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

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝¹⁹ : CommRing A
inst✝¹⁸ : CommRing B
inst✝¹⁷ : Algebra A B
G : Type u_3
inst✝¹⁶ : Group G
inst✝¹⁵ : Finite G
inst✝¹⁴ : MulSemiringAction G B
inst✝¹³ : SMulCommClass G A B
P : Ideal A
Q : Ideal B
inst✝¹² : Q.IsPrime
inst✝¹¹ : Q.LiesOver P
K : Type u_4
L : Type u_5
inst✝¹⁰ : Field K
inst✝⁹ : Field L
inst✝⁸ : Algebra (A ⧸ P) K
inst✝⁷ : Algebra (B ⧸ Q) L
inst✝⁶ : Algebra (A ⧸ P) L
inst✝⁵ : IsScalarTower (A ⧸ P) (B ⧸ Q) L
inst✝⁴ : Algebra K L
inst✝³ : IsScalarTower (A ⧸ P) K L
inst✝² : Algebra.IsInvariant A B G
inst✝¹ : IsFractionRing (A ⧸ P) K
inst✝ : IsFractionRing (B ⧸ Q) L
f : Gal(L/K)
val✝ : Fintype G
this✝ : P.IsPrime
this : Algebra.IsIntegral A B
x y : B ⧸ Q
hx :
  ∀ (g : ↥(MulAction.stabilizer G Q)),
    ((IsFractionRing.stabilizerHom G P Q K L) g) ((algebraMap (B ⧸ Q) L) x / (algebraMap (B ⧸ Q) L) y) =
      (algebraMap (B ⧸ Q) L) x / (algebraMap (B ⧸ Q) L) y
b : B ⧸ Q
a : A ⧸ P
h : x * (algebraMap (A ⧸ P) (B ⧸ Q)) a = b * y
ha : (algebraMap (A ⧸ P) L) a ≠ 0
hy : (algebraMap (B ⧸ Q) L) y ≠ 0
⊢ (algebraMap (B ⧸ Q) L) x / (algebraMap (B ⧸ Q) L) y = (algebraMap (B ⧸ Q) L) b / (algebraMap (A ⧸ P) L) a

Tactic:
div_eq_div_iff hy ha,

Goal state after:
A : Type u_1
B : Type u_2
inst✝¹⁹ : CommRing A
inst✝¹⁸ : CommRing B
inst✝¹⁷ : Algebra A B
G : Type u_3
inst✝¹⁶ : Group G
inst✝¹⁵ : Finite G
inst✝¹⁴ : MulSemiringAction G B
inst✝¹³ : SMulCommClass G A B
P : Ideal A
Q : Ideal B
inst✝¹² : Q.IsPrime
inst✝¹¹ : Q.LiesOver P
K : Type u_4
L : Type u_5
inst✝¹⁰ : Field K
inst✝⁹ : Field L
inst✝⁸ : Algebra (A ⧸ P) K
inst✝⁷ : Algebra (B ⧸ Q) L
inst✝⁶ : Algebra (A ⧸ P) L
inst✝⁵ : IsScalarTower (A ⧸ P) (B ⧸ Q) L
inst✝⁴ : Algebra K L
inst✝³ : IsScalarTower (A ⧸ P) K L
inst✝² : Algebra.IsInvariant A B G
inst✝¹ : IsFractionRing (A ⧸ P) K
inst✝ : IsFractionRing (B ⧸ Q) L
f : Gal(L/K)
val✝ : Fintype G
this✝ : P.IsPrime
this : Algebra.IsIntegral A B
x y : B ⧸ Q
hx :
  ∀ (g : ↥(MulAction.stabilizer G Q)),
    ((IsFractionRing.stabilizerHom G P Q K L) g) ((algebraMap (B ⧸ Q) L) x / (algebraMap (B ⧸ Q) L) y) =
      (algebraMap (B ⧸ Q) L) x / (algebraMap (B ⧸ Q) L) y
b : B ⧸ Q
a : A ⧸ P
h : x * (algebraMap (A ⧸ P) (B ⧸ Q)) a = b * y
ha : (algebraMap (A ⧸ P) L) a ≠ 0
hy : (algebraMap (B ⧸ Q) L) y ≠ 0
⊢ (algebraMap (B ⧸ Q) L) x * (algebraMap (A ⧸ P) L) a = (algebraMap (B ⧸ Q) L) b * (algebraMap (B ⧸ Q) L) y
Module: Mathlib.RingTheory.Invariant.Basic

Goal state before:
pp: case intro
A : Type u_1
B : Type u_2
k : Type u_3
inst✝¹³ : CommRing A
inst✝¹² : CommRing B
inst✝¹¹ : Algebra A B
G : Type u_4
inst✝¹⁰ : Finite G
inst✝⁹ : Group G
inst✝⁸ : MulSemiringAction G B
inst✝⁷ : Algebra.IsInvariant A B G
P✝ : Ideal A
Q : Ideal B
inst✝⁶ : Q.LiesOver P✝
inst✝⁵ : CommRing k
inst✝⁴ : Algebra (A ⧸ P✝) k
inst✝³ : Algebra (B ⧸ Q) k
inst✝² : IsScalarTower (A ⧸ P✝) (B ⧸ Q) k
inst✝¹ : IsDomain k
inst✝ : FaithfulSMul (B ⧸ Q) k
σ : k →ₐ[A ⧸ P✝] k
f : B ⧸ Q →ₐ[A ⧸ P✝] k := IsScalarTower.toAlgHom (A ⧸ P✝) (B ⧸ Q) k
hf : Function.Injective ⇑f
x : B
val✝ : Fintype G
algInst✝¹ : Algebra A k := ((algebraMap (A ⧸ P✝) k).comp (algebraMap A (A ⧸ P✝))).toAlgebra
algInst✝ : Algebra B k := ((algebraMap (B ⧸ Q) k).comp (algebraMap B (B ⧸ Q))).toAlgebra
scalarTowerInst✝¹ : IsScalarTower A (A ⧸ P✝) k := IsScalarTower.of_algebraMap_eq' (Eq.refl (algebraMap A k))
scalarTowerInst✝ : IsScalarTower B (B ⧸ Q) k := IsScalarTower.of_algebraMap_eq' (Eq.refl (algebraMap B k))
this✝² : IsScalarTower A (B ⧸ Q) k
this✝¹ : IsScalarTower A B k
P : Polynomial A
hp : (Polynomial.mapRingHom (algebraMap A B)) P = MulSemiringAction.charpoly G x
this✝ : (Polynomial.aeval x) P = 0
this : (Polynomial.aeval (σ ((algebraMap (B ⧸ Q) k) ((mk Q) x)))) (Polynomial.map (algebraMap A B) P) = 0
⊢ (σ.comp f).toRingHom ((mk Q) x) ∈ f.range

Tactic:
← Polynomial.coe_mapRingHom,

Goal state after:
case intro
A : Type u_1
B : Type u_2
k : Type u_3
inst✝¹³ : CommRing A
inst✝¹² : CommRing B
inst✝¹¹ : Algebra A B
G : Type u_4
inst✝¹⁰ : Finite G
inst✝⁹ : Group G
inst✝⁸ : MulSemiringAction G B
inst✝⁷ : Algebra.IsInvariant A B G
P✝ : Ideal A
Q : Ideal B
inst✝⁶ : Q.LiesOver P✝
inst✝⁵ : CommRing k
inst✝⁴ : Algebra (A ⧸ P✝) k
inst✝³ : Algebra (B ⧸ Q) k
inst✝² : IsScalarTower (A ⧸ P✝) (B ⧸ Q) k
inst✝¹ : IsDomain k
inst✝ : FaithfulSMul (B ⧸ Q) k
σ : k →ₐ[A ⧸ P✝] k
f : B ⧸ Q →ₐ[A ⧸ P✝] k := IsScalarTower.toAlgHom (A ⧸ P✝) (B ⧸ Q) k
hf : Function.Injective ⇑f
x : B
val✝ : Fintype G
algInst✝¹ : Algebra A k := ((algebraMap (A ⧸ P✝) k).comp (algebraMap A (A ⧸ P✝))).toAlgebra
algInst✝ : Algebra B k := ((algebraMap (B ⧸ Q) k).comp (algebraMap B (B ⧸ Q))).toAlgebra
scalarTowerInst✝¹ : IsScalarTower A (A ⧸ P✝) k := IsScalarTower.of_algebraMap_eq' (Eq.refl (algebraMap A k))
scalarTowerInst✝ : IsScalarTower B (B ⧸ Q) k := IsScalarTower.of_algebraMap_eq' (Eq.refl (algebraMap B k))
this✝² : IsScalarTower A (B ⧸ Q) k
this✝¹ : IsScalarTower A B k
P : Polynomial A
hp : (Polynomial.mapRingHom (algebraMap A B)) P = MulSemiringAction.charpoly G x
this✝ : (Polynomial.aeval x) P = 0
this : (Polynomial.aeval (σ ((algebraMap (B ⧸ Q) k) ((mk Q) x)))) ((Polynomial.mapRingHom (algebraMap A B)) P) = 0
⊢ (σ.comp f).toRingHom ((mk Q) x) ∈ f.range
Module: Mathlib.RingTheory.Invariant.Basic

Goal state before:
pp: A : Type u_1
B : Type u_2
k : Type u_3
inst✝¹³ : CommRing A
inst✝¹² : CommRing B
inst✝¹¹ : Algebra A B
G : Type u_4
inst✝¹⁰ : Finite G
inst✝⁹ : Group G
inst✝⁸ : MulSemiringAction G B
inst✝⁷ : Algebra.IsInvariant A B G
P✝ : Ideal A
Q : Ideal B
inst✝⁶ : Q.LiesOver P✝
inst✝⁵ : CommRing k
inst✝⁴ : Algebra (A ⧸ P✝) k
inst✝³ : Algebra (B ⧸ Q) k
inst✝² : IsScalarTower (A ⧸ P✝) (B ⧸ Q) k
inst✝¹ : IsDomain k
inst✝ : FaithfulSMul (B ⧸ Q) k
σ : k →ₐ[A ⧸ P✝] k
f : B ⧸ Q →ₐ[A ⧸ P✝] k := IsScalarTower.toAlgHom (A ⧸ P✝) (B ⧸ Q) k
hf : Function.Injective ⇑f
x : B
val✝ : Fintype G
algInst✝¹ : Algebra A k := ((algebraMap (A ⧸ P✝) k).comp (algebraMap A (A ⧸ P✝))).toAlgebra
algInst✝ : Algebra B k := ((algebraMap (B ⧸ Q) k).comp (algebraMap B (B ⧸ Q))).toAlgebra
scalarTowerInst✝¹ : IsScalarTower A (A ⧸ P✝) k := IsScalarTower.of_algebraMap_eq' (Eq.refl (algebraMap A k))
scalarTowerInst✝ : IsScalarTower B (B ⧸ Q) k := IsScalarTower.of_algebraMap_eq' (Eq.refl (algebraMap B k))
this✝² : IsScalarTower A (B ⧸ Q) k
this✝¹ : IsScalarTower A B k
P : Polynomial A
hp : (Polynomial.mapRingHom (algebraMap A B)) P = MulSemiringAction.charpoly G x
this✝ : (Polynomial.aeval x) P = 0
this : (Polynomial.aeval (σ ((algebraMap (B ⧸ Q) k) ((mk Q) x)))) (MulSemiringAction.charpoly G x) = 0
⊢ ∃ τ, σ ((algebraMap B k) x) = (algebraMap B k) (τ • x)

Tactic:
simpa [MulSemiringAction.charpoly, sub_eq_zero, Finset.prod_eq_zero_iff] using! this

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

Goal state before:
pp: A : Type u_1
B : Type u_2
k : Type u_3
inst✝¹³ : CommRing A
inst✝¹² : CommRing B
inst✝¹¹ : Algebra A B
G : Type u_4
inst✝¹⁰ : Finite G
inst✝⁹ : Group G
inst✝⁸ : MulSemiringAction G B
inst✝⁷ : Algebra.IsInvariant A B G
P : Ideal A
Q : Ideal B
inst✝⁶ : Q.LiesOver P
inst✝⁵ : CommRing k
inst✝⁴ : Algebra (A ⧸ P) k
inst✝³ : Algebra (B ⧸ Q) k
inst✝² : IsScalarTower (A ⧸ P) (B ⧸ Q) k
inst✝¹ : IsDomain k
inst✝ : FaithfulSMul (B ⧸ Q) k
σ : k ≃ₐ[A ⧸ P] k
⊢ ∃ τ, ∀ (x : B ⧸ Q), (algebraMap (B ⧸ Q) k) (τ x) = σ ((algebraMap (B ⧸ Q) k) x)

Tactic:
by
  let f : (B ⧸ Q) →ₐ[A ⧸ P] k := IsScalarTower.toAlgHom _ _ _
  have hf : Function.Injective f := FaithfulSMul.algebraMap_injective _ _
  obtain ⟨τ₁, h₁⟩ := Ideal.Quotient.exists_algHom_fixedPoint_quotient_under G P Q σ.toAlgHom
  obtain ⟨τ₂, h₂⟩ := Ideal.Quotient.exists_algHom_fixedPoint_quotient_under G P Q σ.symm.toAlgHom
  refine ⟨{ __ := τ₁, invFun := τ₂, left_inv := ?_, right_inv := ?_ }, h₁⟩
  · intro x
    obtain ⟨x, rfl⟩ := Ideal.Quotient.mk_surjective x
    obtain ⟨y, e⟩ := Ideal.Quotient.mk_surjective (τ₁ (Ideal.Quotient.mk Q x))
    apply hf
    dsimp [f] at h₁ h₂ ⊢
    refine .trans ?_ (σ.symm_apply_apply _)
    rw [← h₁, ← e, h₂]
  · intro x
    obtain ⟨x, rfl⟩ := Ideal.Quotient.mk_surjective x
    obtain ⟨y, e⟩ := Ideal.Quotient.mk_surjective (τ₂ (Ideal.Quotient.mk Q x))
    apply hf
    dsimp [f] at h₁ h₂ ⊢
    refine .trans ?_ (σ.apply_symm_apply _)
    rw [← h₂, ← e, h₁]

Goal state after:
Module: Mathlib.LinearAlgebra.FreeModule.Finite.CardQuotient

Goal state before:
pp: M : Type u_1
inst✝⁴ : AddCommGroup M
inst✝³ : Free ℤ M
inst✝² : Module.Finite ℤ M
ι : Type u_2
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
b : Basis ι ℤ M
N : Submodule ℤ M
bN : Basis ι ℤ ↥N
e : M ≃ₗ[ℤ] ↥N := b.equiv bN (Equiv.refl ι)
⊢ (b.det (Subtype.val ∘ ⇑bN)).natAbs = Nat.card (M ⧸ N)

Tactic:
calc
    (b.det (N.subtype ∘ bN)).natAbs = (LinearMap.det (N.subtype ∘ₗ (e : M →ₗ[ℤ] N))).natAbs := by
      rw [Basis.det_comp_basis]
    _ = _ := natAbs_det_equiv N e

Goal state after:
Module: Mathlib.Topology.JacobsonSpace

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : JacobsonSpace X
h : (closedPoints X).Finite
this : closedPoints X = Set.univ
inst : Finite X
s : Set X
⊢ IsClosed[inst✝¹] sᶜ

Tactic:
← sᶜ.biUnion_of_singleton

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : JacobsonSpace X
h : (closedPoints X).Finite
this : closedPoints X = Set.univ
inst : Finite X
s : Set X
⊢ IsClosed[inst✝¹] (⋃ x ∈ sᶜ, {x})
Module: Mathlib.RingTheory.LocalRing.Length

Goal state before:
pp: A : Type u_1
B : Type u_2
M : Type u_3
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : IsLocalRing A
inst✝⁶ : IsLocalRing B
inst✝⁵ : Algebra A B
inst✝⁴ : IsLocalHom (algebraMap A B)
inst✝³ : AddCommGroup M
inst✝² : Module A M
inst✝¹ : Module B M
inst✝ : IsScalarTower A B M
p q : Submodule B M
h : p ⋖ q
f : ↥p →ₗ[B] ↥q := inclusion ⋯
⊢ length A ↥q = length A ↥p + length (ResidueField A) (ResidueField B)

Tactic:
have key : IsSimpleModule B (q ⧸ f.range) := by
    rwa [range_inclusion, ← covBy_iff_quot_is_simple h.le]

Goal state after:
A : Type u_1
B : Type u_2
M : Type u_3
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : IsLocalRing A
inst✝⁶ : IsLocalRing B
inst✝⁵ : Algebra A B
inst✝⁴ : IsLocalHom (algebraMap A B)
inst✝³ : AddCommGroup M
inst✝² : Module A M
inst✝¹ : Module B M
inst✝ : IsScalarTower A B M
p q : Submodule B M
h : p ⋖ q
f : ↥p →ₗ[B] ↥q := inclusion ⋯
key : IsSimpleModule B (↥q ⧸ f.range)
⊢ length A ↥q = length A ↥p + length (ResidueField A) (ResidueField B)
Module: Mathlib.RingTheory.RamificationInertia.Inertia

Goal state before:
pp: case pos
R : Type u_1
S : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
q : Ideal S
G : Type u_4
inst✝² : Group G
inst✝¹ : MulSemiringAction G S
inst✝ : SMulCommClass G R S
g : G
hq : q.IsPrime
p : Ideal R := under R q
f₀ : S ≃ₐ[R] S := (MulSemiringAction.toAlgAut G R S) g
this✝ : Algebra (Localization.AtPrime p) (Localization.AtPrime q) := Localization.AtPrime.algebraOfLiesOver p q
this : Algebra (Localization.AtPrime p) (Localization.AtPrime (g • q)) :=
  Localization.AtPrime.algebraOfLiesOver p (g • q)
⊢ (g • q).inertiaDeg R = Module.finrank p.ResidueField q.ResidueField

Tactic:
inertiaDeg_eq p (g • q)

Goal state after:
case pos
R : Type u_1
S : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
q : Ideal S
G : Type u_4
inst✝² : Group G
inst✝¹ : MulSemiringAction G S
inst✝ : SMulCommClass G R S
g : G
hq : q.IsPrime
p : Ideal R := under R q
f₀ : S ≃ₐ[R] S := (MulSemiringAction.toAlgAut G R S) g
this✝ : Algebra (Localization.AtPrime p) (Localization.AtPrime q) := Localization.AtPrime.algebraOfLiesOver p q
this : Algebra (Localization.AtPrime p) (Localization.AtPrime (g • q)) :=
  Localization.AtPrime.algebraOfLiesOver p (g • q)
⊢ Module.finrank p.ResidueField (g • q).ResidueField = Module.finrank p.ResidueField q.ResidueField
Module: Mathlib.RingTheory.LocalRing.ResidueField.Instances

Goal state before:
pp: R✝ : Type u_1
A : Type u_2
B : Type u_3
inst✝¹² : CommRing R✝
inst✝¹¹ : CommRing A
inst✝¹⁰ : CommRing B
inst✝⁹ : Algebra R✝ A
inst✝⁸ : Algebra A B
inst✝⁷ : Algebra R✝ B
inst✝⁶ : IsScalarTower R✝ A B
p : Ideal A
q : Ideal B
inst✝⁵ : q.LiesOver p
R : Type u_4
k : Type u_5
inst✝⁴ : CommRing R
inst✝³ : IsLocalRing R
inst✝² : Field k
inst✝¹ : Algebra R k
inst✝ : Module.Finite R k
⊢ (algebraMap (ResidueField R) k).comp ↑(Ideal.quotEquivOfEq ⋯) =
    (↑(RingEquiv.quotientBot k)).comp (algebraMap (R ⧸ Ideal.comap (algebraMap R k) ⊥) (k ⧸ ⊥))

Tactic:
by ext; rfl

Goal state after:
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 : I ^ 2 = ⊥
f : A →ₐ[R] B ⧸ I
P : Generators R A A := Generators.self R A
hP : Function.Injective ⇑P.toExtension.cotangentComplex
l : P.toExtension.CotangentSpace →ₗ[A] P.toExtension.Cotangent
hl : l ∘ₗ P.toExtension.cotangentComplex = LinearMap.id
g : A →ₐ[R] P.Ring ⧸ RingHom.ker (IsScalarTower.toAlgHom R P.Ring A).toRingHom ^ 2
hg : (IsScalarTower.toAlgHom R P.Ring A).kerSquareLift.comp g = AlgHom.id R A
⊢ ∃ a, (Ideal.Quotient.mkₐ R I).comp a = f

Tactic:
let σ := Function.surjInv (f := algebraMap B (B ⧸ I)) Ideal.Quotient.mk_surjective

Goal state after:
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 : I ^ 2 = ⊥
f : A →ₐ[R] B ⧸ I
P : Generators R A A := Generators.self R A
hP : Function.Injective ⇑P.toExtension.cotangentComplex
l : P.toExtension.CotangentSpace →ₗ[A] P.toExtension.Cotangent
hl : l ∘ₗ P.toExtension.cotangentComplex = LinearMap.id
g : A →ₐ[R] P.Ring ⧸ RingHom.ker (IsScalarTower.toAlgHom R P.Ring A).toRingHom ^ 2
hg : (IsScalarTower.toAlgHom R P.Ring A).kerSquareLift.comp g = AlgHom.id R A
σ : B ⧸ I → B := Function.surjInv ⋯
⊢ ∃ a, (Ideal.Quotient.mkₐ R I).comp a = f
Module: Mathlib.RingTheory.Unramified.Finite

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
M : Type u_3
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : Module S M
inst✝² : IsScalarTower R S M
inst✝¹ : FormallyUnramified R S
inst✝ : EssFiniteType R S
r : S
m : M
⊢ (TensorProduct.AlgebraTensorModule.map LinearMap.id ((lsmul R R M).toLinearMap.flip (r • m))) (elem R S) =
    (TensorProduct.AlgebraTensorModule.map LinearMap.id ((lsmul R R M).toLinearMap.flip m)) (1 ⊗ₜ[R] r * elem R S)

Tactic:
induction elem R S using TensorProduct.induction_on

Goal state after:
case zero
R : Type u_1
S : Type u_2
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
M : Type u_3
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : Module S M
inst✝² : IsScalarTower R S M
inst✝¹ : FormallyUnramified R S
inst✝ : EssFiniteType R S
r : S
m : M
⊢ (TensorProduct.AlgebraTensorModule.map LinearMap.id ((lsmul R R M).toLinearMap.flip (r • m))) 0 =
    (TensorProduct.AlgebraTensorModule.map LinearMap.id ((lsmul R R M).toLinearMap.flip m)) (1 ⊗ₜ[R] r * 0)

case tmul
R : Type u_1
S : Type u_2
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
M : Type u_3
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : Module S M
inst✝² : IsScalarTower R S M
inst✝¹ : FormallyUnramified R S
inst✝ : EssFiniteType R S
r : S
m : M
x✝ y✝ : S
⊢ (TensorProduct.AlgebraTensorModule.map LinearMap.id ((lsmul R R M).toLinearMap.flip (r • m))) (x✝ ⊗ₜ[R] y✝) =
    (TensorProduct.AlgebraTensorModule.map LinearMap.id ((lsmul R R M).toLinearMap.flip m)) (1 ⊗ₜ[R] r * x✝ ⊗ₜ[R] y✝)

case add
R : Type u_1
S : Type u_2
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
M : Type u_3
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : Module S M
inst✝² : IsScalarTower R S M
inst✝¹ : FormallyUnramified R S
inst✝ : EssFiniteType R S
r : S
m : M
x✝ y✝ : S ⊗[R] S
a✝¹ :
  (TensorProduct.AlgebraTensorModule.map LinearMap.id ((lsmul R R M).toLinearMap.flip (r • m))) x✝ =
    (TensorProduct.AlgebraTensorModule.map LinearMap.id ((lsmul R R M).toLinearMap.flip m)) (1 ⊗ₜ[R] r * x✝)
a✝ :
  (TensorProduct.AlgebraTensorModule.map LinearMap.id ((lsmul R R M).toLinearMap.flip (r • m))) y✝ =
    (TensorProduct.AlgebraTensorModule.map LinearMap.id ((lsmul R R M).toLinearMap.flip m)) (1 ⊗ₜ[R] r * y✝)
⊢ (TensorProduct.AlgebraTensorModule.map LinearMap.id ((lsmul R R M).toLinearMap.flip (r • m))) (x✝ + y✝) =
    (TensorProduct.AlgebraTensorModule.map LinearMap.id ((lsmul R R M).toLinearMap.flip m)) (1 ⊗ₜ[R] r * (x✝ + y✝))
Module: Mathlib.RingTheory.Unramified.Finite

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
M : Type u_3
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : Module S M
inst✝² : IsScalarTower R S M
inst✝¹ : FormallyUnramified R S
inst✝ : EssFiniteType R S
r : S
m : M
this : 1 ⊗ₜ[R] r * elem R S = r ⊗ₜ[R] 1 * elem R S
⊢ (TensorProduct.AlgebraTensorModule.map LinearMap.id ((lsmul R R M).toLinearMap.flip m)) (r ⊗ₜ[R] 1 * elem R S) =
    r • (TensorProduct.AlgebraTensorModule.map LinearMap.id ((lsmul R R M).toLinearMap.flip m)) (elem R S)

Tactic:
induction elem R S using TensorProduct.induction_on

Goal state after:
case zero
R : Type u_1
S : Type u_2
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
M : Type u_3
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : Module S M
inst✝² : IsScalarTower R S M
inst✝¹ : FormallyUnramified R S
inst✝ : EssFiniteType R S
r : S
m : M
this : 1 ⊗ₜ[R] r * elem R S = r ⊗ₜ[R] 1 * elem R S
⊢ (TensorProduct.AlgebraTensorModule.map LinearMap.id ((lsmul R R M).toLinearMap.flip m)) (r ⊗ₜ[R] 1 * 0) =
    r • (TensorProduct.AlgebraTensorModule.map LinearMap.id ((lsmul R R M).toLinearMap.flip m)) 0

case tmul
R : Type u_1
S : Type u_2
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
M : Type u_3
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : Module S M
inst✝² : IsScalarTower R S M
inst✝¹ : FormallyUnramified R S
inst✝ : EssFiniteType R S
r : S
m : M
this : 1 ⊗ₜ[R] r * elem R S = r ⊗ₜ[R] 1 * elem R S
x✝ y✝ : S
⊢ (TensorProduct.AlgebraTensorModule.map LinearMap.id ((lsmul R R M).toLinearMap.flip m)) (r ⊗ₜ[R] 1 * x✝ ⊗ₜ[R] y✝) =
    r • (TensorProduct.AlgebraTensorModule.map LinearMap.id ((lsmul R R M).toLinearMap.flip m)) (x✝ ⊗ₜ[R] y✝)

case add
R : Type u_1
S : Type u_2
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
M : Type u_3
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : Module S M
inst✝² : IsScalarTower R S M
inst✝¹ : FormallyUnramified R S
inst✝ : EssFiniteType R S
r : S
m : M
this : 1 ⊗ₜ[R] r * elem R S = r ⊗ₜ[R] 1 * elem R S
x✝ y✝ : S ⊗[R] S
a✝¹ :
  (TensorProduct.AlgebraTensorModule.map LinearMap.id ((lsmul R R M).toLinearMap.flip m)) (r ⊗ₜ[R] 1 * x✝) =
    r • (TensorProduct.AlgebraTensorModule.map LinearMap.id ((lsmul R R M).toLinearMap.flip m)) x✝
a✝ :
  (TensorProduct.AlgebraTensorModule.map LinearMap.id ((lsmul R R M).toLinearMap.flip m)) (r ⊗ₜ[R] 1 * y✝) =
    r • (TensorProduct.AlgebraTensorModule.map LinearMap.id ((lsmul R R M).toLinearMap.flip m)) y✝
⊢ (TensorProduct.AlgebraTensorModule.map LinearMap.id ((lsmul R R M).toLinearMap.flip m)) (r ⊗ₜ[R] 1 * (x✝ + y✝)) =
    r • (TensorProduct.AlgebraTensorModule.map LinearMap.id ((lsmul R R M).toLinearMap.flip m)) (x✝ + y✝)
Module: Mathlib.RingTheory.Unramified.Finite

Goal state before:
pp: case add
R : Type u_1
S : Type u_2
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
M : Type u_3
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : Module S M
inst✝² : IsScalarTower R S M
inst✝¹ : FormallyUnramified R S
inst✝ : EssFiniteType R S
r : S
m : M
this : 1 ⊗ₜ[R] r * elem R S = r ⊗ₜ[R] 1 * elem R S
x✝ y✝ : S ⊗[R] S
a✝¹ :
  (TensorProduct.AlgebraTensorModule.map LinearMap.id ((lsmul R R M).toLinearMap.flip m)) (r ⊗ₜ[R] 1 * x✝) =
    r • (TensorProduct.AlgebraTensorModule.map LinearMap.id ((lsmul R R M).toLinearMap.flip m)) x✝
a✝ :
  (TensorProduct.AlgebraTensorModule.map LinearMap.id ((lsmul R R M).toLinearMap.flip m)) (r ⊗ₜ[R] 1 * y✝) =
    r • (TensorProduct.AlgebraTensorModule.map LinearMap.id ((lsmul R R M).toLinearMap.flip m)) y✝
⊢ (TensorProduct.AlgebraTensorModule.map LinearMap.id ((lsmul R R M).toLinearMap.flip m)) (r ⊗ₜ[R] 1 * (x✝ + y✝)) =
    r • (TensorProduct.AlgebraTensorModule.map LinearMap.id ((lsmul R R M).toLinearMap.flip m)) (x✝ + y✝)

Tactic:
· simp only [map_add, smul_add, mul_add, *]

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

Goal state before:
pp: R : Type u_4
Rₘ : Type u_6
Sₘ : Type u_7
inst✝⁸ : CommRing R
inst✝⁷ : CommRing Rₘ
inst✝⁶ : CommRing Sₘ
M : Submonoid R
inst✝⁵ : Algebra R Sₘ
inst✝⁴ : Algebra R Rₘ
inst✝³ : Algebra Rₘ Sₘ
inst✝² : IsScalarTower R Rₘ Sₘ
inst✝¹ : IsLocalization M Rₘ
inst✝ : FormallySmooth R Sₘ
Q : Type (max u_6 u_7)
x✝¹ : CommRing Q
x✝ : Algebra Rₘ Q
I : Ideal Q
e : I ^ 2 = ⊥
f : Sₘ →ₐ[Rₘ] Q ⧸ I
this : Algebra R Q := ((algebraMap Rₘ Q).comp (algebraMap R Rₘ)).toAlgebra
⊢ ∃ a, (Ideal.Quotient.mkₐ Rₘ I).comp a = f

Tactic:
let : IsScalarTower R Rₘ Q := IsScalarTower.of_algebraMap_eq' rfl

Goal state after:
R : Type u_4
Rₘ : Type u_6
Sₘ : Type u_7
inst✝⁸ : CommRing R
inst✝⁷ : CommRing Rₘ
inst✝⁶ : CommRing Sₘ
M : Submonoid R
inst✝⁵ : Algebra R Sₘ
inst✝⁴ : Algebra R Rₘ
inst✝³ : Algebra Rₘ Sₘ
inst✝² : IsScalarTower R Rₘ Sₘ
inst✝¹ : IsLocalization M Rₘ
inst✝ : FormallySmooth R Sₘ
Q : Type (max u_6 u_7)
x✝¹ : CommRing Q
x✝ : Algebra Rₘ Q
I : Ideal Q
e : I ^ 2 = ⊥
f : Sₘ →ₐ[Rₘ] Q ⧸ I
this✝ : Algebra R Q := ((algebraMap Rₘ Q).comp (algebraMap R Rₘ)).toAlgebra
this : IsScalarTower R Rₘ Q := IsScalarTower.of_algebraMap_eq' rfl
⊢ ∃ a, (Ideal.Quotient.mkₐ Rₘ I).comp a = f
Module: Mathlib.RingTheory.Unramified.Field

Goal state before:
pp: case pos
K : Type u_1
L : Type u_3
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : FormallyUnramified K L
inst✝¹ : EssFiniteType K L
inst✝ : IsPurelyInseparable K L
this✝¹ : Nontrivial (L ⊗[K] L)
n : ℕ
this✝ : ExpChar (L ⊗[K] L) (ringExpChar K)
a✝ : 1 ∈ ⊤
hn : 1 ^ ringExpChar K ^ n ∈ (algebraMap K L).range
this : (1 ⊗ₜ[K] 1 - 1 ⊗ₜ[K] 1) ^ ringExpChar K ^ n = 0
H : 1 ⊗ₜ[K] 1 = 1 ⊗ₜ[K] 1
h' : LinearIndependent K ![1, 1]
h : LinearIndepOn K _root_.id (Set.range ![1, 1])
S : Set L := h.extend ⋯
a : ↑S := ⟨1, ⋯⟩
ha : (Basis.extend h) a = 1
b : ↑S := ⟨1, ⋯⟩
hb : (Basis.extend h) b = 1
e : a = b
⊢ 1 ∈ (algebraMap K L).range

Tactic:
exact ⟨1, map_one _⟩

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

Goal state before:
pp: case h
K : Type u_1
L : Type u_3
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : FormallyUnramified K L
inst✝¹ : EssFiniteType K L
inst✝ : IsPurelyInseparable K L
this✝² : Nontrivial (L ⊗[K] L)
x : L
a✝ : x ∈ ⊤
n : ℕ
hn : x ^ ringExpChar K ^ n ∈ (algebraMap K L).range
this✝¹ : ExpChar (L ⊗[K] L) (ringExpChar K)
this✝ : (1 ⊗ₜ[K] x - x ⊗ₜ[K] 1) ^ ringExpChar K ^ n = 0
H : 1 ⊗ₜ[K] x = x ⊗ₜ[K] 1
a b : K
e : a • 1 + b • x = 0
hab : a = 0 → ¬b = 0
this : IsUnit b
⊢ -(-↑this.unit⁻¹ • b • x) = x

Tactic:
neg_smul,

Goal state after:
case h
K : Type u_1
L : Type u_3
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : FormallyUnramified K L
inst✝¹ : EssFiniteType K L
inst✝ : IsPurelyInseparable K L
this✝² : Nontrivial (L ⊗[K] L)
x : L
a✝ : x ∈ ⊤
n : ℕ
hn : x ^ ringExpChar K ^ n ∈ (algebraMap K L).range
this✝¹ : ExpChar (L ⊗[K] L) (ringExpChar K)
this✝ : (1 ⊗ₜ[K] x - x ⊗ₜ[K] 1) ^ ringExpChar K ^ n = 0
H : 1 ⊗ₜ[K] x = x ⊗ₜ[K] 1
a b : K
e : a • 1 + b • x = 0
hab : a = 0 → ¬b = 0
this : IsUnit b
⊢ - -(↑this.unit⁻¹ • b • x) = x