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

Goal state before:
pp: R : Type u_1
inst✝¹⁰ : CommRing R
A : Type u_2
inst✝⁹ : CommRing A
inst✝⁸ : Algebra R A
B : Type u_3
inst✝⁷ : CommRing B
inst✝⁶ : Algebra R B
inst✝⁵ : Algebra A B
inst✝⁴ : IsScalarTower R A B
inst✝³ : FormallyUnramified R A
inst✝² : FormallyUnramified A B
C : Type u_3
inst✝¹ : CommRing C
inst✝ : Algebra R C
I : Ideal C
hI : I ^ 2 = ⊥
f₁ f₂ : B →ₐ[R] C
e : (Ideal.Quotient.mkₐ R I).comp f₁ = (Ideal.Quotient.mkₐ R I).comp f₂
⊢ ((Ideal.Quotient.mkₐ R I).comp f₂).comp (IsScalarTower.toAlgHom R A B) =
    (Ideal.Quotient.mkₐ R I).comp (f₂.comp (IsScalarTower.toAlgHom R A B))

Tactic:
AlgHom.comp_assoc

Goal state after:
R : Type u_1
inst✝¹⁰ : CommRing R
A : Type u_2
inst✝⁹ : CommRing A
inst✝⁸ : Algebra R A
B : Type u_3
inst✝⁷ : CommRing B
inst✝⁶ : Algebra R B
inst✝⁵ : Algebra A B
inst✝⁴ : IsScalarTower R A B
inst✝³ : FormallyUnramified R A
inst✝² : FormallyUnramified A B
C : Type u_3
inst✝¹ : CommRing C
inst✝ : Algebra R C
I : Ideal C
hI : I ^ 2 = ⊥
f₁ f₂ : B →ₐ[R] C
e : (Ideal.Quotient.mkₐ R I).comp f₁ = (Ideal.Quotient.mkₐ R I).comp f₂
⊢ (Ideal.Quotient.mkₐ R I).comp (f₂.comp (IsScalarTower.toAlgHom R A B)) =
    (Ideal.Quotient.mkₐ R I).comp (f₂.comp (IsScalarTower.toAlgHom R A B))
Module: Mathlib.RingTheory.QuasiFinite.Basic

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : QuasiFinite R S
inst✝¹ : IsArtinianRing R
inst✝ : IsLocalRing R
⊢ Module.Finite R S

Tactic:
let e : (maximalIdeal R).Fiber S ≃ₐ[R] S ⧸ (maximalIdeal R).map (algebraMap R S) :=
    (Algebra.TensorProduct.congr (.symm <| .ofBijective _
      (Ideal.bijective_algebraMap_quotient_residueField (maximalIdeal R))) .refl).trans <|
    (Algebra.TensorProduct.comm _ _ _).trans
    ((Algebra.TensorProduct.quotIdealMapEquivTensorQuot S (maximalIdeal R)).symm.restrictScalars _)

Goal state after:
R : Type u_1
S : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : QuasiFinite R S
inst✝¹ : IsArtinianRing R
inst✝ : IsLocalRing R
e : (maximalIdeal R).Fiber S ≃ₐ[R] S ⧸ Ideal.map (algebraMap R S) (maximalIdeal R) :=
  (TensorProduct.congr
        (AlgEquiv.ofBijective (Ideal.Quotient.liftₐ (maximalIdeal R) (ofId R (maximalIdeal R).ResidueField) ⋯) ⋯).symm
        AlgEquiv.refl).trans
    ((TensorProduct.comm R (R ⧸ maximalIdeal R) S).trans
      (AlgEquiv.restrictScalars R (TensorProduct.quotIdealMapEquivTensorQuot S (maximalIdeal R)).symm))
⊢ Module.Finite R S
Module: Mathlib.RingTheory.QuasiFinite.Basic

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : CommRing T
inst✝⁵ : Algebra R S
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
M : Submonoid S
inst✝¹ : IsLocalization M T
inst✝ : QuasiFinite R S
P : Ideal S
hP : P.IsPrime
p : P.ResidueField
s : S
t : ↥M
ht : ¬(algebraMap S P.ResidueField) ↑t = 0
⊢ (fun x ↦ x * TensorProduct.includeRight ((algebraMap S T) ↑t))
      ((↑S (Algebra.linearMap P.ResidueField (P.Fiber T))) (s • p / (algebraMap S P.ResidueField) ↑t)) =
    (s • p) ⊗ₜ[S] 1

pp: R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : CommRing T
inst✝⁵ : Algebra R S
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
M : Submonoid S
inst✝¹ : IsLocalization M T
inst✝ : QuasiFinite R S
P : Ideal S
hP : P.IsPrime
p : P.ResidueField
s : S
t : ↥M
ht : ¬(algebraMap S P.ResidueField) ↑t = 0
⊢ (s • p) ⊗ₜ[S] 1 = (fun x ↦ x * TensorProduct.includeRight ((algebraMap S T) ↑t)) (p ⊗ₜ[S] IsLocalization.mk' T s t)

Tactic:
· simp [div_mul_cancel₀ _ ht]

Goal state after:
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : CommRing T
inst✝⁵ : Algebra R S
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
M : Submonoid S
inst✝¹ : IsLocalization M T
inst✝ : QuasiFinite R S
P : Ideal S
hP : P.IsPrime
p : P.ResidueField
s : S
t : ↥M
ht : ¬(algebraMap S P.ResidueField) ↑t = 0
⊢ (s • p) ⊗ₜ[S] 1 = (fun x ↦ x * TensorProduct.includeRight ((algebraMap S T) ↑t)) (p ⊗ₜ[S] IsLocalization.mk' T s t)
Module: Mathlib.RingTheory.QuasiFinite.Basic

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R S
inst✝³ : Algebra R T
p : Ideal S
inst✝² : p.IsPrime
inst✝¹ : QuasiFiniteAt R p
f : S →ₐ[R] T
hf : f.SurjectiveOnStalks
q : Ideal T
inst✝ : q.IsPrime
hq : p = Ideal.comap f.toRingHom q
⊢ QuasiFiniteAt R q

Tactic:
subst hq
  refine .of_surjective_algHom ⟨Localization.localRingHom _ q f.toRingHom rfl, ?_⟩ (hf q ‹_›)
  simp [IsScalarTower.algebraMap_apply R S (Localization.AtPrime (q.comap _)),
    IsScalarTower.algebraMap_apply R T (Localization.AtPrime _)]

Goal state after:
Module: Mathlib.RingTheory.QuasiFinite.Basic

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : IsArtinianRing R
p : PrimeSpectrum S
inst✝ : FiniteType R S
H✝ : IsClopen {p}
this✝¹ : IsNoetherianRing S
this✝ : IsJacobsonRing S
e : S
he : IsIdempotentElem e
H : {p} = ↑(PrimeSpectrum.basicOpen e)
hep : e ∉ p.asIdeal
f : Localization.Away e →ₐ[S] Localization.AtPrime p.asIdeal := IsLocalization.Away.liftAlgHom e ⋯
h₁ : Set.range (PrimeSpectrum.comap (algebraMap S (Localization.Away e))) = {p}
this : Subsingleton (PrimeSpectrum (Localization.Away e))
x : S
s : ↥p.asIdeal.primeCompl
⊢ ∃ a, f a = IsLocalization.mk' (Localization.AtPrime p.asIdeal) x s

Tactic:
suffices IsUnit (algebraMap _ (Localization.Away e) s.1) by
      refine ⟨algebraMap _ _ x * this.unit⁻¹, (this.map f).mul_right_cancel ?_⟩
      simp only [← map_mul, mul_assoc, IsUnit.val_inv_mul]
      simp

Goal state after:
R : Type u_1
S : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : IsArtinianRing R
p : PrimeSpectrum S
inst✝ : FiniteType R S
H✝ : IsClopen {p}
this✝¹ : IsNoetherianRing S
this✝ : IsJacobsonRing S
e : S
he : IsIdempotentElem e
H : {p} = ↑(PrimeSpectrum.basicOpen e)
hep : e ∉ p.asIdeal
f : Localization.Away e →ₐ[S] Localization.AtPrime p.asIdeal := IsLocalization.Away.liftAlgHom e ⋯
h₁ : Set.range (PrimeSpectrum.comap (algebraMap S (Localization.Away e))) = {p}
this : Subsingleton (PrimeSpectrum (Localization.Away e))
x : S
s : ↥p.asIdeal.primeCompl
⊢ IsUnit ((algebraMap S (Localization.Away e)) ↑s)
Module: Mathlib.RingTheory.Smooth.Basic

Goal state before:
pp: R : Type u
A : Type v
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
B : Type u_1
inst✝² : CommRing B
inst✝¹ : Algebra R B
inst✝ : FormallySmooth R A
I : Ideal B
hI : IsNilpotent I
g : A →ₐ[R] B ⧸ I
⊢ ∃ f, (Ideal.Quotient.mkₐ R I).comp f = g

Tactic:
revert g
  change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
  revert ‹Algebra R B›
  apply Ideal.IsNilpotent.induction_on (S := B) I hI
  · intro B _ I hI _; exact FormallySmooth.comp_surjective R A I hI
  · intro B _ I J hIJ h₁ h₂ _ g
    let : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
      { (DoubleQuot.quotQuotEquivQuotSup I J).trans
          (Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
        commutes' := fun x => rfl }
    obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
    obtain ⟨g', rfl⟩ := h₁ g'
    replace e := congr_arg this.toAlgHom.comp e
    conv_rhs at e =>
      rw [← AlgHom.comp_assoc, AlgEquiv.comp_symm, AlgHom.id_comp]
    exact ⟨g', e⟩

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

Goal state before:
pp: R : Type u
A : Type v
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
H :
  ∀ ⦃B : Type (max u v)⦄ [inst : CommRing B] [inst_1 : Algebra R B] (I : Ideal B),
    I ^ 2 = ⊥ → Function.Bijective (Ideal.Quotient.mkₐ R I).comp
this : FormallyUnramified R A
⊢ ∀ ⦃B : Type (max u v)⦄ [inst : CommRing B] [inst_1 : Algebra R B] (I : Ideal B),
    I ^ 2 = ⊥ → Function.Surjective (Ideal.Quotient.mkₐ R I).comp

Tactic:
by aesop (add safe Function.Bijective.surjective)

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

Goal state before:
pp: R : Type u
A : Type v
inst✝⁹ : CommRing R
inst✝⁸ : CommRing A
inst✝⁷ : Algebra R A
B : Type u_1
P : Type u_2
C : Type u_3
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R B
inst✝⁴ : CommRing C
inst✝³ : Algebra R C
inst✝² : CommRing P
inst✝¹ : Algebra R P
P₁ : Extension R A
P₂ : Extension R A
inst✝ : FormallySmooth R P₁.Ring
lift : P₁.Ring →ₐ[R] P₂.infinitesimal.Ring :=
  FormallySmooth.liftOfSurjective (IsScalarTower.toAlgHom R P₁.Ring A)
    (IsScalarTower.toAlgHom R P₂.infinitesimal.Ring A) ⋯ ⋯
⊢ ∀ a ∈ P₁.ker ^ 2, lift a = 0

Tactic:
by
        change P₁.ker ^ 2 ≤ RingHom.ker lift
        rw [pow_two, Ideal.mul_le]
        have : ∀ r ∈ P₁.ker, lift r ∈ P₂.infinitesimal.ker :=
          fun r hr ↦ (FormallySmooth.liftOfSurjective_apply _
            (IsScalarTower.toAlgHom R P₂.infinitesimal.Ring A) _ _ r).trans hr
        intro r hr s hs
        rw [RingHom.mem_ker, map_mul, ← Ideal.mem_bot, ← P₂.ker.cotangentIdeal_square,
          ← ker_infinitesimal, pow_two]
        exact Ideal.mul_mem_mul (this r hr) (this s hs)

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

Goal state before:
pp: K : Type u_1
A : Type u_2
inst✝⁶ : Field K
inst✝⁵ : CommRing A
inst✝⁴ : Algebra K A
inst✝³ : FormallyUnramified K A
inst✝² : EssFiniteType K A
inst✝¹ : IsAlgClosed K
inst✝ : IsLocalRing A
this✝ : Module.Finite K A
this : IsArtinianRing A
hA : IsNilpotent (IsLocalRing.maximalIdeal A)
e : K ≃ₐ[K] A ⧸ IsLocalRing.maximalIdeal A :=
  let __spread.0 := ofId K (A ⧸ IsLocalRing.maximalIdeal A);
  let __spread.1 := Equiv.ofBijective ⇑(algebraMap K (A ⧸ IsLocalRing.maximalIdeal A)) ⋯;
  { toFun := (↑↑__spread.0.toRingHom).toFun, invFun := __spread.1.invFun, left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯,
    map_add' := ⋯, commutes' := ⋯ }
e' : A ⊗[K] (A ⧸ IsLocalRing.maximalIdeal A) ≃ₐ[A] A :=
  (TensorProduct.congr AlgEquiv.refl e.symm).trans (TensorProduct.rid K A A)
f : A ⧸ IsLocalRing.maximalIdeal A →ₗ[A] A := e'.toLinearMap ∘ₗ sec K A (A ⧸ IsLocalRing.maximalIdeal A)
hf : (ofId A (A ⧸ IsLocalRing.maximalIdeal A)).toLinearMap ∘ₗ f = LinearMap.id
hf₁ : f 1 • 1 = 1
⊢ Function.Bijective ⇑(algebraMap K A)

Tactic:
have hf₂ : 1 - f 1 ∈ IsLocalRing.maximalIdeal A := by
    rw [← Ideal.Quotient.eq_zero_iff_mem, map_sub, map_one, ← Ideal.Quotient.algebraMap_eq,
     algebraMap_eq_smul_one, hf₁, sub_self]

Goal state after:
K : Type u_1
A : Type u_2
inst✝⁶ : Field K
inst✝⁵ : CommRing A
inst✝⁴ : Algebra K A
inst✝³ : FormallyUnramified K A
inst✝² : EssFiniteType K A
inst✝¹ : IsAlgClosed K
inst✝ : IsLocalRing A
this✝ : Module.Finite K A
this : IsArtinianRing A
hA : IsNilpotent (IsLocalRing.maximalIdeal A)
e : K ≃ₐ[K] A ⧸ IsLocalRing.maximalIdeal A :=
  let __spread.0 := ofId K (A ⧸ IsLocalRing.maximalIdeal A);
  let __spread.1 := Equiv.ofBijective ⇑(algebraMap K (A ⧸ IsLocalRing.maximalIdeal A)) ⋯;
  { toFun := (↑↑__spread.0.toRingHom).toFun, invFun := __spread.1.invFun, left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯,
    map_add' := ⋯, commutes' := ⋯ }
e' : A ⊗[K] (A ⧸ IsLocalRing.maximalIdeal A) ≃ₐ[A] A :=
  (TensorProduct.congr AlgEquiv.refl e.symm).trans (TensorProduct.rid K A A)
f : A ⧸ IsLocalRing.maximalIdeal A →ₗ[A] A := e'.toLinearMap ∘ₗ sec K A (A ⧸ IsLocalRing.maximalIdeal A)
hf : (ofId A (A ⧸ IsLocalRing.maximalIdeal A)).toLinearMap ∘ₗ f = LinearMap.id
hf₁ : f 1 • 1 = 1
hf₂ : 1 - f 1 ∈ IsLocalRing.maximalIdeal A
⊢ Function.Bijective ⇑(algebraMap K A)
Module: Mathlib.RingTheory.Unramified.Field

Goal state before:
pp: K : Type u_1
A : Type u_2
inst✝⁶ : Field K
inst✝⁵ : CommRing A
inst✝⁴ : Algebra K A
inst✝³ : FormallyUnramified K A
inst✝² : EssFiniteType K A
inst✝¹ : IsAlgClosed K
inst✝ : IsLocalRing A
this✝ : Module.Finite K A
this : IsArtinianRing A
hA : IsNilpotent (IsLocalRing.maximalIdeal A)
e : K ≃ₐ[K] A ⧸ IsLocalRing.maximalIdeal A :=
  let __spread.0 := ofId K (A ⧸ IsLocalRing.maximalIdeal A);
  let __spread.1 := Equiv.ofBijective ⇑(algebraMap K (A ⧸ IsLocalRing.maximalIdeal A)) ⋯;
  { toFun := (↑↑__spread.0.toRingHom).toFun, invFun := __spread.1.invFun, left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯,
    map_add' := ⋯, commutes' := ⋯ }
e' : A ⊗[K] (A ⧸ IsLocalRing.maximalIdeal A) ≃ₐ[A] A :=
  (TensorProduct.congr AlgEquiv.refl e.symm).trans (TensorProduct.rid K A A)
f : A ⧸ IsLocalRing.maximalIdeal A →ₗ[A] A := e'.toLinearMap ∘ₗ sec K A (A ⧸ IsLocalRing.maximalIdeal A)
hf : (ofId A (A ⧸ IsLocalRing.maximalIdeal A)).toLinearMap ∘ₗ f = LinearMap.id
hf₁ : f 1 • 1 = 1
hf₂ : 1 - f 1 ∈ IsLocalRing.maximalIdeal A
hf₃ : IsIdempotentElem (1 - f 1)
hf₄ : f 1 = 1
x : A
⊢ (algebraMap K A) ((⇑e.symm ∘ ⇑(algebraMap A (A ⧸ IsLocalRing.maximalIdeal A))) x) = x

Tactic:
have : ⇑(algebraMap K A) = ⇑f ∘ ⇑e := by
    ext k
    conv_rhs => rw [← mul_one k, ← smul_eq_mul, Function.comp_apply, map_smul,
      LinearMap.map_smul_of_tower, map_one, hf₄, ← algebraMap_eq_smul_one]

Goal state after:
K : Type u_1
A : Type u_2
inst✝⁶ : Field K
inst✝⁵ : CommRing A
inst✝⁴ : Algebra K A
inst✝³ : FormallyUnramified K A
inst✝² : EssFiniteType K A
inst✝¹ : IsAlgClosed K
inst✝ : IsLocalRing A
this✝¹ : Module.Finite K A
this✝ : IsArtinianRing A
hA : IsNilpotent (IsLocalRing.maximalIdeal A)
e : K ≃ₐ[K] A ⧸ IsLocalRing.maximalIdeal A :=
  let __spread.0 := ofId K (A ⧸ IsLocalRing.maximalIdeal A);
  let __spread.1 := Equiv.ofBijective ⇑(algebraMap K (A ⧸ IsLocalRing.maximalIdeal A)) ⋯;
  { toFun := (↑↑__spread.0.toRingHom).toFun, invFun := __spread.1.invFun, left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯,
    map_add' := ⋯, commutes' := ⋯ }
e' : A ⊗[K] (A ⧸ IsLocalRing.maximalIdeal A) ≃ₐ[A] A :=
  (TensorProduct.congr AlgEquiv.refl e.symm).trans (TensorProduct.rid K A A)
f : A ⧸ IsLocalRing.maximalIdeal A →ₗ[A] A := e'.toLinearMap ∘ₗ sec K A (A ⧸ IsLocalRing.maximalIdeal A)
hf : (ofId A (A ⧸ IsLocalRing.maximalIdeal A)).toLinearMap ∘ₗ f = LinearMap.id
hf₁ : f 1 • 1 = 1
hf₂ : 1 - f 1 ∈ IsLocalRing.maximalIdeal A
hf₃ : IsIdempotentElem (1 - f 1)
hf₄ : f 1 = 1
x : A
this : ⇑(algebraMap K A) = ⇑f ∘ ⇑e
⊢ (algebraMap K A) ((⇑e.symm ∘ ⇑(algebraMap A (A ⧸ IsLocalRing.maximalIdeal A))) x) = x
Module: Mathlib.RingTheory.Smooth.Kaehler

Goal state before:
pp: R : Type u_1
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
P : Extension R S
⊢ Function.Bijective ⇑(CotangentSpace.map P.toInfinitesimal)

Tactic:
suffices CotangentSpace.map P.toInfinitesimal =
      (tensorKaehlerQuotKerSqEquiv _ _ _).symm.toLinearMap by
    rw [this]; exact (tensorKaehlerQuotKerSqEquiv _ _ _).symm.bijective

Goal state after:
R : Type u_1
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
P : Extension R S
⊢ CotangentSpace.map P.toInfinitesimal = ↑(tensorKaehlerQuotKerSqEquiv R P.Ring S).symm
Module: Mathlib.RingTheory.Unramified.LocalRing

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : IsLocalRing R
inst✝³ : IsLocalRing S
inst✝² : IsLocalHom (algebraMap R S)
inst✝¹ : EssFiniteType R S
inst✝ : Algebra.IsSeparable (ResidueField R) (ResidueField S)
H : Ideal.map (algebraMap R S) (maximalIdeal R) = maximalIdeal S
this✝ : FormallyUnramified (ResidueField R) (ResidueField S)
this : FormallyUnramified R (ResidueField S)
x✝ a x : S
hx : x ∈ Submodule.span S (⇑(algebraMap R S) '' ↑(maximalIdeal R))
a✝ : 1 ⊗ₜ[S] (KaehlerDifferential.D R S) x = 0
⊢ x ∈ maximalIdeal S

Tactic:
← H

Goal state after:
R : Type u_1
S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : IsLocalRing R
inst✝³ : IsLocalRing S
inst✝² : IsLocalHom (algebraMap R S)
inst✝¹ : EssFiniteType R S
inst✝ : Algebra.IsSeparable (ResidueField R) (ResidueField S)
H : Ideal.map (algebraMap R S) (maximalIdeal R) = maximalIdeal S
this✝ : FormallyUnramified (ResidueField R) (ResidueField S)
this : FormallyUnramified R (ResidueField S)
x✝ a x : S
hx : x ∈ Submodule.span S (⇑(algebraMap R S) '' ↑(maximalIdeal R))
a✝ : 1 ⊗ₜ[S] (KaehlerDifferential.D R S) x = 0
⊢ x ∈ Ideal.map (algebraMap R S) (maximalIdeal R)
Module: Mathlib.RingTheory.Etale.Kaehler

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R S
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
M : Submonoid S
inst✝ : IsLocalization M T
P : Extension R S := (Generators.self R S).toExtension
M' : Submonoid P.Ring := Submonoid.comap (algebraMap P.Ring S) M
fQ : Localization M' →ₐ[R] T := IsLocalization.liftAlgHom ⋯
Q : Extension R T := Extension.ofSurjective fQ ⋯
f : P.Hom Q :=
  { toRingHom := algebraMap P.Ring (Localization M'), toRingHom_algebraMap := ⋯, algebraMap_toRingHom := ⋯ }
this✝⁵ : FormallySmooth R P.Ring
this✝⁴ : FormallySmooth P.Ring (Localization M')
this✝³ : FormallySmooth R Q.Ring
this✝² : Module.Flat S T
this✝¹ : Algebra P.Ring Q.Ring := inferInstance
this✝ : Algebra P.Ring T := ((algebraMap S T).comp (algebraMap P.Ring S)).toAlgebra
this : Algebra (Localization M') T := fQ.toAlgebra
⊢ T ⊗[S] H1Cotangent R S ≃ₗ[T] H1Cotangent R T

Tactic:
haveI : IsScalarTower P.Ring S T := .of_algebraMap_eq' rfl

Goal state after:
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R S
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
M : Submonoid S
inst✝ : IsLocalization M T
P : Extension R S := (Generators.self R S).toExtension
M' : Submonoid P.Ring := Submonoid.comap (algebraMap P.Ring S) M
fQ : Localization M' →ₐ[R] T := IsLocalization.liftAlgHom ⋯
Q : Extension R T := Extension.ofSurjective fQ ⋯
f : P.Hom Q :=
  { toRingHom := algebraMap P.Ring (Localization M'), toRingHom_algebraMap := ⋯, algebraMap_toRingHom := ⋯ }
this✝⁶ : FormallySmooth R P.Ring
this✝⁵ : FormallySmooth P.Ring (Localization M')
this✝⁴ : FormallySmooth R Q.Ring
this✝³ : Module.Flat S T
this✝² : Algebra P.Ring Q.Ring := inferInstance
this✝¹ : Algebra P.Ring T := ((algebraMap S T).comp (algebraMap P.Ring S)).toAlgebra
this✝ : Algebra (Localization M') T := fQ.toAlgebra
this : IsScalarTower P.Ring S T
⊢ T ⊗[S] H1Cotangent R S ≃ₗ[T] H1Cotangent R T
Module: Mathlib.RingTheory.RamificationInertia.Basic

Goal state before:
pp: R : Type u_1
inst✝⁵ : CommRing R
p : Ideal R
inst✝⁴ : p.IsPrime
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : Algebra.QuasiFinite R S
inst✝ : Fintype ↑(p.primesOver S)
⊢ ∑ q, (↑q).ramificationIdx R * (↑q).inertiaDeg R = finrank p.ResidueField (p.Fiber S)

Tactic:
let := Fintype.ofFinite (PrimeSpectrum (p.Fiber S))
  rw [IsArtinianRing.finrank_eq_sum_primeSpectrum, ← (primesOverOrderIsoFiber R S p).symm.sum_comp]
  apply Finset.sum_congr rfl
  intro q _
  simp_rw [toEquiv_symm, coe_symm_toEquiv, coe_primesOverOrderIsoFiber_symm_apply]
  set r := q.1.comap Algebra.TensorProduct.includeRight
  let := Localization.AtPrime.algebraOfLiesOver p r
  rw [ramificationIdx_eq p r, inertiaDeg_eq p r]
  let Rp := Localization.AtPrime p
  let Sq := Localization.AtPrime q.1
  let Sr := Localization.AtPrime r
  let κp := p.ResidueField
  let κr := r.ResidueField
  let A := Sr ⧸ p.map (algebraMap R Sr)
  suffices length Sr A * finrank κp κr = finrank κp Sq by simpa using congr_arg ENat.toNat this
  calc length Sr A * finrank κp κr = length Sr A * length κp κr := by rw [length_eq_finrank]
    _ = length Rp A := (length_restrictScalars Rp Sr A).symm
    _ = length Rp Sq := (Fiber.localizationAlgEquivQuotient p q.1).toLinearEquiv.length_eq.symm
    _ = length κp Sq := length_eq_of_surjective residue_surjective
    _ = finrank κp Sq := length_eq_finrank κp Sq

Goal state after:
Module: Mathlib.Data.NNReal.Defs

Goal state before:
pp: Γ₀ : Type u_1
inst✝ : LinearOrderedCommGroupWithZero Γ₀
h : Nontrivial Γ₀ˣ
f : Γ₀ →*₀ ℝ≥0
hf : StrictMono ⇑f
r : ℝ≥0
hr : 0 < r
g : Γ₀ˣ
hg1 : g ≠ 1
u : Γ₀ˣ := if g < 1 then g else g⁻¹
hu : u = if g < 1 then g else g⁻¹
⊢ f ↑u < 1

Tactic:
rw [hu]

Goal state after:
Γ₀ : Type u_1
inst✝ : LinearOrderedCommGroupWithZero Γ₀
h : Nontrivial Γ₀ˣ
f : Γ₀ →*₀ ℝ≥0
hf : StrictMono ⇑f
r : ℝ≥0
hr : 0 < r
g : Γ₀ˣ
hg1 : g ≠ 1
u : Γ₀ˣ := if g < 1 then g else g⁻¹
hu : u = if g < 1 then g else g⁻¹
⊢ f ↑(if g < 1 then g else g⁻¹) < 1
Module: Mathlib.Data.ENNReal.Basic

Goal state before:
pp: n : ℕ
⊢ (↑n).toNNReal = (↑n).toNNReal

Tactic:
by
  rw [Real.toNNReal_of_nonneg (by positivity), ENNReal.toNNReal_natCast, mk_natCast]

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Seminorm

Goal state before:
pp: R : Type u_1
R' : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
inst✝⁴ : AddGroup E
inst✝³ : SMul R ℝ
inst✝² : SMul R ℝ≥0
inst✝¹ : IsScalarTower R ℝ≥0 ℝ
inst✝ : DecidableEq E
x y : E
⊢ (if x + y = 0 then 0 else 1) ≤ (if x = 0 then 0 else 1) + if y = 0 then 0 else 1

Tactic:
by_cases hx : x = 0
        · rw [if_pos hx, hx, zero_add, zero_add]
        · rw [if_neg hx]
          refine le_add_of_le_of_nonneg ?_ ?_ <;> split_ifs <;> norm_num

Goal state after:
Module: Mathlib.Algebra.Module.Torsion.PrimaryComponent

Goal state before:
pp: A : Type u_1
inst✝⁵ : CommRing A
inst✝⁴ : IsDedekindDomain A
M₁ : Type u_5
M₂ : Type u_6
inst✝³ : AddCommGroup M₁
inst✝² : AddCommGroup M₂
inst✝¹ : Module A M₁
inst✝ : Module A M₂
hM₁ : IsTorsion A M₁
P : HeightOneSpectrum A
φ : M₁ →ₗ[A] M₂
hf✝ : Surjective ⇑φ
b : M₁
hy : φ b ∈ primaryComponent M₂ P.asIdeal
f : Π₀ (i : HeightOneSpectrum A), ↥(primaryComponent M₁ i.asIdeal)
hf : ((DFinsupp.lsum ℕ) fun i ↦ (primaryComponent M₁ i.asIdeal).subtype) f = b
⊢ P ∉ f.support → φ ↑(f P) = 0

Tactic:
by aesop

Goal state after:
Module: Mathlib.Order.Interval.Set.OrdConnectedComponent

Goal state before:
pp: case inr.inr
α✝ : Type u_1
inst✝¹ : LinearOrder α✝
s✝ t✝ : Set α✝
α : Type u_1
inst✝ : LinearOrder α
s t : Set α
a : α
has : a ∈ s
b : α
hbt : b ∈ t
hab : a ≤ b
hsub : [[a, b]] ⊆ (s.ordSeparatingSet t).ordConnectedSectionᶜ
x : { x // x ∈ s.ordSeparatingSet t }
ha : [[a, ↑x]] ⊆ tᶜ
hb : [[b, ↑x]] ⊆ sᶜ
hax : a ≤ ↑x
hxb : ↑x ≤ b
y : α
hy : y ∈ (s.ordSeparatingSet t).ordConnectedComponent ↑x
⊢ ¬y < a ∧ y ≤ b

Tactic:
← not_lt

Goal state after:
case inr.inr
α✝ : Type u_1
inst✝¹ : LinearOrder α✝
s✝ t✝ : Set α✝
α : Type u_1
inst✝ : LinearOrder α
s t : Set α
a : α
has : a ∈ s
b : α
hbt : b ∈ t
hab : a ≤ b
hsub : [[a, b]] ⊆ (s.ordSeparatingSet t).ordConnectedSectionᶜ
x : { x // x ∈ s.ordSeparatingSet t }
ha : [[a, ↑x]] ⊆ tᶜ
hb : [[b, ↑x]] ⊆ sᶜ
hax : a ≤ ↑x
hxb : ↑x ≤ b
y : α
hy : y ∈ (s.ordSeparatingSet t).ordConnectedComponent ↑x
⊢ ¬y < a ∧ ¬b < y
Module: Mathlib.Data.ENNReal.Operations

Goal state before:
pp: a : ℝ≥0∞
⊢ a * a < ∞ ↔ a < ∞

Tactic:
rw [ENNReal.mul_lt_top_iff, and_self, or_self, or_iff_left_iff_imp]

Goal state after:
a : ℝ≥0∞
⊢ a = 0 → a < ∞
Module: Mathlib.Data.ENNReal.Operations

Goal state before:
pp: case coe
a b : ℝ≥0
h : ↑b ≤ ↑a
x✝ : ℝ≥0
⊢ ↑a - ↑x✝ - (↑a - ↑b) = ↑b - ↑x✝

Tactic:
· norm_cast
    rw [tsub_tsub_tsub_cancel_left]
    exact mod_cast h

Goal state after:
Module: Mathlib.Data.EReal.Basic

Goal state before:
pp: x : EReal
hx : 0 < x
h'x : x ≠ ⊤
⊢ x ≠ ⊤ ∧ x ≠ ⊥

Tactic:
by aesop

Goal state after:
Module: Mathlib.Data.EReal.Basic

Goal state before:
pp: x : EReal
hx : x < 0
h'x : x ≠ ⊥
⊢ x ≠ ⊤ ∧ x ≠ ⊥

Tactic:
by aesop

Goal state after:
Module: Mathlib.Data.EReal.Basic

Goal state before:
pp: y : EReal
hy0 : 0 < y
right✝ : y < ⊤
⊢ y ≠ ⊤ ∧ y ≠ ⊥

Tactic:
by aesop

Goal state after:
Module: Mathlib.Data.EReal.Basic

Goal state before:
pp: y : EReal
left✝ : ⊥ < y
hy0 : y < 0
⊢ y ≠ ⊤ ∧ y ≠ ⊥

Tactic:
by aesop

Goal state after:
Module: Mathlib.Data.EReal.Basic

Goal state before:
pp: x : ℝ
⊢ Real.toEReal '' Ici x = Ico ↑x ⊤

Tactic:
refine (image_comp WithBot.some WithTop.some _).trans ?_
  rw [WithTop.image_coe_Ici, WithBot.image_coe_Ico]
  rfl

Goal state after: