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

Goal state before:
pp: K : Type u_1
L : Type u_2
M : Type u_3
N : Type u_4
inst✝⁸ : CommRing K
inst✝⁷ : CommRing L
inst✝⁶ : CommRing M
inst✝⁵ : CommRing N
i : K →+* L
j : K →+* M
k : K →+* N
f : L →+* M
g : L →+* N
p : ℕ
inst✝⁴ : ExpChar M p
inst✝³ : ExpChar K p
inst✝² : PerfectRing M p
inst✝¹ : IsPRadical i p
inst✝ : ExpChar L p
x1 x2 : L
n1 : ℕ
y1 : K
h1 : i y1 = x1 ^ p ^ n1
n2 : ℕ
y2 : K
h2 : i y2 = x2 ^ p ^ n2
⊢ liftAux i j p (x1 + x2) = liftAux i j p x1 + liftAux i j p x2

Tactic:
liftAux_apply i j p _ _ _ h1,

Goal state after:
K : Type u_1
L : Type u_2
M : Type u_3
N : Type u_4
inst✝⁸ : CommRing K
inst✝⁷ : CommRing L
inst✝⁶ : CommRing M
inst✝⁵ : CommRing N
i : K →+* L
j : K →+* M
k : K →+* N
f : L →+* M
g : L →+* N
p : ℕ
inst✝⁴ : ExpChar M p
inst✝³ : ExpChar K p
inst✝² : PerfectRing M p
inst✝¹ : IsPRadical i p
inst✝ : ExpChar L p
x1 x2 : L
n1 : ℕ
y1 : K
h1 : i y1 = x1 ^ p ^ n1
n2 : ℕ
y2 : K
h2 : i y2 = x2 ^ p ^ n2
⊢ liftAux i j p (x1 + x2) = (iterateFrobeniusEquiv M p n1).symm (j y1) + liftAux i j p x2
Module: Mathlib.FieldTheory.PerfectClosure

Goal state before:
pp: K : Type u
inst✝² : CommRing K
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : CharP K p
x✝¹ : PerfectClosure K p
x : ℕ × K
x✝ : IsNilpotent (mk K p x)
n m : ℕ
h : (iterateFrobenius K p m) (x.2 ^ p ^ n) = 0
⊢ (iterateFrobenius K p (n + m)) x.2 = 0

Tactic:
simpa only [iterateFrobenius_def, pow_add, pow_mul] using h

Goal state after:
Module: Mathlib.FieldTheory.Isaacs

Goal state before:
pp: F : Type u_1
E : Type u_2
K : Type u_3
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Field K
inst✝¹ : Algebra F E
inst✝ : Algebra F K
alg : Algebra.IsAlgebraic F E
h : ∀ (x : E), ∃ y, (aeval y) (minpoly F x) = 0
S : Finset E
p : K[X] := ∏ x ∈ S, Polynomial.map (algebraMap F K) (minpoly F x)
K' : Type u_3 := p.SplittingField
splits : ∀ s ∈ S, (Polynomial.map (algebraMap F K') (minpoly F s)).Splits
K₀ : IntermediateField F K' := restrictScalars F ⊥
FS : IntermediateField F E := adjoin F ↑S
⊢ ↥(adjoin F ↑S) →ₐ[F] K

Tactic:
let Ω := FS →ₐ[F] K'

Goal state after:
F : Type u_1
E : Type u_2
K : Type u_3
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Field K
inst✝¹ : Algebra F E
inst✝ : Algebra F K
alg : Algebra.IsAlgebraic F E
h : ∀ (x : E), ∃ y, (aeval y) (minpoly F x) = 0
S : Finset E
p : K[X] := ∏ x ∈ S, Polynomial.map (algebraMap F K) (minpoly F x)
K' : Type u_3 := p.SplittingField
splits : ∀ s ∈ S, (Polynomial.map (algebraMap F K') (minpoly F s)).Splits
K₀ : IntermediateField F K' := restrictScalars F ⊥
FS : IntermediateField F E := adjoin F ↑S
Ω : Type (max u_2 u_3) := ↥FS →ₐ[F] K'
⊢ ↥(adjoin F ↑S) →ₐ[F] K
Module: Mathlib.FieldTheory.JacobsonNoether

Goal state before:
pp: D : Type u_1
inst✝¹ : DivisionRing D
inst✝ : Algebra.IsAlgebraic (↥k) D
p : ℕ
hchar : ExpChar D p
a : D
hinsep : ∀ (x : D), IsSeparable (↥k) x → x ∈ k
⊢ ∃ n, a ^ p ^ n ∈ k

Tactic:
by
  have := (@isPurelyInseparable_iff_pow_mem k D _ _ _ _ p (ExpChar.expChar_center_iff.2 hchar)).1
  have pure : IsPurelyInseparable k D := ⟨Algebra.IsAlgebraic.isIntegral, fun x hx ↦ by
    rw [RingHom.mem_range, Subtype.exists]
    exact ⟨x, ⟨hinsep x hx, rfl⟩⟩⟩
  obtain ⟨n, ⟨m, hm⟩⟩ := this pure a
  have := Subalgebra.range_subset (R := k) ⟨(k).toSubsemiring, fun r ↦ r.2⟩
  exact ⟨n, Set.mem_of_subset_of_mem this <| Set.mem_range.2 ⟨m, hm⟩⟩

Goal state after:
Module: Mathlib.GroupTheory.CosetCover

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq (Subgroup G)
n : ℕ
ih :
  ∀ m < n,
    ∀ {ι : Type u_2} {H : ι → Subgroup G} {g : ι → G} {s : Finset ι},
      ⋃ i ∈ s, g i • ↑(H i) = Set.univ →
        ∀ j ∈ s,
          ⋃ i ∈ {x ∈ s | H x = H j}, g i • ↑(H i) ≠ Set.univ →
            m = (Finset.image H s).card → ∃ i ∈ s, H i ≠ H j ∧ (H i).FiniteIndex
ι : Type u_2
H : ι → Subgroup G
g : ι → G
s : Finset ι
j : ι
hcovers : ⋃ i ∈ {x ∈ s | H x = H j} ∪ {a ∈ s | ¬H a = H j}, g i • ↑(H i) = Set.univ
hj : j ∈ s
hcovers' : ⋃ i ∈ {x ∈ s | H x = H j}, g i • ↑(H i) ≠ Set.univ
hn : n = (Finset.image H s).card
x : G
hx : ∀ (y : G), y • ↑(H j) ⊆ ⋃ i ∈ {x ∈ s | H x ≠ H j}, (y * x⁻¹ * g i) • ↑(H i)
κ : Type u_2 := ↥({x ∈ s | H x ≠ H j}) × Option ↥({x ∈ s | H x = H j})
f : κ → G :=
  fun x_1 ↦
    match x_1 with
    | (k₁, some k₂) => g ↑k₂ * x⁻¹ * g ↑k₁
    | (k₁, none) => g ↑k₁
K : κ → Subgroup G := fun k ↦ H ↑k.1
hK' : ∀ (k : κ), K k ∈ (Finset.image H s).erase (H j)
hK : ∀ (k : κ), K k ≠ H j
y : G
⊢ ∃ i, ∃ (_ : i ∈ Finset.univ), y ∈ f i • ↑(K i)

Tactic:
Finset.set_biUnion_union

Goal state after:
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq (Subgroup G)
n : ℕ
ih :
  ∀ m < n,
    ∀ {ι : Type u_2} {H : ι → Subgroup G} {g : ι → G} {s : Finset ι},
      ⋃ i ∈ s, g i • ↑(H i) = Set.univ →
        ∀ j ∈ s,
          ⋃ i ∈ {x ∈ s | H x = H j}, g i • ↑(H i) ≠ Set.univ →
            m = (Finset.image H s).card → ∃ i ∈ s, H i ≠ H j ∧ (H i).FiniteIndex
ι : Type u_2
H : ι → Subgroup G
g : ι → G
s : Finset ι
j : ι
hcovers : (⋃ x ∈ {x ∈ s | H x = H j}, g x • ↑(H x)) ∪ ⋃ x ∈ {a ∈ s | ¬H a = H j}, g x • ↑(H x) = Set.univ
hj : j ∈ s
hcovers' : ⋃ i ∈ {x ∈ s | H x = H j}, g i • ↑(H i) ≠ Set.univ
hn : n = (Finset.image H s).card
x : G
hx : ∀ (y : G), y • ↑(H j) ⊆ ⋃ i ∈ {x ∈ s | H x ≠ H j}, (y * x⁻¹ * g i) • ↑(H i)
κ : Type u_2 := ↥({x ∈ s | H x ≠ H j}) × Option ↥({x ∈ s | H x = H j})
f : κ → G :=
  fun x_1 ↦
    match x_1 with
    | (k₁, some k₂) => g ↑k₂ * x⁻¹ * g ↑k₁
    | (k₁, none) => g ↑k₁
K : κ → Subgroup G := fun k ↦ H ↑k.1
hK' : ∀ (k : κ), K k ∈ (Finset.image H s).erase (H j)
hK : ∀ (k : κ), K k ≠ H j
y : G
⊢ ∃ i, ∃ (_ : i ∈ Finset.univ), y ∈ f i • ↑(K i)
Module: Mathlib.RingTheory.MvPolynomial

Goal state before:
pp: σ : Type u
K : Type v
inst✝ : Field K
I : Ideal (MvPolynomial σ K)
hI : I ≠ ⊤
x : K
hx : ((Ideal.Quotient.mk I).comp C) x = 0
⊢ x = 0

Tactic:
rw [RingHom.comp_apply, Ideal.Quotient.eq_zero_iff_mem] at hx

Goal state after:
σ : Type u
K : Type v
inst✝ : Field K
I : Ideal (MvPolynomial σ K)
hI : I ≠ ⊤
x : K
hx : C x ∈ I
⊢ x = 0
Module: Mathlib.GroupTheory.CosetCover

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
ι : Type u_2
H : ι → Subgroup G
g : ι → G
s : Finset ι
inst✝ : DecidablePred FiniteIndex
hcovers : ⋃ i ∈ {i ∈ s | (H i).FiniteIndex} ∪ {a ∈ s | ¬(H a).FiniteIndex}, g i • ↑(H i) = Set.univ
D : Subgroup G := ⨅ k ∈ {i ∈ s | (H i).FiniteIndex}, H k
hD : D.FiniteIndex
hD_le : ∀ {i : ι}, i ∈ s → (H i).FiniteIndex → D ≤ H i
t : (i : ι) → i ∈ s → (H i).FiniteIndex → Finset ↥(H i)
ht :
  ∀ (i : ι) (hi : i ∈ s) (hfi : (H i).FiniteIndex),
    IsComplement ↑(t i hi hfi) ↑(D.subgroupOf (H i)) ∧ ⋃ g ∈ t i hi hfi, ↑g • ↑D = ↑(H i)
κ : Type (max u_2 u_1) := (i : ↥s) × ↥(if h : (H ↑i).FiniteIndex then t ↑i ⋯ h else {1})
f : κ → G := fun k ↦ g ↑k.fst * ↑↑k.snd
K : κ → Subgroup G := fun k ↦ if (H ↑k.fst).FiniteIndex then D else H ↑k.fst
⊢ (⋃ x ∈ {k | (H ↑k.fst).FiniteIndex}, f x • ↑(K x)) ∪ ⋃ x ∈ {a | ¬(H ↑a.fst).FiniteIndex}, f x • ↑(K x) =
    ⋃ i ∈ {i ∈ s | (H i).FiniteIndex} ∪ {a ∈ s | ¬(H a).FiniteIndex}, g i • ↑(H i)

Tactic:
Finset.set_biUnion_union

Goal state after:
G : Type u_1
inst✝¹ : Group G
ι : Type u_2
H : ι → Subgroup G
g : ι → G
s : Finset ι
inst✝ : DecidablePred FiniteIndex
hcovers : ⋃ i ∈ {i ∈ s | (H i).FiniteIndex} ∪ {a ∈ s | ¬(H a).FiniteIndex}, g i • ↑(H i) = Set.univ
D : Subgroup G := ⨅ k ∈ {i ∈ s | (H i).FiniteIndex}, H k
hD : D.FiniteIndex
hD_le : ∀ {i : ι}, i ∈ s → (H i).FiniteIndex → D ≤ H i
t : (i : ι) → i ∈ s → (H i).FiniteIndex → Finset ↥(H i)
ht :
  ∀ (i : ι) (hi : i ∈ s) (hfi : (H i).FiniteIndex),
    IsComplement ↑(t i hi hfi) ↑(D.subgroupOf (H i)) ∧ ⋃ g ∈ t i hi hfi, ↑g • ↑D = ↑(H i)
κ : Type (max u_2 u_1) := (i : ↥s) × ↥(if h : (H ↑i).FiniteIndex then t ↑i ⋯ h else {1})
f : κ → G := fun k ↦ g ↑k.fst * ↑↑k.snd
K : κ → Subgroup G := fun k ↦ if (H ↑k.fst).FiniteIndex then D else H ↑k.fst
⊢ (⋃ x ∈ {k | (H ↑k.fst).FiniteIndex}, f x • ↑(K x)) ∪ ⋃ x ∈ {a | ¬(H ↑a.fst).FiniteIndex}, f x • ↑(K x) =
    (⋃ x ∈ {i ∈ s | (H i).FiniteIndex}, g x • ↑(H x)) ∪ ⋃ x ∈ {a ∈ s | ¬(H a).FiniteIndex}, g x • ↑(H x)
Module: Mathlib.FieldTheory.JacobsonNoether

Goal state before:
pp: D : Type u_1
inst✝¹ : DivisionRing D
inst✝ : Algebra.IsAlgebraic (↥k) D
H : k ≠ ⊤
p : ℕ
hp : ExpChar D p
insep : ∀ (x : D), IsSeparable (↥k) x → x ∈ k
a : D
ha : a ∉ k
ha₀ : a ≠ 0
b : D
hb1 : ((ad (↥k) D) a) b ≠ 0
n : ℕ
hn : 0 < n
c : D := (⇑((ad (↥k) D) a))^[n] b
hb : c ≠ 0 ∧ (⇑((ad (↥k) D) a))^[n + 1] b = 0
hc_def : c = (⇑((ad (↥k) D) a))^[n] b
x✝ : Invertible c := { invOf := c⁻¹, invOf_mul_self := ⋯, mul_invOf_self := ⋯ }
hc : a * c = c * a
d : D := c⁻¹ * a * (⇑((ad (↥k) D) a))^[n - 1] b
hd_def : d = c⁻¹ * a * (⇑((ad (↥k) D) a))^[n - 1] b
hc' : c⁻¹ * a = a * c⁻¹
c_eq : a * (⇑((ad (↥k) D) a))^[n - 1] b - (⇑((ad (↥k) D) a))^[n - 1] b * a = c
eq1 : c⁻¹ * a * (⇑((ad (↥k) D) a))^[n - 1] b - c⁻¹ * (⇑((ad (↥k) D) a))^[n - 1] b * a = 1
deq : a * d - d * a = a
⊢ False

Tactic:
apply_fun (a⁻¹ * ·) at deq

Goal state after:
D : Type u_1
inst✝¹ : DivisionRing D
inst✝ : Algebra.IsAlgebraic (↥k) D
H : k ≠ ⊤
p : ℕ
hp : ExpChar D p
insep : ∀ (x : D), IsSeparable (↥k) x → x ∈ k
a : D
ha : a ∉ k
ha₀ : a ≠ 0
b : D
hb1 : ((ad (↥k) D) a) b ≠ 0
n : ℕ
hn : 0 < n
c : D := (⇑((ad (↥k) D) a))^[n] b
hb : c ≠ 0 ∧ (⇑((ad (↥k) D) a))^[n + 1] b = 0
hc_def : c = (⇑((ad (↥k) D) a))^[n] b
x✝ : Invertible c := { invOf := c⁻¹, invOf_mul_self := ⋯, mul_invOf_self := ⋯ }
hc : a * c = c * a
d : D := c⁻¹ * a * (⇑((ad (↥k) D) a))^[n - 1] b
hd_def : d = c⁻¹ * a * (⇑((ad (↥k) D) a))^[n - 1] b
hc' : c⁻¹ * a = a * c⁻¹
c_eq : a * (⇑((ad (↥k) D) a))^[n - 1] b - (⇑((ad (↥k) D) a))^[n - 1] b * a = c
eq1 : c⁻¹ * a * (⇑((ad (↥k) D) a))^[n - 1] b - c⁻¹ * (⇑((ad (↥k) D) a))^[n - 1] b * a = 1
deq : a⁻¹ * (a * d - d * a) = a⁻¹ * a
⊢ False
Module: Mathlib.FieldTheory.KummerExtension

Goal state before:
pp: K : Type u
inst✝ : Field K
n : ℕ
hζ : (primitiveRoots n K).Nonempty
a : K
H : Irreducible (X ^ n - C a)
η : ↥(rootsOfUnity n K)
⊢ eval₂ (↑(Algebra.ofId K K[n√a])) (↑↑η • root (X ^ n - C a)) (X ^ n - C a) = 0

Tactic:
have := (mem_rootsOfUnity' _ _).mp η.prop
    change aeval _ _ = _
    rw [map_sub, map_pow, aeval_C, aeval_X, Algebra.smul_def, mul_pow, root_X_pow_sub_C_pow,
      AdjoinRoot.algebraMap_eq, ← map_pow, this, map_one, one_mul, sub_self]

Goal state after:
Module: Mathlib.FieldTheory.KummerExtension

Goal state before:
pp: K : Type u
inst✝¹ : Field K
n : ℕ
hζ : (primitiveRoots n K).Nonempty
a : K
H : Irreducible (X ^ n - C a)
inst✝ : NeZero n
σ : Gal(K[n√a]/K)
this✝¹ : Fact (Irreducible (X ^ n - C a)) := { out := H }
this✝ : IsDomain K[n√a] := inferInstance
this : DecidableEq K := Classical.decEq K
⊢ (if a = 0 then 1 else σ (root (X ^ n - C a)) / root (X ^ n - C a)) ^ n = 1

Tactic:
split

Goal state after:
case isTrue
K : Type u
inst✝¹ : Field K
n : ℕ
hζ : (primitiveRoots n K).Nonempty
a : K
H : Irreducible (X ^ n - C a)
inst✝ : NeZero n
σ : Gal(K[n√a]/K)
this✝¹ : Fact (Irreducible (X ^ n - C a)) := { out := H }
this✝ : IsDomain K[n√a] := inferInstance
this : DecidableEq K := Classical.decEq K
h✝ : a = 0
⊢ 1 ^ n = 1

case isFalse
K : Type u
inst✝¹ : Field K
n : ℕ
hζ : (primitiveRoots n K).Nonempty
a : K
H : Irreducible (X ^ n - C a)
inst✝ : NeZero n
σ : Gal(K[n√a]/K)
this✝¹ : Fact (Irreducible (X ^ n - C a)) := { out := H }
this✝ : IsDomain K[n√a] := inferInstance
this : DecidableEq K := Classical.decEq K
h✝ : ¬a = 0
⊢ (σ (root (X ^ n - C a)) / root (X ^ n - C a)) ^ n = 1
Module: Mathlib.GroupTheory.CosetCover

Goal state before:
pp: case pos
G : Type u_1
inst✝¹ : Group G
ι : Type u_2
H : ι → Subgroup G
g : ι → G
s : Finset ι
hcovers : ⋃ i ∈ s, g i • ↑(H i) = Set.univ
inst✝ : DecidablePred FiniteIndex
D : Subgroup G := ⨅ k ∈ {i ∈ s | (H i).FiniteIndex}, H k
hD : D.FiniteIndex
hD_le : ∀ {i : ι}, i ∈ s → (H i).FiniteIndex → D ≤ H i
t : (i : ι) → i ∈ s → (H i).FiniteIndex → Finset ↥(H i)
ht :
  ∀ (i : ι) (hi : i ∈ s) (hfi : (H i).FiniteIndex),
    IsComplement ↑(t i hi hfi) ↑(D.subgroupOf (H i)) ∧ ⋃ g ∈ t i hi hfi, ↑g • ↑D = ↑(H i)
κ : Type (max u_2 u_1) := (i : ↥s) × ↥(if h : (H ↑i).FiniteIndex then t ↑i ⋯ h else {1})
f : κ → G := fun k ↦ g ↑k.fst * ↑↑k.snd
K : κ → Subgroup G := fun k ↦ if (H ↑k.fst).FiniteIndex then D else H ↑k.fst
k : κ
hkfi : (H ↑k.fst).FiniteIndex
hk : K k = D
hcovers' : ⋃ i ∈ {x | K x = D}, f i • ↑D = Set.univ
hHD : ∀ (i : ι), ¬(H i).FiniteIndex → H i ≠ D
i : ↥s
x✝ : i ∈ s.attach
hfi : (H ↑i).FiniteIndex
⊢ ∑ s_1, ↑(if K ⟨i, s_1⟩ = D then 1 else 0) = (↑(H ↑i).index)⁻¹ * ↑D.index

Tactic:
rw [← relIndex_mul_index (hD_le i.2 hfi), Nat.cast_mul, mul_comm,
        mul_inv_cancel_right₀ (Nat.cast_ne_zero.mpr hfi.index_ne_zero)]

Goal state after:
case pos
G : Type u_1
inst✝¹ : Group G
ι : Type u_2
H : ι → Subgroup G
g : ι → G
s : Finset ι
hcovers : ⋃ i ∈ s, g i • ↑(H i) = Set.univ
inst✝ : DecidablePred FiniteIndex
D : Subgroup G := ⨅ k ∈ {i ∈ s | (H i).FiniteIndex}, H k
hD : D.FiniteIndex
hD_le : ∀ {i : ι}, i ∈ s → (H i).FiniteIndex → D ≤ H i
t : (i : ι) → i ∈ s → (H i).FiniteIndex → Finset ↥(H i)
ht :
  ∀ (i : ι) (hi : i ∈ s) (hfi : (H i).FiniteIndex),
    IsComplement ↑(t i hi hfi) ↑(D.subgroupOf (H i)) ∧ ⋃ g ∈ t i hi hfi, ↑g • ↑D = ↑(H i)
κ : Type (max u_2 u_1) := (i : ↥s) × ↥(if h : (H ↑i).FiniteIndex then t ↑i ⋯ h else {1})
f : κ → G := fun k ↦ g ↑k.fst * ↑↑k.snd
K : κ → Subgroup G := fun k ↦ if (H ↑k.fst).FiniteIndex then D else H ↑k.fst
k : κ
hkfi : (H ↑k.fst).FiniteIndex
hk : K k = D
hcovers' : ⋃ i ∈ {x | K x = D}, f i • ↑D = Set.univ
hHD : ∀ (i : ι), ¬(H i).FiniteIndex → H i ≠ D
i : ↥s
x✝ : i ∈ s.attach
hfi : (H ↑i).FiniteIndex
⊢ ∑ s_1, ↑(if K ⟨i, s_1⟩ = D then 1 else 0) = ↑(D.relIndex (H ↑i))
Module: Mathlib.GroupTheory.CosetCover

Goal state before:
pp: case refine_3
G : Type u_1
inst✝¹ : Group G
ι : Type u_2
H : ι → Subgroup G
g : ι → G
s : Finset ι
hcovers : ⋃ i ∈ s, g i • ↑(H i) = Set.univ
inst✝ : DecidablePred FiniteIndex
D : Subgroup G := ⨅ k ∈ {i ∈ s | (H i).FiniteIndex}, H k
hD : D.FiniteIndex
hD_le : ∀ {i : ι}, i ∈ s → (H i).FiniteIndex → D ≤ H i
t : (i : ι) → i ∈ s → (H i).FiniteIndex → Finset ↥(H i)
ht :
  ∀ (i : ι) (hi : i ∈ s) (hfi : (H i).FiniteIndex),
    IsComplement ↑(t i hi hfi) ↑(D.subgroupOf (H i)) ∧ ⋃ g ∈ t i hi hfi, ↑g • ↑D = ↑(H i)
κ : Type (max u_2 u_1) := (i : ↥s) × ↥(if h : (H ↑i).FiniteIndex then t ↑i ⋯ h else {1})
f : κ → G := fun k ↦ g ↑k.fst * ↑↑k.snd
K : κ → Subgroup G := fun k ↦ if (H ↑k.fst).FiniteIndex then D else H ↑k.fst
k : κ
hkfi : (H ↑k.fst).FiniteIndex
hk : K k = D
hcovers' : ⋃ i ∈ {x | K x = D}, f i • ↑D = Set.univ
hHD : ∀ (i : ι), ¬(H i).FiniteIndex → H i ≠ D
hdensity : ∑ i ∈ s, (↑(H i).index)⁻¹ = ↑{x | K x = D}.card * (↑D.index)⁻¹
⊢ ∑ i ∈ s, (↑(H i).index)⁻¹ = 1 → (↑({i ∈ s | (H i).FiniteIndex})).PairwiseDisjoint fun i ↦ g i • ↑(H i)

Tactic:
· rw [hdensity, mul_inv_eq_one₀ (Nat.cast_ne_zero.mpr hD.index_ne_zero),
      Nat.cast_inj, Finset.coe_filter]
    intro h i hi j hj hij c hi' hj' x hx
    have hdisjoint := pairwiseDisjoint_leftCoset_cover_const_of_index_eq hcovers' h.symm
    -- We know the `f k • K k` are pairwise disjoint and need to prove that the `g i • H i` are.
    rw [Set.mem_ofPred_eq] at hi hj
    have hk' (i) (hi : i ∈ s ∧ (H i).FiniteIndex) (hi' : c ≤ g i • (H i : Set G)) :
        ∃ (k : κ), k.1.1 = i ∧ K k = D ∧ x ∈ f k • (D : Set G) := by
      rw [← (ht i hi.1 hi.2).2] at hi'
      suffices ∃ r : H i, r ∈ t i hi.1 hi.2 ∧ x ∈ (g i * r) • (D : Set G) by
        have ⟨r, hr, hxr⟩ := this
        refine ⟨⟨⟨i, hi.1⟩, ⟨r, dif_pos hi.2 ▸ hr⟩⟩, rfl, ?_⟩
        simpa [K, f, if_pos hi.2] using! hxr
      simpa [Set.mem_smul_set_iff_inv_smul_mem, smul_eq_mul, mul_assoc] using! hi' hx
    have ⟨k₁, hik₁, hk₁, hxk₁⟩ := hk' i hi hi'
    have ⟨k₂, hjk₂, hk₂, hxk₂⟩ := hk' j hj hj'
    rw [← Set.singleton_subset_iff] at hxk₁ hxk₂ ⊢
    exact hdisjoint
      (Finset.mem_filter.mpr ⟨Finset.mem_univ k₁, hk₁⟩)
      (Finset.mem_filter.mpr ⟨Finset.mem_univ k₂, hk₂⟩)
      (ne_of_apply_ne Sigma.fst (ne_of_apply_ne Subtype.val (hik₁ ▸ hjk₂ ▸ hij)))
      hxk₁ hxk₂

Goal state after:
Module: Mathlib.FieldTheory.Minpoly.ConjRootClass

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra.IsSeparable K L
inst✝¹ : Normal K L
c : ConjRootClass K L
inst✝ : Fintype ↑c.carrier
⊢ map (algebraMap K L) c.minpoly = ∏ x ∈ c.carrier.toFinset, (X - C x)

Tactic:
simp_rw [← rootSet_minpoly_eq_carrier, Finset.prod_eq_multiset_prod, rootSet_def,
    Finset.toFinset_coe, Multiset.toFinset_val]

Goal state after:
K : Type u_1
L : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra.IsSeparable K L
inst✝¹ : Normal K L
c : ConjRootClass K L
inst✝ : Fintype ↑c.carrier
⊢ map (algebraMap K L) c.minpoly = (Multiset.map (fun x ↦ X - C x) (c.minpoly.aroots L).dedup).prod
Module: Mathlib.RingTheory.LinearDisjoint

Goal state before:
pp: R : Type u
S : Type v
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
A B : Subalgebra R S
H : A.LinearDisjoint B
H' : A ⊔ B = ⊤
inst✝¹ : Free R ↥B
inst✝ : Module.Finite R ↥B
x : ↥B
⊢ ∑ x_1, (((Algebra.leftMulMatrix (Free.chooseBasis R ↥B)) x).map ⇑(algebraMap R ↥A)).diag x_1 =
    ∑ x_1, (algebraMap R ↥A) (((Algebra.leftMulMatrix (Free.chooseBasis R ↥B)) x).diag x_1)

Tactic:
Matrix.diag_apply,

Goal state after:
R : Type u
S : Type v
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
A B : Subalgebra R S
H : A.LinearDisjoint B
H' : A ⊔ B = ⊤
inst✝¹ : Free R ↥B
inst✝ : Module.Finite R ↥B
x : ↥B
⊢ ∑ x_1, ((Algebra.leftMulMatrix (Free.chooseBasis R ↥B)) x).map (⇑(algebraMap R ↥A)) x_1 x_1 =
    ∑ x_1, (algebraMap R ↥A) ((Algebra.leftMulMatrix (Free.chooseBasis R ↥B)) x x_1 x_1)
Module: Mathlib.RingTheory.LinearDisjoint

Goal state before:
pp: R : Type u
inst✝⁸ : CommRing R
A : Type v
inst✝⁷ : CommRing A
B : Type w
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra R B
inst✝³ : Flat R A
inst✝² : Flat R B
inst✝¹ : Algebra.Transcendental R A
inst✝ : Algebra.Transcendental R B
H : IsField (A ⊗[R] B)
this✝³ : Field (A ⊗[R] B) := H.toField
a : A
b : B
ha : Function.Injective ⇑(algebraMap R A)
hb : Function.Injective ⇑(algebraMap R B)
fa : A →ₐ[R] A ⊗[R] B := Algebra.TensorProduct.includeLeft
fb : B →ₐ[R] A ⊗[R] B := Algebra.TensorProduct.includeRight
hfa : Function.Injective ⇑fa
hfb : Function.Injective ⇑fb
this✝² : IsDomain A
this✝¹ : IsDomain B
this✝ : IsDomain R
this : Flat R ↥(toSubmodule fa.range)
key1 : Module.rank R ↥(fa.range ⊓ fb.range) ≤ 1
ga : R[X] →ₐ[R] A := aeval a
gb : R[X] →ₐ[R] B := aeval b
gab : R[X] →ₐ[R] A ⊗[R] B := fa.comp ga
hta : Function.Injective ⇑ga
htb : Function.Injective ⇑gb
⊢ False

Tactic:
have htab : Function.Injective gab := hfa.comp hta

Goal state after:
R : Type u
inst✝⁸ : CommRing R
A : Type v
inst✝⁷ : CommRing A
B : Type w
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra R B
inst✝³ : Flat R A
inst✝² : Flat R B
inst✝¹ : Algebra.Transcendental R A
inst✝ : Algebra.Transcendental R B
H : IsField (A ⊗[R] B)
this✝³ : Field (A ⊗[R] B) := H.toField
a : A
b : B
ha : Function.Injective ⇑(algebraMap R A)
hb : Function.Injective ⇑(algebraMap R B)
fa : A →ₐ[R] A ⊗[R] B := Algebra.TensorProduct.includeLeft
fb : B →ₐ[R] A ⊗[R] B := Algebra.TensorProduct.includeRight
hfa : Function.Injective ⇑fa
hfb : Function.Injective ⇑fb
this✝² : IsDomain A
this✝¹ : IsDomain B
this✝ : IsDomain R
this : Flat R ↥(toSubmodule fa.range)
key1 : Module.rank R ↥(fa.range ⊓ fb.range) ≤ 1
ga : R[X] →ₐ[R] A := aeval a
gb : R[X] →ₐ[R] B := aeval b
gab : R[X] →ₐ[R] A ⊗[R] B := fa.comp ga
hta : Function.Injective ⇑ga
htb : Function.Injective ⇑gb
htab : Function.Injective ⇑gab
⊢ False
Module: Mathlib.FieldTheory.LinearDisjoint

Goal state before:
pp: F : Type u
E : Type v
inst✝⁵ : Field F
inst✝⁴ : Field E
inst✝³ : Algebra F E
A B : IntermediateField F E
inst✝² : IsGalois F ↥A
inst✝¹ : FiniteDimensional F ↥A
inst✝ : FiniteDimensional F ↥B
h : A ⊓ B = ⊥
C : IntermediateField F E := A ⊔ B
A' : IntermediateField F ↥C := restrict ⋯
B' : IntermediateField F ↥C := restrict ⋯
hA : IntermediateField.map C.val A' = A
hB : IntermediateField.map C.val B' = B
⊢ A.LinearDisjoint ↥B

Tactic:
suffices A'.LinearDisjoint B' from hA ▸ hB ▸ LinearDisjoint.map this C.val

Goal state after:
F : Type u
E : Type v
inst✝⁵ : Field F
inst✝⁴ : Field E
inst✝³ : Algebra F E
A B : IntermediateField F E
inst✝² : IsGalois F ↥A
inst✝¹ : FiniteDimensional F ↥A
inst✝ : FiniteDimensional F ↥B
h : A ⊓ B = ⊥
C : IntermediateField F E := A ⊔ B
A' : IntermediateField F ↥C := restrict ⋯
B' : IntermediateField F ↥C := restrict ⋯
hA : IntermediateField.map C.val A' = A
hB : IntermediateField.map C.val B' = B
⊢ A'.LinearDisjoint ↥B'
Module: Mathlib.FieldTheory.PurelyInseparable.PerfectClosure

Goal state before:
pp: case haeval
F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
q : ℕ
hF : ExpChar F q
inst✝ : ExpChar E q
n : ℕ
a : E
hsep : IsSeparable F a
hai : IsIntegral F a
hapi : IsIntegral F ((iterateFrobenius E q n) a)
⊢ (Polynomial.aeval (a ^ q ^ n)) (Polynomial.map (iterateFrobenius F q n) (minpoly F a)) = 0

Tactic:
simpa using! Eq.symm <|
      (minpoly F a).map_aeval_eq_aeval_map (RingHom.iterateFrobenius_comm _ q n) a

Goal state after:
Module: Mathlib.FieldTheory.Relrank

Goal state before:
pp: E : Type v
inst✝ : Field E
A : Subfield E
⊢ A.relfinrank ⊤ = finrank (↥A) E

Tactic:
simp [relfinrank_eq_toNat_relrank, finrank]

Goal state after:
Module: Mathlib.FieldTheory.Relrank

Goal state before:
pp: F : Type u
E : Type v
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
A : IntermediateField F E
⊢ A.relfinrank ⊤ = finrank (↥A) E

Tactic:
simp [relfinrank_eq_toNat_relrank, finrank]

Goal state after:
Module: Mathlib.FieldTheory.Relrank

Goal state before:
pp: F : Type u
E : Type v
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
A : IntermediateField F E
⊢ ⊥.relfinrank A = finrank F ↥A

Tactic:
simp [relfinrank_eq_toNat_relrank, finrank]

Goal state after:
Module: Mathlib.Geometry.Diffeology.Basic

Goal state before:
pp: X : Type u_1
inst✝⁸ : DiffeologicalSpace X
Y : Type u_2
inst✝⁷ : NormedAddCommGroup X
inst✝⁶ : NormedSpace ℝ X
inst✝⁵ : IsContDiffCompatible X
inst✝⁴ : NormedAddCommGroup Y
inst✝³ : NormedSpace ℝ Y
inst✝² : DiffeologicalSpace Y
inst✝¹ : IsContDiffCompatible Y
inst✝ : FiniteDimensional ℝ X
f : X → Y
hf : DSmooth f
⊢ ContDiff ℝ ∞ f

Tactic:
by
  let g := toEuclidean (E := X)
  rw [← Function.comp_id f, ← g.symm_comp_self]
  exact (hf _ _ g.symm.contDiff.isPlot).contDiff.comp g.contDiff

Goal state after:
Module: Mathlib.Geometry.Euclidean.Altitude

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
n : ℕ
inst✝ : NeZero n
s : Simplex ℝ P n
i : Fin (n + 1)
h :
  finrank ℝ ↥(vectorSpan ℝ (s.points '' {i}ᶜ)) +
      finrank ℝ ↥((vectorSpan ℝ (s.points '' {i}ᶜ))ᗮ ⊓ vectorSpan ℝ (Set.range s.points)) =
    finrank ℝ ↥(vectorSpan ℝ (Set.range s.points))
hn : n - 1 + 1 = n
⊢ n + 1 - 1 = n - 1 + 1

Tactic:
hn,

Goal state after:
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
n : ℕ
inst✝ : NeZero n
s : Simplex ℝ P n
i : Fin (n + 1)
h :
  finrank ℝ ↥(vectorSpan ℝ (s.points '' {i}ᶜ)) +
      finrank ℝ ↥((vectorSpan ℝ (s.points '' {i}ᶜ))ᗮ ⊓ vectorSpan ℝ (Set.range s.points)) =
    finrank ℝ ↥(vectorSpan ℝ (Set.range s.points))
hn : n - 1 + 1 = n
⊢ n + 1 - 1 = n
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.Basic

Goal state before:
pp: V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
hx : x ≠ 0
hy : y ≠ 0
⊢ o.oangle (-x) y = o.oangle x y + ↑π

Tactic:
simp only [oangle, map_neg]
  convert! Complex.arg_neg_coe_angle _
  exact o.kahler_ne_zero hx hy

Goal state after:
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.Basic

Goal state before:
pp: V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
hx : x ≠ 0
hy : y ≠ 0
⊢ o.oangle x (-y) = o.oangle x y + ↑π

Tactic:
simp only [oangle, map_neg]
  convert! Complex.arg_neg_coe_angle _
  exact o.kahler_ne_zero hx hy

Goal state after: