K
inst✝¹ : CommRing A
inst✝ : Algebra K A
⊢ Etale K A → ∃ I, ∃ (_ : Finite I), ∃ Ai x x_1 x_2, ∀ (i : I), Module.Finite K (Ai i) ∧ Algebra.IsSeparable K (Ai i)

Tactic:
intro H

Goal state after:
case mp
K : Type u_1
A : Type u
inst✝² : Field K
inst✝¹ : CommRing A
inst✝ : Algebra K A
H : Etale K A
⊢ ∃ I, ∃ (_ : Finite I), ∃ Ai x x_1 x_2, ∀ (i : I), Module.Finite K (Ai i) ∧ Algebra.IsSeparable K (Ai i)
Module: Mathlib.RingTheory.Kaehler.TensorProduct

Goal state before:
pp: case tmul.tmul.tmul
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
this : Algebra.IsPushout R A S B
e₁ : B ⊗[A] Ω[A⁄R] ≃ₗ[A] Ω[A⁄R] ⊗[R] S :=
  AlgebraTensorModule.congr (↑(Algebra.IsPushout.equiv R A S B).symm) (LinearEquiv.refl A Ω[A⁄R]) ≪≫ₗ
      TensorProduct.comm A (A ⊗[R] S) Ω[A⁄R] ≪≫ₗ
    AlgebraTensorModule.cancelBaseChange R A A Ω[A⁄R] S
e₂ : B ⊗[A] Ω[A⁄R] ≃ₗ[R] Ω[B⁄S] :=
  LinearEquiv.restrictScalars R e₁ ≪≫ₗ TensorProduct.comm R Ω[A⁄R] S ≪≫ₗ
    LinearEquiv.restrictScalars R (tensorKaehlerEquivBase R S A B)
a : A
b : S
y : Ω[A⁄R]
x : A
z : S
⊢ b • z • a • x • (map R S A B) y = ((algebraMap A B) a * (algebraMap S B) b) • z • x • (map R S A B) y

Tactic:
simp only [← mul_smul, ← @algebraMap_smul S _ B, ← @algebraMap_smul A _ B]

Goal state after:
case tmul.tmul.tmul
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
this : Algebra.IsPushout R A S B
e₁ : B ⊗[A] Ω[A⁄R] ≃ₗ[A] Ω[A⁄R] ⊗[R] S :=
  AlgebraTensorModule.congr (↑(Algebra.IsPushout.equiv R A S B).symm) (LinearEquiv.refl A Ω[A⁄R]) ≪≫ₗ
      TensorProduct.comm A (A ⊗[R] S) Ω[A⁄R] ≪≫ₗ
    AlgebraTensorModule.cancelBaseChange R A A Ω[A⁄R] S
e₂ : B ⊗[A] Ω[A⁄R] ≃ₗ[R] Ω[B⁄S] :=
  LinearEquiv.restrictScalars R e₁ ≪≫ₗ TensorProduct.comm R Ω[A⁄R] S ≪≫ₗ
    LinearEquiv.restrictScalars R (tensorKaehlerEquivBase R S A B)
a : A
b : S
y : Ω[A⁄R]
x : A
z : S
⊢ ((algebraMap S B) b * ((algebraMap S B) z * ((algebraMap A B) a * (algebraMap A B) x))) • (map R S A B) y =
    ((algebraMap A B) a * (algebraMap S B) b * ((algebraMap S B) z * (algebraMap A B) x)) • (map R S A B) y
Module: Mathlib.RingTheory.AdicCompletion.Exactness

Goal state before:
pp: R : Type u
inst✝⁶ : CommRing R
I : Ideal R
M : Type u
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : IsNoetherianRing R
inst✝ : Module.Finite R N
f : M →ₗ[R] N
hf : Function.Injective ⇑f
k : ℕ
hk : ∀ n ≥ k, I ^ n • ⊤ ⊓ f.range = I ^ (n - k) • (I ^ k • ⊤ ⊓ f.range)
x : AdicCompletion I M
a : AdicCauchySequence I M
hx : (map I f) ((mk I M) a) = 0
n : ℕ
x✝ : n ≥ 42
⊢ f (↑a (n + k)) ∈ I ^ (n + k - k) • (I ^ k • ⊤ ⊓ f.range)

Tactic:
rw [← hk (n + k) (show n + k ≥ k by lia)]

Goal state after:
R : Type u
inst✝⁶ : CommRing R
I : Ideal R
M : Type u
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : IsNoetherianRing R
inst✝ : Module.Finite R N
f : M →ₗ[R] N
hf : Function.Injective ⇑f
k : ℕ
hk : ∀ n ≥ k, I ^ n • ⊤ ⊓ f.range = I ^ (n - k) • (I ^ k • ⊤ ⊓ f.range)
x : AdicCompletion I M
a : AdicCauchySequence I M
hx : (map I f) ((mk I M) a) = 0
n : ℕ
x✝ : n ≥ 42
⊢ f (↑a (n + k)) ∈ I ^ (n + k) • ⊤ ⊓ f.range