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

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
p : ℕ
x : R
⊢ x ∈ pNilradical R p ↔ ∃ n, x ^ p ^ n = 0

Tactic:
by_cases hp : 1 < p

Goal state after:
case pos
R : Type u_1
inst✝ : CommSemiring R
p : ℕ
x : R
hp : 1 < p
⊢ x ∈ pNilradical R p ↔ ∃ n, x ^ p ^ n = 0

case neg
R : Type u_1
inst✝ : CommSemiring R
p : ℕ
x : R
hp : ¬1 < p
⊢ x ∈ pNilradical R p ↔ ∃ n, x ^ p ^ n = 0
Module: Mathlib.FieldTheory.PerfectClosure

Goal state before:
pp: case mp.trans
K : Type u
inst✝² : CommRing K
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : CharP K p
x✝ y✝ x y z : ℕ × K
H1 : Relation.EqvGen (R K p) x y
H2 : Relation.EqvGen (R K p) y z
z1 : ℕ
ih1 : (⇑(frobenius K p))^[y.1 + z1] x.2 = (⇑(frobenius K p))^[x.1 + z1] y.2
z2 : ℕ
ih2 : (⇑(frobenius K p))^[z.1 + z2] y.2 = (⇑(frobenius K p))^[y.1 + z2] z.2
⊢ (⇑(frobenius K p))^[x.1 + z1] ((⇑(frobenius K p))^[y.1 + z2] z.2) = (⇑(frobenius K p))^[x.1 + (z2 + (y.1 + z1))] z.2

Tactic:
← iterate_add_apply

Goal state after:
case mp.trans
K : Type u
inst✝² : CommRing K
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : CharP K p
x✝ y✝ x y z : ℕ × K
H1 : Relation.EqvGen (R K p) x y
H2 : Relation.EqvGen (R K p) y z
z1 : ℕ
ih1 : (⇑(frobenius K p))^[y.1 + z1] x.2 = (⇑(frobenius K p))^[x.1 + z1] y.2
z2 : ℕ
ih2 : (⇑(frobenius K p))^[z.1 + z2] y.2 = (⇑(frobenius K p))^[y.1 + z2] z.2
⊢ (⇑(frobenius K p))^[x.1 + z1 + (y.1 + z2)] z.2 = (⇑(frobenius K p))^[x.1 + (z2 + (y.1 + z1))] z.2
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
⊢ (iterateFrobeniusEquiv M p (n1 + n2)).symm ((iterateFrobeniusEquiv M p n2) (j y1)) *
      (iterateFrobeniusEquiv M p (n1 + n2)).symm (j y2 ^ p ^ n1) =
    (iterateFrobeniusEquiv M p n1).symm (j y1) * (iterateFrobeniusEquiv M p n2).symm (j y2)

Tactic:
iterateFrobeniusEquiv_symm_add_apply

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
⊢ (iterateFrobeniusEquiv M p n1).symm ((iterateFrobeniusEquiv M p n2).symm ((iterateFrobeniusEquiv M p n2) (j y1))) *
      (iterateFrobeniusEquiv M p (n1 + n2)).symm (j y2 ^ p ^ n1) =
    (iterateFrobeniusEquiv M p n1).symm (j y1) * (iterateFrobeniusEquiv M p n2).symm (j y2)
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
⊢ (iterateFrobeniusEquiv M p (n1 + n2)).symm ((iterateFrobeniusEquiv M p n2) (j y1)) +
      (iterateFrobeniusEquiv M p (n1 + n2)).symm (j y2 ^ p ^ n1) =
    (iterateFrobeniusEquiv M p n1).symm (j y1) + (iterateFrobeniusEquiv M p n2).symm (j y2)

Tactic:
iterateFrobeniusEquiv_symm_add_apply

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
⊢ (iterateFrobeniusEquiv M p n1).symm ((iterateFrobeniusEquiv M p n2).symm ((iterateFrobeniusEquiv M p n2) (j y1))) +
      (iterateFrobeniusEquiv M p (n1 + n2)).symm (j y2 ^ p ^ n1) =
    (iterateFrobeniusEquiv M p n1).symm (j y1) + (iterateFrobeniusEquiv M p n2).symm (j y2)
Module: Mathlib.GroupTheory.CosetCover

Goal state before:
pp: G : Type u_1
inst✝ : Group G
ι : Type u_2
s : Finset ι
H : Subgroup G
g : ι → G
hcovers : ⋃ i ∈ s, g i • ↑H = Set.univ
hind : H.index = s.card
this✝ : Fintype (G ⧸ H)
this : Function.Bijective fun x ↦ ↑(g ↑x)
⊢ (↑s).PairwiseDisjoint fun x ↦ g x • ↑H

Tactic:
intro i hi j hj h' c hi' hj' x hx
    specialize hi' hx
    specialize hj' hx
    rw [mem_leftCoset_iff, SetLike.mem_coe, ← QuotientGroup.eq] at hi' hj'
    rw [ne_eq, ← Subtype.mk.injEq (p := (· ∈ (s : Set ι))) i hi j hj] at h'
    exact h' <| this.injective <| by simp only [hi', hj']

Goal state after:
Module: Mathlib.GroupTheory.CosetCover

Goal state before:
pp: case refine_1
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
hcovers : ⋃ i ∈ s, g i • ↑(H i) = Set.univ
j : ι
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 : ∀ i ∈ s, H i = H j → ↑(g i) ≠ ↑x
y z : G
hz : z ∈ y • ↑(H j)
i : ι
hi : i ∈ s
hmem : ↑(g i) = ↑(x * (y⁻¹ * z))
hij : H i = H j
⊢ ↑(g i) = ↑x

pp: case refine_2
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
hcovers : ⋃ i ∈ s, g i • ↑(H i) = Set.univ
j : ι
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 : ∀ i ∈ s, H i = H j → ↑(g i) ≠ ↑x
y z : G
hz : z ∈ y • ↑(H j)
i : ι
hi : i ∈ s
hmem : ↑(g i) = ↑(x * (y⁻¹ * z))
⊢ z ∈ (y * x⁻¹ * g i) • ↑(H i)

Tactic:
· rwa [hmem, eq_comm, QuotientGroup.eq, hij, inv_mul_cancel_left,
          ← SetLike.mem_coe, ← mem_leftCoset_iff]

Goal state after:
case refine_2
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
hcovers : ⋃ i ∈ s, g i • ↑(H i) = Set.univ
j : ι
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 : ∀ i ∈ s, H i = H j → ↑(g i) ≠ ↑x
y z : G
hz : z ∈ y • ↑(H j)
i : ι
hi : i ∈ s
hmem : ↑(g i) = ↑(x * (y⁻¹ * z))
⊢ z ∈ (y * x⁻¹ * g i) • ↑(H i)
Module: Mathlib.GroupTheory.CosetCover

Goal state before:
pp: case inr
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
hy✝ : y ∈ ⋃ x ∈ {a ∈ s | ¬H a = H j}, g x • ↑(H x)
i : ι
hi : i ∈ {a ∈ s | ¬H a = H j}
hy : y ∈ g i • ↑(H i)
⊢ ∃ i, ∃ (_ : i ∈ Finset.univ), y ∈ f i • ↑(K i)

Tactic:
exact ⟨⟨⟨i, hi⟩, none⟩, Finset.mem_univ _, hy⟩

Goal state after:
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
⊢ ⋃ k ∈ Finset.univ, f k • ↑(K k) = Set.univ

Tactic:
← hcovers,

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
⊢ ⋃ k ∈ Finset.univ, f k • ↑(K k) = ⋃ i ∈ {i ∈ s | (H i).FiniteIndex} ∪ {a ∈ s | ¬(H a).FiniteIndex}, g i • ↑(H i)
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
r : ℕ
hr : d ^ p ^ r ∈ k
deq : 1 = 0
⊢ False

Tactic:
exact one_ne_zero deq

Goal state after:
Module: Mathlib.FieldTheory.KummerExtension

Goal state before:
pp: case splits'
K : Type u
inst✝ : Field K
n : ℕ
hζ : (primitiveRoots n K).Nonempty
a : K
H : Irreducible (X ^ n - C a)
this✝ : Fact (Irreducible (X ^ n - C a))
this : Algebra K K[n√a] := ⋯
⊢ (Polynomial.map (algebraMap K K[n√a]) (X ^ n - C a)).Splits

pp: case adjoin_rootSet'
K : Type u
inst✝ : Field K
n : ℕ
hζ : (primitiveRoots n K).Nonempty
a : K
H : Irreducible (X ^ n - C a)
this✝ : Fact (Irreducible (X ^ n - C a))
this : Algebra K K[n√a] := ⋯
⊢ Algebra.adjoin K ((X ^ n - C a).rootSet K[n√a]) = ⊤

Tactic:
· rw [Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_C,
      Polynomial.map_X]
    have ⟨_, hζ⟩ := hζ
    rw [mem_primitiveRoots (Nat.pos_of_ne_zero <| ne_zero_of_irreducible_X_pow_sub_C H)] at hζ
    exact X_pow_sub_C_splits_of_isPrimitiveRoot (hζ.map_of_injective (algebraMap K _).injective)
      (root_X_pow_sub_C_pow n a)

Goal state after:
case adjoin_rootSet'
K : Type u
inst✝ : Field K
n : ℕ
hζ : (primitiveRoots n K).Nonempty
a : K
H : Irreducible (X ^ n - C a)
this✝ : Fact (Irreducible (X ^ n - C a))
this : Algebra K K[n√a] := ⋯
⊢ Algebra.adjoin K ((X ^ n - C a).rootSet K[n√a]) = ⊤
Module: Mathlib.GroupTheory.CosetCover

Goal state before:
pp: case neg
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 [of_not_not (FiniteIndex.mk.mt hfi), Nat.cast_zero, inv_zero, zero_mul]
      simpa [K, hfi] using! hHD i hfi

Goal state after:
Module: Mathlib.FieldTheory.KummerExtension

Goal state before:
pp: K : Type u
inst✝⁴ : Field K
n : ℕ
a : K
H : Irreducible (X ^ n - C a)
L : Type u_1
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : IsSplittingField K L (X ^ n - C a)
α : L
hα : α ^ n = (algebraMap K L) a
inst✝ : NeZero n
ζ : K
hζ : IsPrimitiveRoot ζ n
m : ℤ
hn : 0 < n
⊢ (autEquivRootsOfUnity ⋯ H L) ((autEquivZmod H L hζ).symm (Multiplicative.ofAdd ↑m)) • α = ζ ^ m • α

Tactic:
simp [MulEquiv.subgroupCongr_symm_apply, Subgroup.smul_def, Units.smul_def, autEquivZmod]

Goal state after:
Module: Mathlib.FieldTheory.KummerExtension

Goal state before:
pp: K : Type u
inst✝⁵ : Field K
L : Type u_1
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : FiniteDimensional K L
hK : (primitiveRoots (finrank K L) K).Nonempty
inst✝¹ : IsGalois K L
inst✝ : IsCyclic Gal(L/K)
ζ : K
hζ : ζ ∈ primitiveRoots (finrank K L) K
⊢ ∃ α, α ^ finrank K L ∈ Set.range ⇑(algebraMap K L) ∧ K⟮α⟯ = ⊤

Tactic:
mem_primitiveRoots finrank_pos

Goal state after:
K : Type u
inst✝⁵ : Field K
L : Type u_1
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : FiniteDimensional K L
hK : (primitiveRoots (finrank K L) K).Nonempty
inst✝¹ : IsGalois K L
inst✝ : IsCyclic Gal(L/K)
ζ : K
hζ : IsPrimitiveRoot ζ (finrank K L)
⊢ ∃ α, α ^ finrank K L ∈ Set.range ⇑(algebraMap K L) ∧ K⟮α⟯ = ⊤
Module: Mathlib.FieldTheory.KummerExtension

Goal state before:
pp: K : Type u
inst✝⁵ : Field K
L : Type u_1
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : FiniteDimensional K L
hK : (primitiveRoots (finrank K L) K).Nonempty
inst✝¹ : IsGalois K L
inst✝ : IsCyclic Gal(L/K)
ζ : K
hζ : IsPrimitiveRoot ζ (finrank K L)
σ : Gal(L/K)
hσ : Function.Surjective fun x ↦ σ ^ x
hσ' : orderOf σ = Nat.card Gal(L/K)
⊢ (minpoly K σ.toLinearMap).IsRoot ζ

Tactic:
rw [IsGalois.card_aut_eq_finrank] at hσ'

Goal state after:
K : Type u
inst✝⁵ : Field K
L : Type u_1
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : FiniteDimensional K L
hK : (primitiveRoots (finrank K L) K).Nonempty
inst✝¹ : IsGalois K L
inst✝ : IsCyclic Gal(L/K)
ζ : K
hζ : IsPrimitiveRoot ζ (finrank K L)
σ : Gal(L/K)
hσ : Function.Surjective fun x ↦ σ ^ x
hσ' : orderOf σ = finrank K L
⊢ (minpoly K σ.toLinearMap).IsRoot ζ
Module: Mathlib.FieldTheory.KummerExtension

Goal state before:
pp: case splits'
K : Type u
inst✝³ : Field K
L : Type u_1
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : FiniteDimensional K L
hK : (primitiveRoots (finrank K L) K).Nonempty
a : K
α : L
ha : α ^ finrank K L = (algebraMap K L) a
hα : K⟮α⟯ = ⊤
w✝ : K
hζ : w✝ ∈ primitiveRoots (finrank K L) K
⊢ (X ^ finrank K L - C ((algebraMap K L) a)).Splits

Tactic:
mem_primitiveRoots finrank_pos

Goal state after:
case splits'
K : Type u
inst✝³ : Field K
L : Type u_1
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : FiniteDimensional K L
hK : (primitiveRoots (finrank K L) K).Nonempty
a : K
α : L
ha : α ^ finrank K L = (algebraMap K L) a
hα : K⟮α⟯ = ⊤
w✝ : K
hζ : IsPrimitiveRoot w✝ (finrank K L)
⊢ (X ^ finrank K L - C ((algebraMap K L) a)).Splits
Module: Mathlib.FieldTheory.KummerExtension

Goal state before:
pp: case adjoin_rootSet'
K : Type u
inst✝³ : Field K
L : Type u_1
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : FiniteDimensional K L
hK : (primitiveRoots (finrank K L) K).Nonempty
a : K
α : L
ha : α ^ finrank K L = (algebraMap K L) a
hα : K⟮α⟯ = ⊤
⊢ ⊤ ≤ Algebra.adjoin K ((X ^ finrank K L - C a).rootSet L)

Tactic:
← IntermediateField.top_toSubalgebra,

Goal state after:
case adjoin_rootSet'
K : Type u
inst✝³ : Field K
L : Type u_1
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : FiniteDimensional K L
hK : (primitiveRoots (finrank K L) K).Nonempty
a : K
α : L
ha : α ^ finrank K L = (algebraMap K L) a
hα : K⟮α⟯ = ⊤
⊢ ⊤.toSubalgebra ≤ Algebra.adjoin K ((X ^ finrank K L - C a).rootSet L)
Module: Mathlib.FieldTheory.Minpoly.ConjRootClass

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : Algebra.IsAlgebraic K L
c : ConjRootClass K L
⊢ c.minpoly.Monic

Tactic:
induction c
  rw [minpoly_mk]
  exact minpoly.monic (Algebra.IsIntegral.isIntegral _)

Goal state after:
Module: Mathlib.FieldTheory.LinearDisjoint

Goal state before:
pp: case e'_6
F : Type u
E : Type v
inst✝²⁰ : Field F
inst✝¹⁹ : Field E
inst✝¹⁸ : Algebra F E
L : Type w
inst✝¹⁷ : Field L
inst✝¹⁶ : Algebra F L
inst✝¹⁵ : Algebra L E
inst✝¹⁴ : IsScalarTower F L E
L' : Type u_1
inst✝¹³ : Field L'
inst✝¹² : Algebra F L'
inst✝¹¹ : Algebra L' E
inst✝¹⁰ : IsScalarTower F L' E
H : (IsScalarTower.toAlgHom F L E).fieldRange.LinearDisjoint (IsScalarTower.toAlgHom F L' E).range
K : Type u_2
inst✝⁹ : Field K
inst✝⁸ : Algebra F K
inst✝⁷ : Algebra L K
inst✝⁶ : IsScalarTower F L K
inst✝⁵ : Algebra L' K
inst✝⁴ : IsScalarTower F L' K
inst✝³ : Algebra E K
inst✝² : IsScalarTower F E K
inst✝¹ : IsScalarTower L E K
inst✝ : IsScalarTower L' E K
this :
  ((IsScalarTower.toAlgHom F E K).comp (IsScalarTower.toAlgHom F L E)).range.LinearDisjoint
    ((IsScalarTower.toAlgHom F E K).comp (IsScalarTower.toAlgHom F L' E)).range
x✝ : L
⊢ (IsScalarTower.toAlgHom F L K) x✝ = ((IsScalarTower.toAlgHom F E K).comp (IsScalarTower.toAlgHom F L E)) x✝

Tactic:
exact IsScalarTower.algebraMap_apply _ E K _

Goal state after:
Module: Mathlib.FieldTheory.LinearDisjoint

Goal state before:
pp: case e'_7
F : Type u
E : Type v
inst✝²⁰ : Field F
inst✝¹⁹ : Field E
inst✝¹⁸ : Algebra F E
L : Type w
inst✝¹⁷ : Field L
inst✝¹⁶ : Algebra F L
inst✝¹⁵ : Algebra L E
inst✝¹⁴ : IsScalarTower F L E
L' : Type u_1
inst✝¹³ : Field L'
inst✝¹² : Algebra F L'
inst✝¹¹ : Algebra L' E
inst✝¹⁰ : IsScalarTower F L' E
H : (IsScalarTower.toAlgHom F L E).fieldRange.LinearDisjoint (IsScalarTower.toAlgHom F L' E).range
K : Type u_2
inst✝⁹ : Field K
inst✝⁸ : Algebra F K
inst✝⁷ : Algebra L K
inst✝⁶ : IsScalarTower F L K
inst✝⁵ : Algebra L' K
inst✝⁴ : IsScalarTower F L' K
inst✝³ : Algebra E K
inst✝² : IsScalarTower F E K
inst✝¹ : IsScalarTower L E K
inst✝ : IsScalarTower L' E K
this :
  ((IsScalarTower.toAlgHom F E K).comp (IsScalarTower.toAlgHom F L E)).range.LinearDisjoint
    ((IsScalarTower.toAlgHom F E K).comp (IsScalarTower.toAlgHom F L' E)).range
x✝ : L'
⊢ (IsScalarTower.toAlgHom F L' K) x✝ = ((IsScalarTower.toAlgHom F E K).comp (IsScalarTower.toAlgHom F L' E)) x✝

Tactic:
exact IsScalarTower.algebraMap_apply _ E K _

Goal state after:
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✝⁴ : 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✝³ : IsDomain A
this✝² : IsDomain B
this✝¹ : 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
algInst✝¹ : Algebra R[X] A := ga.toAlgebra
algInst✝ : Algebra R[X] B := gb.toAlgebra
f : A ⊗[R] B →ₐ[R] A ⊗[R[X]] B := Algebra.TensorProduct.mapOfCompatibleSMul R[X] R R A B
this : Nontrivial (A ⊗[R[X]] B)
hf : Function.Injective ⇑f
⊢ R[a ⊗ₜ[R] 1] ≤ fa.range ⊓ fb.range

Tactic:
Algebra.adjoin_le_iff,

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✝⁴ : 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✝³ : IsDomain A
this✝² : IsDomain B
this✝¹ : 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
algInst✝¹ : Algebra R[X] A := ga.toAlgebra
algInst✝ : Algebra R[X] B := gb.toAlgebra
f : A ⊗[R] B →ₐ[R] A ⊗[R[X]] B := Algebra.TensorProduct.mapOfCompatibleSMul R[X] R R A B
this : Nontrivial (A ⊗[R[X]] B)
hf : Function.Injective ⇑f
⊢ {a ⊗ₜ[R] 1} ⊆ ↑(fa.range ⊓ fb.range)
Module: Mathlib.FieldTheory.PurelyInseparable.Exponent

Goal state before:
pp: F : Type u_1
K : Type u_2
L : Type u_3
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : HasExponent K L
p : ℕ
inst✝ : ExpChar K p
n : ℕ
hn : exponent K L ≤ n
⊢ (algebraMap K L) (iterateFrobeniusAux K L p n 0) = 0

Tactic:
algebraMap_iterateFrobeniusAux K p hn 0,

Goal state after:
F : Type u_1
K : Type u_2
L : Type u_3
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : HasExponent K L
p : ℕ
inst✝ : ExpChar K p
n : ℕ
hn : exponent K L ≤ n
⊢ 0 ^ p ^ n = 0
Module: Mathlib.FieldTheory.PurelyInseparable.PerfectClosure

Goal state before:
pp: F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
K : Type w
inst✝¹ : Field K
inst✝ : Algebra F K
i : E →ₐ[F] K
x : E
⊢ i x ∈ perfectClosure F K ↔ x ∈ perfectClosure F E

Tactic:
simp_rw [mem_perfectClosure_iff]
  refine ⟨fun ⟨n, y, h⟩ ↦ ⟨n, y, ?_⟩, fun ⟨n, y, h⟩ ↦ ⟨n, y, ?_⟩⟩
  · apply_fun i using i.injective
    rwa [AlgHom.commutes, map_pow]
  simpa only [AlgHom.commutes, map_pow] using congr_arg i h

Goal state after:
Module: Mathlib.FieldTheory.RatFunc.Degree

Goal state before:
pp: case pos
K : Type u
inst✝ : Field K
x : K⟮X⟯
hx : x = 0
⊢ x⁻¹.intDegree = -x.intDegree

Tactic:
simp [hx, eq_neg_iff_add_eq_zero, ← intDegree_mul (inv_ne_zero hx) hx]

Goal state after:
Module: Mathlib.FieldTheory.RatFunc.Degree

Goal state before:
pp: case neg
K : Type u
inst✝ : Field K
x : K⟮X⟯
hx : ¬x = 0
⊢ x⁻¹.intDegree = -x.intDegree

Tactic:
simp [hx, eq_neg_iff_add_eq_zero, ← intDegree_mul (inv_ne_zero hx) hx]

Goal state after: