mathlib-initiative/mathlib-tactics huggingface dataset:
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✝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
⊢ (⇑f ∘ ⇑e) ((⇑e.symm ∘ ⇑(algebraMap A (A ⧸ IsLocalRing.maximalIdeal A))) x) = x

Tactic:
simp only [Function.comp_apply, AlgEquiv.apply_symm_apply, algebraMap_eq_smul_one,
    map_smul, hf₄, smul_eq_mul, mul_one]

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

Goal state before:
pp: K : Type u_1
A : Type u_2
inst✝⁴ : Field K
inst✝³ : CommRing A
inst✝² : Algebra K A
inst✝¹ : FormallyUnramified K A
inst✝ : EssFiniteType K A
⊢ IsReduced A

Tactic:
by
  constructor
  intro x hx
  let f := (Algebra.TensorProduct.includeRight (R := K) (A := AlgebraicClosure K) (B := A))
  have : Function.Injective f := by
    have : ⇑f = (LinearMap.rTensor A (Algebra.ofId K (AlgebraicClosure K)).toLinearMap).comp
        (Algebra.TensorProduct.lid K A).symm.toLinearMap := by
      ext x; simp [f]
    rw [this]
    suffices Function.Injective
        (LinearMap.rTensor A (Algebra.ofId K (AlgebraicClosure K)).toLinearMap) by
      exact this.comp (Algebra.TensorProduct.lid K A).symm.injective
    apply Module.Flat.rTensor_preserves_injective_linearMap
    exact (algebraMap K _).injective
  apply this
  rw [map_zero]
  apply eq_zero_of_localization
  intro M hM
  have hy := (hx.map f).map (algebraMap _ (Localization.AtPrime M))
  generalize algebraMap _ (Localization.AtPrime M) (f x) = y at *
  have := EssFiniteType.of_isLocalization (Localization.AtPrime M) M.primeCompl
  have := of_isLocalization (Rₘ := Localization.AtPrime M) M.primeCompl
  have := EssFiniteType.comp (AlgebraicClosure K) (AlgebraicClosure K ⊗[K] A)
    (Localization.AtPrime M)
  have := comp (AlgebraicClosure K) (AlgebraicClosure K ⊗[K] A)
    (Localization.AtPrime M)
  let := (isField_of_isAlgClosed_of_isLocalRing (AlgebraicClosure K)
    (A := Localization.AtPrime M)).toField
  exact hy.eq_zero

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

Goal state before:
pp: A : Type u_2
inst✝⁵ : CommRing A
B : Type u_4
inst✝⁴ : CommRing B
inst✝³ : Algebra A B
inst✝² : EssFiniteType A B
inst✝¹ : FormallyUnramified A B
p : Ideal A
inst✝ : p.IsMaximal
this : Field (A ⧸ p) := Ideal.Quotient.field p
⊢ IsReduced (B ⧸ Ideal.map (algebraMap A B) p)

Tactic:
exact Algebra.FormallyUnramified.isReduced_of_field (A ⧸ p) (B ⧸ p.map (algebraMap A B))

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

Goal state before:
pp: R : Type u₁
S : Type u₂
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
T : Type u₃
inst✝³ : CommRing T
inst✝² : Algebra R T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
ι : Type w₁
σ : Type w₂
τ : Type w₅
Q : Generators S T ι
P : Generators R S σ
W : Generators R T τ
f : W.Hom Q
⊢ Function.Exact
    ⇑(↑T (Extension.H1Cotangent.map f.toExtensionHom) ∘ₗ
        Extension.H1Cotangent.map ((Q.comp P).defaultHom W).toExtensionHom)
    ⇑(δ Q P)

Tactic:
rw [← Extension.H1Cotangent.map_comp, Extension.H1Cotangent.map_eq _ (Q.ofComp P).toExtensionHom]

Goal state after:
R : Type u₁
S : Type u₂
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
T : Type u₃
inst✝³ : CommRing T
inst✝² : Algebra R T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
ι : Type w₁
σ : Type w₂
τ : Type w₅
Q : Generators S T ι
P : Generators R S σ
W : Generators R T τ
f : W.Hom Q
⊢ Function.Exact ⇑(Extension.H1Cotangent.map (Q.ofComp P).toExtensionHom) ⇑(δ Q P)
Module: Mathlib.RingTheory.Etale.Kaehler

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : CommRing T
inst✝³ : Algebra R S
inst✝² : Algebra R T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
P : Extension R S
Q : Extension R T
f : P.Hom Q
alg : Algebra P.Ring Q.Ring
halg : algebraMap P.Ring Q.Ring = f.toRingHom
H : Function.Bijective ⇑(LinearMap.liftBaseChange Q.Ring (f.mapKer halg))
this✝¹ : Algebra P.Ring T := ((algebraMap S T).comp (algebraMap P.Ring S)).toAlgebra
this✝ : IsScalarTower P.Ring S T
this : IsScalarTower P.Ring Q.Ring T
e : Q.Ring ⊗[P.Ring] ↥P.ker ≃ₗ[Q.Ring] ↥Q.ker :=
  LinearEquiv.ofBijective (LinearMap.liftBaseChange Q.Ring (f.mapKer halg)) H
f' : ↥Q.ker →ₗ[Q.Ring] T ⊗[S] P.Cotangent :=
  LinearMap.liftBaseChange Q.Ring (↑P.Ring ((TensorProduct.mk S T P.Cotangent) 1) ∘ₗ Cotangent.mk) ∘ₗ ↑e.symm
x y : Q.Ring ⊗[P.Ring] ↥P.ker
⊢ ↑(e x) • (LinearMap.liftBaseChange Q.Ring (↑P.Ring ((TensorProduct.mk S T P.Cotangent) 1) ∘ₗ Cotangent.mk)) y = 0

Tactic:
induction x with

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

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
inst✝⁵ : EssFiniteType R S
p : Ideal R
inst✝⁴ : p.IsPrime
q : Ideal S
inst✝³ : q.IsPrime
inst✝² : q.LiesOver p
inst✝¹ : Algebra (Localization.AtPrime p) (Localization.AtPrime q)
inst✝ : Localization.AtPrime.IsLiesOverAlgebra p q
⊢ IsUnramifiedAt R q ↔
    Algebra.IsSeparable p.ResidueField q.ResidueField ∧
      Ideal.map (algebraMap R (Localization.AtPrime q)) p = maximalIdeal (Localization.AtPrime q)

Tactic:
have : EssFiniteType (Localization.AtPrime p) (Localization.AtPrime q) := .of_comp R _ _

Goal state after:
R : Type u_1
S : Type u_2
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
inst✝⁵ : EssFiniteType R S
p : Ideal R
inst✝⁴ : p.IsPrime
q : Ideal S
inst✝³ : q.IsPrime
inst✝² : q.LiesOver p
inst✝¹ : Algebra (Localization.AtPrime p) (Localization.AtPrime q)
inst✝ : Localization.AtPrime.IsLiesOverAlgebra p q
this : EssFiniteType (Localization.AtPrime p) (Localization.AtPrime q)
⊢ IsUnramifiedAt R q ↔
    Algebra.IsSeparable p.ResidueField q.ResidueField ∧
      Ideal.map (algebraMap R (Localization.AtPrime q)) p = maximalIdeal (Localization.AtPrime q)
Module: Mathlib.RingTheory.Unramified.LocalRing

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
p : Ideal R
inst✝⁵ : p.IsPrime
q : Ideal S
inst✝⁴ : q.IsPrime
hq : p.primesOver S = {q}
inst✝³ : Module.Finite R S
inst✝² : q.LiesOver p
inst✝¹ : Algebra (Localization.AtPrime p) (Localization.AtPrime q)
inst✝ : AtPrime.IsLiesOverAlgebra p q
s✝ : Finset S
hs : Submodule.span R ↑s✝ = ⊤
x s : S
hsq : s ∈ q.primeCompl
r : R
hr : r ∉ p
t : S
e' : (algebraMap R S) r = s * t
⊢ (algebraMap S (Localization.AtPrime q)) s *
      ((algebraMap S (Localization.AtPrime q)) t * IsLocalization.mk' (Localization.AtPrime q) x ⟨s, hsq⟩) ∈
    Submodule.span (Localization.AtPrime p) (Set.range ⇑(algebraMap S (Localization.AtPrime q)))

Tactic:
mul_left_comm,

Goal state after:
R : Type u_1
S : Type u_2
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
p : Ideal R
inst✝⁵ : p.IsPrime
q : Ideal S
inst✝⁴ : q.IsPrime
hq : p.primesOver S = {q}
inst✝³ : Module.Finite R S
inst✝² : q.LiesOver p
inst✝¹ : Algebra (Localization.AtPrime p) (Localization.AtPrime q)
inst✝ : AtPrime.IsLiesOverAlgebra p q
s✝ : Finset S
hs : Submodule.span R ↑s✝ = ⊤
x s : S
hsq : s ∈ q.primeCompl
r : R
hr : r ∉ p
t : S
e' : (algebraMap R S) r = s * t
⊢ (algebraMap S (Localization.AtPrime q)) t *
      ((algebraMap S (Localization.AtPrime q)) s * IsLocalization.mk' (Localization.AtPrime q) x ⟨s, hsq⟩) ∈
    Submodule.span (Localization.AtPrime p) (Set.range ⇑(algebraMap S (Localization.AtPrime q)))
Module: Mathlib.RingTheory.Etale.Kaehler

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : CommRing T
inst✝³ : Algebra R S
inst✝² : Algebra R T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
P : Extension R S
Q : Extension R T
f : P.Hom Q
alg : Algebra P.Ring Q.Ring
halg : algebraMap P.Ring Q.Ring = f.toRingHom
H : Function.Bijective ⇑(LinearMap.liftBaseChange Q.Ring (f.mapKer halg))
x : T ⊗[S] P.Cotangent
⊢ (tensorCotangentInvFun f halg H) ((LinearMap.liftBaseChange T (Cotangent.map f)) x) = x

Tactic:
induction x with

Goal state after:
Module: Mathlib.RingTheory.RamificationInertia.Ramification

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
p : Ideal R
q : Ideal S
inst✝¹ : q.LiesOver p
inst✝ : q.IsPrime
⊢ q.ramificationIdx R =
    (Module.length (Localization.AtPrime q)
        (Localization.AtPrime q ⧸ map (algebraMap R (Localization.AtPrime q)) p)).toNat

Tactic:
by
  rw [ramificationIdx_def, over_def q p]

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

Goal state before:
pp: case tmul
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : CommRing T
inst✝³ : Algebra R S
inst✝² : Algebra R T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
P : Extension R S
Q : Extension R T
f : P.Hom Q
alg : Algebra P.Ring Q.Ring
halg : algebraMap P.Ring Q.Ring = f.toRingHom
H : Function.Bijective ⇑(LinearMap.liftBaseChange Q.Ring (f.mapKer halg))
a : Q.Ring
b : ↥P.ker
⊢ (LinearMap.liftBaseChange T (Cotangent.map f))
      ((tensorCotangentInvFun f halg H)
        (Cotangent.mk ((LinearMap.liftBaseChange Q.Ring (f.mapKer halg)) (a ⊗ₜ[P.Ring] b)))) =
    Cotangent.mk ((LinearMap.liftBaseChange Q.Ring (f.mapKer halg)) (a ⊗ₜ[P.Ring] b))

Tactic:
simp only [LinearMap.liftBaseChange_tmul, map_smul]

Goal state after:
case tmul
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : CommRing T
inst✝³ : Algebra R S
inst✝² : Algebra R T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
P : Extension R S
Q : Extension R T
f : P.Hom Q
alg : Algebra P.Ring Q.Ring
halg : algebraMap P.Ring Q.Ring = f.toRingHom
H : Function.Bijective ⇑(LinearMap.liftBaseChange Q.Ring (f.mapKer halg))
a : Q.Ring
b : ↥P.ker
⊢ (LinearMap.liftBaseChange T (Cotangent.map f))
      ((tensorCotangentInvFun f halg H) (a • Cotangent.mk ((f.mapKer halg) b))) =
    a • Cotangent.mk ((f.mapKer halg) b)
Module: Mathlib.RingTheory.Etale.Kaehler

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : CommRing T
inst✝³ : Algebra R S
inst✝² : Algebra R T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
P : Extension R S
Q : Extension R T
f : P.Hom Q
alg : Algebra P.Ring Q.Ring
halg : algebraMap P.Ring Q.Ring = f.toRingHom
H : Function.Bijective ⇑(LinearMap.liftBaseChange Q.Ring (f.mapKer halg))
x : Q.Ring ⊗[P.Ring] ↥P.ker
⊢ (LinearMap.liftBaseChange T (Cotangent.map f))
      ((tensorCotangentInvFun f halg H) (Cotangent.mk ((LinearMap.liftBaseChange Q.Ring (f.mapKer halg)) x))) =
    Cotangent.mk ((LinearMap.liftBaseChange Q.Ring (f.mapKer halg)) x)

Tactic:
induction x with

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

Goal state before:
pp: case h
R : Type u₁
S : Type u₂
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
T : Type u₃
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
ι : Type w₁
σ : Type w₂
Q : Generators S T ι
P : Generators R S σ
inst✝ : Flat S T
x : T ⊗[S] P.toExtension.H1Cotangent
x_in :
  (liftBaseChange T (Extension.Cotangent.map (Q.toComp P).toExtensionHom)) ((lTensor T Extension.h1Cotangentι) x) ∈
    (Q.comp P).toExtension.cotangentComplex.ker
⊢ (liftBaseChange T (Extension.H1Cotangent.map (Q.toComp P).toExtensionHom)) x =
    ⟨(liftBaseChange T (Extension.Cotangent.map (Q.toComp P).toExtensionHom)) ((lTensor T Extension.h1Cotangentι) x),
      x_in⟩

Tactic:
induction x with

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

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
p : Ideal R
inst✝² : p.IsPrime
q : Ideal S
inst✝¹ : q.IsPrime
inst✝ : q.LiesOver p
H : Function.Injective ⇑(localRingHom p q (algebraMap R S) ⋯)
s : Finset R
hs : Ideal.span ↑s = RingHom.ker (algebraMap R S)
this : ∀ (x : ↥s), (algebraMap R (Localization.AtPrime p)) ↑x = 0
m : ↥s → ↥p.primeCompl
hm : ∀ (x : ↥s), ↑(m x) * ↑x = 0
⊢ RingHom.ker (algebraMap R S) ≤ RingHom.ker (algebraMap R (Away ↑(∏ i, m i)))

Tactic:
rw [← hs, Ideal.span_le]

Goal state after:
R : Type u_1
S : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
p : Ideal R
inst✝² : p.IsPrime
q : Ideal S
inst✝¹ : q.IsPrime
inst✝ : q.LiesOver p
H : Function.Injective ⇑(localRingHom p q (algebraMap R S) ⋯)
s : Finset R
hs : Ideal.span ↑s = RingHom.ker (algebraMap R S)
this : ∀ (x : ↥s), (algebraMap R (Localization.AtPrime p)) ↑x = 0
m : ↥s → ↥p.primeCompl
hm : ∀ (x : ↥s), ↑(m x) * ↑x = 0
⊢ ↑s ⊆ ↑(RingHom.ker (algebraMap R (Away ↑(∏ i, m i))))
Module: Mathlib.RingTheory.Unramified.LocalRing

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
p : Ideal R
inst✝² : p.IsPrime
q : Ideal S
inst✝¹ : q.IsPrime
inst✝ : q.LiesOver p
H : Function.Injective ⇑(localRingHom p q (algebraMap R S) ⋯)
s : Finset R
hs : Ideal.span ↑s = RingHom.ker (algebraMap R S)
this : ∀ (x : ↥s), (algebraMap R (Localization.AtPrime p)) ↑x = 0
m : ↥s → ↥p.primeCompl
hm : ∀ (x : ↥s), ↑(m x) * ↑x = 0
x : R
hxs : x ∈ ↑s
⊢ ↑(m ⟨x, hxs⟩) * (↑(∏ i ∈ {⟨x, hxs⟩}ᶜ, m i) * x) = 0

Tactic:
mul_left_comm,

Goal state after:
R : Type u_1
S : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
p : Ideal R
inst✝² : p.IsPrime
q : Ideal S
inst✝¹ : q.IsPrime
inst✝ : q.LiesOver p
H : Function.Injective ⇑(localRingHom p q (algebraMap R S) ⋯)
s : Finset R
hs : Ideal.span ↑s = RingHom.ker (algebraMap R S)
this : ∀ (x : ↥s), (algebraMap R (Localization.AtPrime p)) ↑x = 0
m : ↥s → ↥p.primeCompl
hm : ∀ (x : ↥s), ↑(m x) * ↑x = 0
x : R
hxs : x ∈ ↑s
⊢ ↑(∏ i ∈ {⟨x, hxs⟩}ᶜ, m i) * (↑(m ⟨x, hxs⟩) * x) = 0
Module: Mathlib.RingTheory.Unramified.LocalRing

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
p : Ideal R
inst✝² : p.IsPrime
q : Ideal S
inst✝¹ : q.IsPrime
inst✝ : q.LiesOver p
H✝ : Function.Injective ⇑(localRingHom p q (algebraMap R S) ⋯)
s✝ : Finset R
hs : Ideal.span ↑s✝ = RingHom.ker (algebraMap R S)
this : ∀ (x : ↥s✝), (algebraMap R (Localization.AtPrime p)) ↑x = 0
m : ↥s✝ → ↥p.primeCompl
hm : ∀ (x : ↥s✝), ↑(m x) * ↑x = 0
H : RingHom.ker (algebraMap R S) ≤ RingHom.ker (algebraMap R (Away ↑(∏ i, m i)))
r' s : R
e : r' = ↑(∏ i, m i) * s
x : R
w✝ : ↥(Submonoid.powers r')
n : ℕ
hn : (algebraMap R S) (r' ^ n * x) = 0
k : ℕ
hk : ↑(∏ i, m i) ^ k * (r' ^ n * x) = 0
⊢ ↑(∏ i, m i) ^ k * (s ^ k * (r' ^ n * x)) = 0

Tactic:
mul_left_comm,

Goal state after:
R : Type u_1
S : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
p : Ideal R
inst✝² : p.IsPrime
q : Ideal S
inst✝¹ : q.IsPrime
inst✝ : q.LiesOver p
H✝ : Function.Injective ⇑(localRingHom p q (algebraMap R S) ⋯)
s✝ : Finset R
hs : Ideal.span ↑s✝ = RingHom.ker (algebraMap R S)
this : ∀ (x : ↥s✝), (algebraMap R (Localization.AtPrime p)) ↑x = 0
m : ↥s✝ → ↥p.primeCompl
hm : ∀ (x : ↥s✝), ↑(m x) * ↑x = 0
H : RingHom.ker (algebraMap R S) ≤ RingHom.ker (algebraMap R (Away ↑(∏ i, m i)))
r' s : R
e : r' = ↑(∏ i, m i) * s
x : R
w✝ : ↥(Submonoid.powers r')
n : ℕ
hn : (algebraMap R S) (r' ^ n * x) = 0
k : ℕ
hk : ↑(∏ i, m i) ^ k * (r' ^ n * x) = 0
⊢ s ^ k * (↑(∏ i, m i) ^ k * (r' ^ n * x)) = 0
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
x : H1Cotangent R S
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 ⋯
⊢ (IsScalarTower.toAlgHom R Q.Ring T).comp (MvPolynomial.aeval Q.σ) =
    IsScalarTower.toAlgHom R (Generators.self R T).toExtension.Ring T

Tactic:
ext i

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
x : H1Cotangent R S
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 ⋯
i : T
⊢ ((IsScalarTower.toAlgHom R Q.Ring T).comp (MvPolynomial.aeval Q.σ)) (MvPolynomial.X i) =
    (IsScalarTower.toAlgHom R (Generators.self R T).toExtension.Ring T) (MvPolynomial.X i)
Module: Mathlib.Algebra.Order.Archimedean.Real.Basic

Goal state before:
pp: case refine_1
s : Set ℝ
L : ℝ
hL : L ∈ s
U : ℝ
hU : U ∈ upperBounds s
this : ∀ (d : ℕ), BddAbove {m | ∃ y ∈ s, ↑m ≤ y * ↑d}
f : ℕ → ℤ
hf : ∀ (d : ℕ), f d ∈ {m | ∃ y ∈ s, ↑m ≤ y * ↑d} ∧ ∀ z ∈ {m | ∃ y ∈ s, ↑m ≤ y * ↑d}, z ≤ f d
hf₁ : ∀ n > 0, ∃ y ∈ s, ↑(↑(f n) / ↑n) ≤ y
hf₂ : ∀ n > 0, ∀ y ∈ s, y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)
hg : IsCauSeq abs fun n ↦ ↑(f n) / ↑n
g : CauSeq ℚ abs := ⟨fun n ↦ ↑(f n) / ↑n, hg⟩
x : ℝ
xS : x ∈ s
z : ℝ
xz : z < x
K : ℕ
hK : (x - z)⁻¹ < ↑K
n : ℕ
nK : n ≥ K
⊢ z ≤ ↑(↑g n)

Tactic:
replace xz := sub_pos.2 xz

Goal state after:
case refine_1
s : Set ℝ
L : ℝ
hL : L ∈ s
U : ℝ
hU : U ∈ upperBounds s
this : ∀ (d : ℕ), BddAbove {m | ∃ y ∈ s, ↑m ≤ y * ↑d}
f : ℕ → ℤ
hf : ∀ (d : ℕ), f d ∈ {m | ∃ y ∈ s, ↑m ≤ y * ↑d} ∧ ∀ z ∈ {m | ∃ y ∈ s, ↑m ≤ y * ↑d}, z ≤ f d
hf₁ : ∀ n > 0, ∃ y ∈ s, ↑(↑(f n) / ↑n) ≤ y
hf₂ : ∀ n > 0, ∀ y ∈ s, y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)
hg : IsCauSeq abs fun n ↦ ↑(f n) / ↑n
g : CauSeq ℚ abs := ⟨fun n ↦ ↑(f n) / ↑n, hg⟩
x : ℝ
xS : x ∈ s
z : ℝ
K : ℕ
hK : (x - z)⁻¹ < ↑K
n : ℕ
nK : n ≥ K
xz : 0 < x - z
⊢ z ≤ ↑(↑g n)
Module: Mathlib.Algebra.Order.Archimedean.Real.Basic

Goal state before:
pp: case refine_2.refine_1
f : ℝ → ℝ
x : ℝ
hf : BddBelow (f '' Ioi x)
hf_mono : Monotone f
q : ↑(Ioi x)
hq : x < ↑q
y : ℚ
hxy : x < ↑y
hyq : ↑y < ↑q
⊢ BddBelow (range fun q ↦ f ↑↑q)

Tactic:
refine ⟨hf.some, fun z => ?_⟩

Goal state after:
case refine_2.refine_1
f : ℝ → ℝ
x : ℝ
hf : BddBelow (f '' Ioi x)
hf_mono : Monotone f
q : ↑(Ioi x)
hq : x < ↑q
y : ℚ
hxy : x < ↑y
hyq : ↑y < ↑q
z : ℝ
⊢ (z ∈ range fun q ↦ f ↑↑q) → Set.Nonempty.some hf ≤ z
Module: Mathlib.Data.NNReal.Defs

Goal state before:
pp: case inr
s : Set ℝ
h : s.OrdConnected
x : ℝ
hx : x ∈ s
y : ℝ
hy : y ∈ s
z : ℝ≥0
hz : z ∈ Icc x.toNNReal y.toNNReal
hy₀ : 0 ≤ y
⊢ z ∈ toNNReal '' s

Tactic:
lift y to ℝ≥0 using hy₀

Goal state after:
case inr
s : Set ℝ
h : s.OrdConnected
x : ℝ
hx : x ∈ s
z y : ℝ≥0
hy : ↑y ∈ s
hz : z ∈ Icc x.toNNReal (↑y).toNNReal
⊢ z ∈ toNNReal '' s
Module: Mathlib.RingTheory.DedekindDomain.Factorization

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
I : Ideal R
h0 : I ≠ 0
⊢ ⨅ i, i.maxPowDividing I = ∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I

Tactic:
rw [finprod_def, dif_pos (Ideal.hasFiniteMulSupport h0), Ideal.prod_eq_iInf_of_pairwise_isCoprime]

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
I : Ideal R
h0 : I ≠ 0
⊢ ⨅ i, i.maxPowDividing I = ⨅ i ∈ Finite.toFinset ⋯, i.maxPowDividing I

R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
I : Ideal R
h0 : I ≠ 0
⊢ (↑(Finite.toFinset ⋯)).Pairwise (IsCoprime on fun i ↦ i.maxPowDividing I)
Module: Mathlib.Algebra.Order.Archimedean.Real.Basic

Goal state before:
pp: a : ℝ
ha : 1 < a
k : ℕ
posk : 0 < k
hk : 1 / ↑k + 1 < a
⊢ (1 / ↑k + 1) ^ (k * (2 * k)) = (1 / ↑k + 1) ^ (2 * k ^ 2)

Tactic:
mul_left_comm,

Goal state after:
a : ℝ
ha : 1 < a
k : ℕ
posk : 0 < k
hk : 1 / ↑k + 1 < a
⊢ (1 / ↑k + 1) ^ (2 * (k * k)) = (1 / ↑k + 1) ^ (2 * k ^ 2)
Module: Mathlib.RingTheory.DedekindDomain.Factorization

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
I : FractionalIdeal R⁰ K
hI : I ≠ 0
a : R
J : Ideal R
haJ : I = spanSingleton R⁰ ((algebraMap R K) a)⁻¹ * ↑J
hJ_ne_zero : J ≠ 0
hJ : ∏ᶠ (v : HeightOneSpectrum R), ↑v.asIdeal ^ ↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) = ↑J
ha_ne_zero : span {a} ≠ 0
ha :
  ∏ᶠ (v : HeightOneSpectrum R), ↑v.asIdeal ^ ↑((Associates.mk v.asIdeal).count (Associates.mk (span {a})).factors) =
    ↑(span {a})
⊢ ∏ᶠ (v : HeightOneSpectrum R),
      ↑v.asIdeal ^
        (↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) -
          ↑((Associates.mk v.asIdeal).count (Associates.mk (span {a})).factors)) =
    spanSingleton R⁰ ((algebraMap R K) a)⁻¹ * ↑J

Tactic:
← div_spanSingleton,

Goal state after:
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
I : FractionalIdeal R⁰ K
hI : I ≠ 0
a : R
J : Ideal R
haJ : I = spanSingleton R⁰ ((algebraMap R K) a)⁻¹ * ↑J
hJ_ne_zero : J ≠ 0
hJ : ∏ᶠ (v : HeightOneSpectrum R), ↑v.asIdeal ^ ↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) = ↑J
ha_ne_zero : span {a} ≠ 0
ha :
  ∏ᶠ (v : HeightOneSpectrum R), ↑v.asIdeal ^ ↑((Associates.mk v.asIdeal).count (Associates.mk (span {a})).factors) =
    ↑(span {a})
⊢ ∏ᶠ (v : HeightOneSpectrum R),
      ↑v.asIdeal ^
        (↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) -
          ↑((Associates.mk v.asIdeal).count (Associates.mk (span {a})).factors)) =
    ↑J / spanSingleton R⁰ ((algebraMap R K) a)
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✝² : Group E
inst✝¹ : Group F
inst✝ : Group G
p✝ q✝ : GroupSeminorm E
f : F →* E
p q : GroupSeminorm E
x : E
⊢ p x⁻¹ + q x⁻¹ = p x + q x

Tactic:
by simp_rw [map_inv_eq_map p, map_inv_eq_map q]

Goal state after:
Module: Mathlib.Data.ENNReal.Real

Goal state before:
pp: case inr.inl
p : ℝ≥0∞
hp : 0 < p
hpq : p ≤ ∞
⊢ p = 0 ∧ ∞ = 0 ∨
    p = 0 ∧ ∞ = ∞ ∨
      p = 0 ∧ 0 < ∞.toReal ∨ p = ∞ ∧ ∞ = ∞ ∨ 0 < p.toReal ∧ ∞ = ∞ ∨ 0 < p.toReal ∧ 0 < ∞.toReal ∧ p.toReal ≤ ∞.toReal

pp: case inr.inr
p q : ℝ≥0∞
hpq : p ≤ q
hp : 0 < p
hq : q < ∞
⊢ p = 0 ∧ q = 0 ∨
    p = 0 ∧ q = ∞ ∨
      p = 0 ∧ 0 < q.toReal ∨ p = ∞ ∧ q = ∞ ∨ 0 < p.toReal ∧ q = ∞ ∨ 0 < p.toReal ∧ 0 < q.toReal ∧ p.toReal ≤ q.toReal

Tactic:
· simpa using p.trichotomy

Goal state after:
case inr.inr
p q : ℝ≥0∞
hpq : p ≤ q
hp : 0 < p
hq : q < ∞
⊢ p = 0 ∧ q = 0 ∨
    p = 0 ∧ q = ∞ ∨
      p = 0 ∧ 0 < q.toReal ∨ p = ∞ ∧ q = ∞ ∨ 0 < p.toReal ∧ q = ∞ ∨ 0 < p.toReal ∧ 0 < q.toReal ∧ p.toReal ≤ q.toReal