mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.FieldTheory.Finite.Basic

Goal state before:
pp: case neg
K : Type u_1
inst✝¹ : Field K
inst✝ : Fintype K
i : ℕ
h : i < q - 1
hi : ¬i = 0
⊢ ∑ x, x ^ i = 0

Tactic:
have hiq : ¬q - 1 ∣ i := by contrapose! h; exact Nat.le_of_dvd (Nat.pos_of_ne_zero hi) h
    let φ : Kˣ ↪ K := ⟨fun x ↦ x, Units.val_injective⟩
    have : univ.map φ = univ \ {0} := by
      ext x
      simpa only [mem_map, mem_univ, Function.Embedding.coeFn_mk, true_and, mem_sdiff,
        mem_singleton, φ] using! isUnit_iff_ne_zero
    calc
      ∑ x : K, x ^ i = ∑ x ∈ univ \ {(0 : K)}, x ^ i := by
        rw [← sum_sdiff ({0} : Finset K).subset_univ, sum_singleton, zero_pow hi, add_zero]
      _ = ∑ x : Kˣ, (x ^ i : K) := by simp [φ, ← this, univ.sum_map φ]
      _ = 0 := by rw [sum_pow_units K i, if_neg]; exact hiq

Goal state after:
Module: Mathlib.FieldTheory.Finite.Basic

Goal state before:
pp: case succ
K : Type u_1
inst✝³ : Field K
inst✝² : Fintype K
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : CharP K p
x : K
n : ℕ
hn : (frobenius K p ^ n) x = x ^ p ^ n
⊢ (frobenius K p * frobenius K p ^ n) x = x ^ (p ^ n * p)

Tactic:
pow_mul,

Goal state after:
case succ
K : Type u_1
inst✝³ : Field K
inst✝² : Fintype K
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : CharP K p
x : K
n : ℕ
hn : (frobenius K p ^ n) x = x ^ p ^ n
⊢ (frobenius K p * frobenius K p ^ n) x = (x ^ p ^ n) ^ p
Module: Mathlib.RingTheory.Invariant.Basic

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝¹⁶ : CommRing A
inst✝¹⁵ : CommRing B
inst✝¹⁴ : Algebra A B
G : Type u_3
inst✝¹³ : Group G
inst✝¹² : MulSemiringAction G B
inst✝¹¹ : SMulCommClass G A B
P : Ideal A
Q : Ideal B
inst✝¹⁰ : Q.LiesOver P
K : Type u_4
L : Type u_5
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra (A ⧸ P) K
inst✝⁶ : Algebra (B ⧸ Q) L
inst✝⁵ : Algebra (A ⧸ P) L
inst✝⁴ : IsScalarTower (A ⧸ P) (B ⧸ Q) L
inst✝³ : Algebra K L
inst✝² : IsScalarTower (A ⧸ P) K L
inst✝¹ : IsFractionRing (A ⧸ P) K
inst✝ : IsFractionRing (B ⧸ Q) L
⊢ (Ideal.Quotient.stabilizerHom Q P G).ker = Ideal.inertia (↥(MulAction.stabilizer G Q)) Q

pp: case hg
A : Type u_1
B : Type u_2
inst✝¹⁶ : CommRing A
inst✝¹⁵ : CommRing B
inst✝¹⁴ : Algebra A B
G : Type u_3
inst✝¹³ : Group G
inst✝¹² : MulSemiringAction G B
inst✝¹¹ : SMulCommClass G A B
P : Ideal A
Q : Ideal B
inst✝¹⁰ : Q.LiesOver P
K : Type u_4
L : Type u_5
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra (A ⧸ P) K
inst✝⁶ : Algebra (B ⧸ Q) L
inst✝⁵ : Algebra (A ⧸ P) L
inst✝⁴ : IsScalarTower (A ⧸ P) (B ⧸ Q) L
inst✝³ : Algebra K L
inst✝² : IsScalarTower (A ⧸ P) K L
inst✝¹ : IsFractionRing (A ⧸ P) K
inst✝ : IsFractionRing (B ⧸ Q) L
⊢ Function.Injective ⇑(fieldEquivOfAlgEquivHom K L)

Tactic:
Ideal.Quotient.ker_stabilizerHom

Goal state after:
A : Type u_1
B : Type u_2
inst✝¹⁶ : CommRing A
inst✝¹⁵ : CommRing B
inst✝¹⁴ : Algebra A B
G : Type u_3
inst✝¹³ : Group G
inst✝¹² : MulSemiringAction G B
inst✝¹¹ : SMulCommClass G A B
P : Ideal A
Q : Ideal B
inst✝¹⁰ : Q.LiesOver P
K : Type u_4
L : Type u_5
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra (A ⧸ P) K
inst✝⁶ : Algebra (B ⧸ Q) L
inst✝⁵ : Algebra (A ⧸ P) L
inst✝⁴ : IsScalarTower (A ⧸ P) (B ⧸ Q) L
inst✝³ : Algebra K L
inst✝² : IsScalarTower (A ⧸ P) K L
inst✝¹ : IsFractionRing (A ⧸ P) K
inst✝ : IsFractionRing (B ⧸ Q) L
⊢ Ideal.inertia (↥(MulAction.stabilizer G Q)) Q = Ideal.inertia (↥(MulAction.stabilizer G Q)) Q

case hg
A : Type u_1
B : Type u_2
inst✝¹⁶ : CommRing A
inst✝¹⁵ : CommRing B
inst✝¹⁴ : Algebra A B
G : Type u_3
inst✝¹³ : Group G
inst✝¹² : MulSemiringAction G B
inst✝¹¹ : SMulCommClass G A B
P : Ideal A
Q : Ideal B
inst✝¹⁰ : Q.LiesOver P
K : Type u_4
L : Type u_5
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra (A ⧸ P) K
inst✝⁶ : Algebra (B ⧸ Q) L
inst✝⁵ : Algebra (A ⧸ P) L
inst✝⁴ : IsScalarTower (A ⧸ P) (B ⧸ Q) L
inst✝³ : Algebra K L
inst✝² : IsScalarTower (A ⧸ P) K L
inst✝¹ : IsFractionRing (A ⧸ P) K
inst✝ : IsFractionRing (B ⧸ Q) L
⊢ Function.Injective ⇑(fieldEquivOfAlgEquivHom K L)
Module: Mathlib.FieldTheory.Finite.Basic

Goal state before:
pp: p : ℕ
inst✝ : Fact (Prime p)
x : ℤ
⊢ ∃ a b, a ≤ p / 2 ∧ b ≤ p / 2 ∧ ↑a ^ 2 + ↑b ^ 2 ≡ x [ZMOD ↑p]

Tactic:
by
  rcases ZMod.sq_add_sq p x with ⟨a, b, hx⟩
  refine ⟨a.valMinAbs.natAbs, b.valMinAbs.natAbs, ZMod.natAbs_valMinAbs_le _,
    ZMod.natAbs_valMinAbs_le _, ?_⟩
  rw [← a.coe_valMinAbs, ← b.coe_valMinAbs] at hx
  push_cast
  rw [sq_abs, sq_abs, ← ZMod.intCast_eq_intCast_iff]
  exact mod_cast hx

Goal state after:
Module: Mathlib.FieldTheory.Finite.Basic

Goal state before:
pp: case neg
p x y : ℕ
hp : Nat.Prime p
h : (p - 1) * ((x - y) / (p - 1)) = x - y
hxy : y ≤ x
hy : 0 < y
n : ℤ
hn : IsCoprime (↑p) n
⊢ n ^ ((p - 1) * ((x - y) / (p - 1))) * n ^ y ≡ n ^ y [ZMOD ↑p]

Tactic:
pow_mul,

Goal state after:
case neg
p x y : ℕ
hp : Nat.Prime p
h : (p - 1) * ((x - y) / (p - 1)) = x - y
hxy : y ≤ x
hy : 0 < y
n : ℤ
hn : IsCoprime (↑p) n
⊢ (n ^ (p - 1)) ^ ((x - y) / (p - 1)) * n ^ y ≡ n ^ y [ZMOD ↑p]
Module: Mathlib.FieldTheory.Finite.Basic

Goal state before:
pp: case succ
p a m d : ℕ
hd : (1 + p * a) ^ p ^ m = p ^ m * d + 1
⊢ (1 + p * a) ^ (p ^ m * p) ≡ 1 [MOD p ^ m * p]

Tactic:
pow_mul,

Goal state after:
case succ
p a m d : ℕ
hd : (1 + p * a) ^ p ^ m = p ^ m * d + 1
⊢ ((1 + p * a) ^ p ^ m) ^ p ≡ 1 [MOD p ^ m * p]
Module: Mathlib.FieldTheory.Finite.Basic

Goal state before:
pp: F : Type u_3
inst✝² : Field F
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : CharP F p
⊢ Nat.card ↥⊥ = p

Tactic:
rw [← fieldRange_castHom_eq_bot p,
    ← Nat.card_eq_of_bijective _ (RingHom.rangeRestrictField_bijective _), Nat.card_zmod]

Goal state after:
Module: Mathlib.FieldTheory.Finite.Basic

Goal state before:
pp: F : Type u_3
inst✝¹ : Field F
inst✝ : Fintype F
⊢ ringChar F = 2 ↔ Fintype.card F % 2 = 0

Tactic:
by
  rcases FiniteField.card F (ringChar F) with ⟨n, hp, h⟩
  rw [h, ← Nat.even_iff, Nat.even_pow, hp.even_iff]
  simp

Goal state after:
Module: Mathlib.FieldTheory.Finite.Basic

Goal state before:
pp: F : Type u_3
inst✝¹ : Field F
inst✝ : Fintype F
hF : ringChar F ≠ 2
a : F
ha : a ≠ 0
h₁ : a ^ (Fintype.card F / 2 * 2) = 1
⊢ a ^ (Fintype.card F / 2) = 1 ∨ a ^ (Fintype.card F / 2) = -1

Tactic:
pow_mul,

Goal state after:
F : Type u_3
inst✝¹ : Field F
inst✝ : Fintype F
hF : ringChar F ≠ 2
a : F
ha : a ≠ 0
h₁ : (a ^ (Fintype.card F / 2)) ^ 2 = 1
⊢ a ^ (Fintype.card F / 2) = 1 ∨ a ^ (Fintype.card F / 2) = -1
Module: Mathlib.FieldTheory.Finite.Basic

Goal state before:
pp: case mpr
F : Type u_3
inst✝¹ : Field F
inst✝ : Fintype F
hF : ringChar F ≠ 2
g : Fˣ
hg : ∀ (x : Fˣ), x ∈ Subgroup.zpowers g
hodd : 2 * (Nat.card F / 2) = Nat.card F - 1
this : 0 < Nat.card F / 2
m : ℕ
h : (fun x ↦ g ^ x) (2 * m) ^ (Nat.card F / 2) = 1
key : 2 * (Nat.card F / 2) ∣ 2 * m * (Nat.card F / 2)
⊢ g ^ (m * 2) = g ^ m * g ^ m

Tactic:
pow_mul,

Goal state after:
case mpr
F : Type u_3
inst✝¹ : Field F
inst✝ : Fintype F
hF : ringChar F ≠ 2
g : Fˣ
hg : ∀ (x : Fˣ), x ∈ Subgroup.zpowers g
hodd : 2 * (Nat.card F / 2) = Nat.card F - 1
this : 0 < Nat.card F / 2
m : ℕ
h : (fun x ↦ g ^ x) (2 * m) ^ (Nat.card F / 2) = 1
key : 2 * (Nat.card F / 2) ∣ 2 * m * (Nat.card F / 2)
⊢ (g ^ m) ^ 2 = g ^ m * g ^ m
Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm

Goal state before:
pp: S : Type u_1
inst✝ : CommRing S
I J : Ideal S
coprime : IsCoprime I J
⊢ Nat.card (S ⧸ I * J) = Nat.card (S ⧸ I) * Nat.card (S ⧸ J)

Tactic:
Nat.card_congr (Ideal.quotientMulEquivQuotientProd I J coprime).toEquiv,

Goal state after:
S : Type u_1
inst✝ : CommRing S
I J : Ideal S
coprime : IsCoprime I J
⊢ Nat.card ((S ⧸ I) × S ⧸ J) = Nat.card (S ⧸ I) * Nat.card (S ⧸ J)
Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm

Goal state before:
pp: S : Type u_1
inst✝² : CommRing S
inst✝¹ : IsDedekindDomain S
inst✝ : Free ℤ S
I J : Ideal S
⊢ cardQuot (I * J) = cardQuot I * cardQuot J

Tactic:
by rw [cardQuot_mul]

Goal state after:
Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm

Goal state before:
pp: case neg
S : Type u_1
inst✝⁵ : CommRing S
inst✝⁴ : IsDedekindDomain S
inst✝³ : Free ℤ S
inst✝² : Module.Finite ℤ S
I : Ideal S
E : Type u_2
inst✝¹ : EquivLike E S ↥I
inst✝ : AddEquivClass E S ↥I
e : E
hI : ¬I = ⊥
⊢ (LinearMap.det (↑ℤ (Submodule.subtype I) ∘ₗ (↑e).toIntLinearMap)).natAbs = absNorm I

Tactic:
exact Submodule.natAbs_det_equiv (I.restrictScalars ℤ) e

Goal state after:
Module: Mathlib.GroupTheory.Submonoid.Inverses

Goal state before:
pp: M : Type u_1
inst✝ : CommMonoid M
S : Submonoid M
a : ↥S.leftInv
b : M
⊢ ↑(S.fromLeftInv a) = b ↔ ↑a * b = 1

Tactic:
rw [← IsUnit.mul_right_inj (leftInv_le_isUnit _ a.prop), S.mul_fromLeftInv, eq_comm]

Goal state after:
Module: Mathlib.RingTheory.Localization.InvSubmonoid

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
M : Submonoid R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : IsLocalization M S
inst✝ : Monoid.FG ↥M
this : Monoid.FG ↥(invSubmonoid M S)
⊢ Algebra.FiniteType R S

Tactic:
Monoid.fg_iff_submonoid_fg

Goal state after:
R : Type u_1
inst✝⁴ : CommRing R
M : Submonoid R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : IsLocalization M S
inst✝ : Monoid.FG ↥M
this : (invSubmonoid M S).FG
⊢ Algebra.FiniteType R S
Module: Mathlib.RingTheory.HopkinsLevitzki

Goal state before:
pp: R : Type u_3
inst✝² : CommRing R
inst✝¹ : IsNoetherianRing R
inst✝ : Ring.KrullDimLE 0 R
eq : Ring.jacobson R = nilradical R
Spec : Set (Ideal R) := {I | I.IsPrime}
this✝¹ : Finite ↑Spec
this✝ : ∀ (I : ↑Spec), (↑I).IsPrime
this : ∀ (I : ↑Spec), IsSemisimpleRing (R ⧸ ↑I)
⊢ IsSemisimpleRing (R ⧸ Ring.jacobson R)

Tactic:
rw [eq, nilradical_eq_sInf, sInf_eq_iInf']

Goal state after:
R : Type u_3
inst✝² : CommRing R
inst✝¹ : IsNoetherianRing R
inst✝ : Ring.KrullDimLE 0 R
eq : Ring.jacobson R = nilradical R
Spec : Set (Ideal R) := {I | I.IsPrime}
this✝¹ : Finite ↑Spec
this✝ : ∀ (I : ↑Spec), (↑I).IsPrime
this : ∀ (I : ↑Spec), IsSemisimpleRing (R ⧸ ↑I)
⊢ IsSemisimpleRing (R ⧸ ⨅ a, ↑a)
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 ⋯
key : IsSimpleModule B (↥q ⧸ f.range)
m : Ideal B
hm : m.IsMaximal
e : (↥q ⧸ f.range) ≃ₗ[B] B ⧸ maximalIdeal B
g : ↥q →ₗ[B] ResidueField B := ↑e ∘ₗ f.range.mkQ
this✝¹ : Function.Injective ⇑f
this✝ : Function.Surjective ⇑g
this : Function.Exact ⇑f ⇑g
⊢ length A ↥q = length A ↥p + length (ResidueField A) (ResidueField B)

Tactic:
length_eq_add_of_exact (f.restrictScalars A) (g.restrictScalars A)
    (by simpa) (by simpa) (by simpa),

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)
m : Ideal B
hm : m.IsMaximal
e : (↥q ⧸ f.range) ≃ₗ[B] B ⧸ maximalIdeal B
g : ↥q →ₗ[B] ResidueField B := ↑e ∘ₗ f.range.mkQ
this✝¹ : Function.Injective ⇑f
this✝ : Function.Surjective ⇑g
this : Function.Exact ⇑f ⇑g
⊢ length A ↥p + length A (ResidueField B) = length A ↥p + length (ResidueField A) (ResidueField B)
Module: Mathlib.RingTheory.Unramified.Finite

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : EssFiniteType R S
this✝ : ∀ (t : S ⊗[R] S), (TensorProduct.lmul' R) t = 1 ↔ 1 - t ∈ KaehlerDifferential.ideal R S
t e : S ⊗[R] S
ht₁ : ∀ (s : S), (1 ⊗ₜ[R] s - s ⊗ₜ[R] 1) * (1 - e) = 0
ht₂ : e ∈ Ideal.span (Set.range fun s ↦ 1 ⊗ₜ[R] s - s ⊗ₜ[R] 1)
this : e ∈ (S ⊗[R] S ∙ (1 - e)).annihilator
⊢ IsIdempotentElem e

Tactic:
by
        simpa [IsIdempotentElem, mul_sub, sub_eq_zero, eq_comm,
          Submodule.mem_annihilator_span_singleton] using this

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

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : FiniteType R S
⊢ QuasiFinite R S ↔ ∀ (I : Ideal R), I.IsPrime → (I.primesOver S).Finite

Tactic:
by
  rw [iff_finite_comap_preimage_singleton,
    (PrimeSpectrum.equivSubtype R).forall_congr_left, Subtype.forall]
  refine forall₂_congr fun I hI ↦ ?_
  rw [← Set.finite_image_iff (Function.Injective.injOn fun _ _ ↦ PrimeSpectrum.ext)]
  congr!
  ext J
  simp [(PrimeSpectrum.equivSubtype S).exists_congr_left, PrimeSpectrum.ext_iff, eq_comm,
    PrimeSpectrum.equivSubtype, Ideal.primesOver, and_comm, Ideal.liesOver_iff, Ideal.under]

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

Goal state before:
pp: case hg
R : Type u_1
S : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : FormallyUnramified R S
inst✝¹ : EssFiniteType R S
inst✝ : Free R S
I : Type u_2 := Free.ChooseBasisIndex R S
b : Basis I R S := Free.chooseBasis R S
f : I →₀ S
hf : elem R S = f.sum fun i x ↦ x ⊗ₜ[R] b i
x : S
a : I → I →₀ R := fun i ↦ b.repr (b i * x)
F : I →₀ I →₀ R := Finsupp.onFinset f.support (fun j ↦ b.repr (x * f j)) ⋯
hG : ∀ j ∉ f.support.biUnion fun i ↦ (a i).support, b.repr (f.sum fun i y ↦ (a i) j • y) = 0
G : I →₀ I →₀ R :=
  Finsupp.onFinset (f.support.biUnion fun i ↦ (a i).support) (fun j ↦ b.repr (f.sum fun i y ↦ (a i) j • y)) ⋯
this : ∀ (i : I), ((b.repr (x * f i)).sum fun j k ↦ k • b j ⊗ₜ[R] b i) = (x * f i) ⊗ₜ[R] b i
⊢ ∀ (a : I), (Finsupp.sum 0 fun b' c ↦ c • b b' ⊗ₜ[R] b a) = 0

pp: case hg
R : Type u_1
S : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : FormallyUnramified R S
inst✝¹ : EssFiniteType R S
inst✝ : Free R S
I : Type u_2 := Free.ChooseBasisIndex R S
b : Basis I R S := Free.chooseBasis R S
f : I →₀ S
hf : elem R S = f.sum fun i x ↦ x ⊗ₜ[R] b i
x : S
a : I → I →₀ R := fun i ↦ b.repr (b i * x)
F : I →₀ I →₀ R := Finsupp.onFinset f.support (fun j ↦ b.repr (x * f j)) ⋯
hG : ∀ j ∉ f.support.biUnion fun i ↦ (a i).support, b.repr (f.sum fun i y ↦ (a i) j • y) = 0
G : I →₀ I →₀ R :=
  Finsupp.onFinset (f.support.biUnion fun i ↦ (a i).support) (fun j ↦ b.repr (f.sum fun i y ↦ (a i) j • y)) ⋯
this : ∀ (i : I), ((b.repr (x * f i)).sum fun j k ↦ k • b j ⊗ₜ[R] b i) = (x * f i) ⊗ₜ[R] b i
⊢ ∀ (a : I), (Finsupp.sum 0 fun b' c ↦ c • b b' ⊗ₜ[R] b a) = 0

Tactic:
· intro; simp

Goal state after:
case hg
R : Type u_1
S : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : FormallyUnramified R S
inst✝¹ : EssFiniteType R S
inst✝ : Free R S
I : Type u_2 := Free.ChooseBasisIndex R S
b : Basis I R S := Free.chooseBasis R S
f : I →₀ S
hf : elem R S = f.sum fun i x ↦ x ⊗ₜ[R] b i
x : S
a : I → I →₀ R := fun i ↦ b.repr (b i * x)
F : I →₀ I →₀ R := Finsupp.onFinset f.support (fun j ↦ b.repr (x * f j)) ⋯
hG : ∀ j ∉ f.support.biUnion fun i ↦ (a i).support, b.repr (f.sum fun i y ↦ (a i) j • y) = 0
G : I →₀ I →₀ R :=
  Finsupp.onFinset (f.support.biUnion fun i ↦ (a i).support) (fun j ↦ b.repr (f.sum fun i y ↦ (a i) j • y)) ⋯
this : ∀ (i : I), ((b.repr (x * f i)).sum fun j k ↦ k • b j ⊗ₜ[R] b i) = (x * f i) ⊗ₜ[R] b i
⊢ ∀ (a : I), (Finsupp.sum 0 fun b' c ↦ c • b b' ⊗ₜ[R] b a) = 0
Module: Mathlib.RingTheory.Unramified.Finite

Goal state before:
pp: case hg
R : Type u_1
S : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : FormallyUnramified R S
inst✝¹ : EssFiniteType R S
inst✝ : Free R S
I : Type u_2 := Free.ChooseBasisIndex R S
b : Basis I R S := Free.chooseBasis R S
f : I →₀ S
hf : elem R S = f.sum fun i x ↦ x ⊗ₜ[R] b i
x : S
a : I → I →₀ R := fun i ↦ b.repr (b i * x)
F : I →₀ I →₀ R := Finsupp.onFinset f.support (fun j ↦ b.repr (x * f j)) ⋯
hG : ∀ j ∉ f.support.biUnion fun i ↦ (a i).support, b.repr (f.sum fun i y ↦ (a i) j • y) = 0
G : I →₀ I →₀ R :=
  Finsupp.onFinset (f.support.biUnion fun i ↦ (a i).support) (fun j ↦ b.repr (f.sum fun i y ↦ (a i) j • y)) ⋯
this : ∀ (i : I), ((b.repr (x * f i)).sum fun j k ↦ k • b j ⊗ₜ[R] b i) = (x * f i) ⊗ₜ[R] b i
⊢ ∀ (a : I), (Finsupp.sum 0 fun b' c ↦ c • b b' ⊗ₜ[R] b a) = 0

Tactic:
· intro; simp

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✝² : IsNoetherianRing S
this✝¹ : 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
this : IsUnit ((algebraMap S (Localization.Away e)) ↑s)
⊢ f ((algebraMap S (Localization.Away e)) x * ↑this.unit⁻¹) * f ((algebraMap S (Localization.Away e)) ↑s) =
    IsLocalization.mk' (Localization.AtPrime p.asIdeal) x s * f ((algebraMap S (Localization.Away e)) ↑s)

Tactic:
simp only [← map_mul, mul_assoc, IsUnit.val_inv_mul]

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✝² : IsNoetherianRing S
this✝¹ : 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
this : IsUnit ((algebraMap S (Localization.Away e)) ↑s)
⊢ f ((algebraMap S (Localization.Away e)) x * 1) =
    IsLocalization.mk' (Localization.AtPrime p.asIdeal) x s * f ((algebraMap S (Localization.Away e)) ↑s)
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)
⊢ (ofId A (A ⧸ IsLocalRing.maximalIdeal A)).toLinearMap ∘ₗ e'.toLinearMap ∘ₗ sec K A (A ⧸ IsLocalRing.maximalIdeal A) =
    LinearMap.id

Tactic:
← LinearMap.comp_assoc,

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)
⊢ ((ofId A (A ⧸ IsLocalRing.maximalIdeal A)).toLinearMap ∘ₗ e'.toLinearMap) ∘ₗ
      sec K A (A ⧸ IsLocalRing.maximalIdeal A) =
    LinearMap.id
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
x : A
hx : IsNilpotent x
f : A →ₐ[K] AlgebraicClosure K ⊗[K] A := TensorProduct.includeRight
this✝⁴ : Function.Injective ⇑f
M : Ideal (AlgebraicClosure K ⊗[K] A)
hM : M.IsMaximal
y : Localization.AtPrime M
hy : IsNilpotent y
this✝³ : EssFiniteType (AlgebraicClosure K ⊗[K] A) (Localization.AtPrime M)
this✝² : FormallyUnramified (AlgebraicClosure K ⊗[K] A) (Localization.AtPrime M)
this✝¹ : EssFiniteType (AlgebraicClosure K) (Localization.AtPrime M)
this✝ : FormallyUnramified (AlgebraicClosure K) (Localization.AtPrime M)
this : Field (Localization.AtPrime M) := ⋯.toField
⊢ y = 0

Tactic:
exact hy.eq_zero

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

Goal state before:
pp: K : Type u_1
L : Type u_3
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : FormallyUnramified K L
inst✝ : EssFiniteType K L
this : Module.Finite K L
⊢ separableClosure K L = ⊤

Tactic:
have := of_restrictScalars K (separableClosure K L) L

Goal state after:
K : Type u_1
L : Type u_3
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : FormallyUnramified K L
inst✝ : EssFiniteType K L
this✝ : Module.Finite K L
this : FormallyUnramified (↥(separableClosure K L)) L
⊢ separableClosure K L = ⊤
Module: Mathlib.RingTheory.Etale.Kaehler

Goal state before:
pp: case add
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
y✝ x y : Q.Ring ⊗[P.Ring] ↥P.ker
a✝¹ : ↑(e x) • (LinearMap.liftBaseChange Q.Ring (↑P.Ring ((TensorProduct.mk S T P.Cotangent) 1) ∘ₗ Cotangent.mk)) y✝ = 0
a✝ : ↑(e y) • (LinearMap.liftBaseChange Q.Ring (↑P.Ring ((TensorProduct.mk S T P.Cotangent) 1) ∘ₗ Cotangent.mk)) y✝ = 0
⊢ ↑(e (x + y)) • (LinearMap.liftBaseChange Q.Ring (↑P.Ring ((TensorProduct.mk S T P.Cotangent) 1) ∘ₗ Cotangent.mk)) y✝ =
    0

Tactic:
| add x y _ _ =>

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

Goal state before:
pp: case add
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 y : T ⊗[S] P.Cotangent
a✝¹ : (tensorCotangentInvFun f halg H) ((LinearMap.liftBaseChange T (Cotangent.map f)) x) = x
a✝ : (tensorCotangentInvFun f halg H) ((LinearMap.liftBaseChange T (Cotangent.map f)) y) = y
⊢ (tensorCotangentInvFun f halg H) ((LinearMap.liftBaseChange T (Cotangent.map f)) (x + y)) = x + y

Tactic:
| add x y _ _ =>

Goal state after: