mathlib-initiative/mathlib-tactics huggingface dataset:
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
this : IsPurelyInseparable (↥k) D → ∀ (x : D), ∃ n, x ^ p ^ n ∈ (algebraMap (↥k) D).range
x : D
hx : IsSeparable (↥k) x
⊢ ∃ a, ∃ (b : a ∈ k), (algebraMap (↥k) D) ⟨a, b⟩ = x

Tactic:
]

Goal state after:
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
this : IsPurelyInseparable (↥k) D → ∀ (x : D), ∃ n, x ^ p ^ n ∈ (algebraMap (↥k) D).range
x : D
hx : IsSeparable (↥k) x
⊢ ∃ a, ∃ (b : a ∈ k), (algebraMap (↥k) D) ⟨a, b⟩ = x
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
this : IsPurelyInseparable (↥k) D → ∀ (x : D), ∃ n, x ^ p ^ n ∈ (algebraMap (↥k) D).range
pure : IsPurelyInseparable (↥k) D
⊢ ∃ n, a ^ p ^ n ∈ k

Tactic:
obtain ⟨n, ⟨m, hm⟩⟩ := this pure a

Goal state after:
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
this : IsPurelyInseparable (↥k) D → ∀ (x : D), ∃ n, x ^ p ^ n ∈ (algebraMap (↥k) D).range
pure : IsPurelyInseparable (↥k) D
n : ℕ
m : ↥k
hm : (algebraMap (↥k) D) m = a ^ p ^ n
⊢ ∃ n, a ^ p ^ n ∈ k
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 : (⋃ 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)

Tactic:
]

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.FieldTheory.JacobsonNoether

Goal state before:
pp: D : Type u_1
inst✝² : DivisionRing D
inst✝¹ : Algebra.IsAlgebraic (↥k) D
p : ℕ
inst✝ : ExpChar D p
a : D
ha : a ∉ k
hinsep : ∀ (x : D), IsSeparable (↥k) x → x ∈ k
hn : a ∈ k
⊢ False

Tactic:
]

Goal state after:
D : Type u_1
inst✝² : DivisionRing D
inst✝¹ : Algebra.IsAlgebraic (↥k) D
p : ℕ
inst✝ : ExpChar D p
a : D
ha : a ∉ k
hinsep : ∀ (x : D), IsSeparable (↥k) x → x ∈ k
hn : a ∈ k
⊢ False
Module: Mathlib.FieldTheory.Laurent

Goal state before:
pp: R : Type u
inst✝ : CommRing R
r : R
p : R[X]
hp : p ∈ R[X]⁰
⊢ ∀ (x : R[X]), x * (taylor r) p = 0 → x = 0

Tactic:
]

Goal state after:
R : Type u
inst✝ : CommRing R
r : R
p : R[X]
hp : p ∈ R[X]⁰
⊢ ∀ (x : R[X]), x * (taylor r) p = 0 → x = 0
Module: Mathlib.FieldTheory.Laurent

Goal state before:
pp: R : Type u
inst✝ : CommRing R
r : R
p : R[X]
hp : p ∈ R[X]⁰
x : R[X]
hx : x = 0
this : x = (taylor (r - r)) x
⊢ x = 0

Tactic:
]

Goal state after:
R : Type u
inst✝ : CommRing R
r : R
p : R[X]
hp : p ∈ R[X]⁰
x : R[X]
hx : x = 0
this : x = (taylor (r - r)) x
⊢ x = 0
Module: Mathlib.FieldTheory.Laurent

Goal state before:
pp: R : Type u
inst✝¹ : CommRing R
r : R
p : R[X]
inst✝ : IsDomain R
⊢ (algebraMap R[X] R⟮X⟯) ((taylor r) p) / 1 = (algebraMap R[X] R⟮X⟯) ((taylor r) p) / 1

Tactic:
]

Goal state after:
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
ha : a ∉ k
hinsep : ∀ (x : D), IsSeparable (↥k) x → x ∈ k
m : ℕ
hm : 1 ≤ m ∧ a ^ p ^ m ∈ k
n : ℕ
hn : p ^ m ≤ n
x : D
⊢ a ^ p ^ m * x - a ^ p ^ m * x = 0

Tactic:
]

Goal state after:
D : Type u_1
inst✝¹ : DivisionRing D
inst✝ : Algebra.IsAlgebraic (↥k) D
p : ℕ
hchar : ExpChar D p
a : D
ha : a ∉ k
hinsep : ∀ (x : D), IsSeparable (↥k) x → x ∈ k
m : ℕ
hm : 1 ≤ m ∧ a ^ p ^ m ∈ k
n : ℕ
hn : p ^ m ≤ n
x : D
⊢ a ^ p ^ m * x - a ^ p ^ m * x = 0
Module: Mathlib.FieldTheory.Laurent

Goal state before:
pp: R : Type u
inst✝¹ : CommRing R
r : R
inst✝ : IsDomain R
⊢ (algebraMap R[X] R⟮X⟯) Polynomial.X + C r = (algebraMap R[X] R⟮X⟯) Polynomial.X + C r

Tactic:
]

Goal state after:
Module: Mathlib.FieldTheory.Laurent

Goal state before:
pp: R : Type u
inst✝¹ : CommRing R
r : R
inst✝ : IsDomain R
x : R
⊢ (algebraMap R[X] R⟮X⟯) (Polynomial.C x) = (algebraMap R[X] R⟮X⟯) (Polynomial.C x)

Tactic:
]

Goal state after:
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
ha : a ∉ k
hinsep : ∀ (x : D), IsSeparable (↥k) x → x ∈ k
m : ℕ
hm : 1 ≤ m ∧ a ^ p ^ m ∈ k
n : ℕ
hn : p ^ m ≤ n
inter : (⇑((ad (↥k) D) a))^[p ^ m] = 0
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Algebraic.LinearIndependent

Goal state before:
pp: F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
x : E
H : ∀ (p : F[X]), (aeval x) p = 0 → p = 0
⊢ LinearIndependent F fun a ↦ (x - (algebraMap F E) a)⁻¹

Tactic:
]

Goal state after:
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
x : E
H : ∀ (p : F[X]), (aeval x) p = 0 → p = 0
⊢ LinearIndependent F fun a ↦ (x - (algebraMap F E) a)⁻¹
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 : ∃ x, ¬x * a = a * x
ha₀ : a ≠ 0
⊢ ∃ b, ((ad (↥k) D) a) b ≠ 0

Tactic:
]

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 : ∃ x, ¬x * a = a * x
ha₀ : a ≠ 0
⊢ ∃ b, ((ad (↥k) D) a) b ≠ 0
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
m : ℕ
hm2 : ∀ (n : ℕ), p ^ m ≤ n → (⇑((ad (↥k) D) a))^[n] = 0
⊢ 0 = 0

Tactic:
]

Goal state after:
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
m : ℕ
hm2 : ∀ (n : ℕ), p ^ m ≤ n → (⇑((ad (↥k) D) a))^[n] = 0
⊢ (⇑((ad (↥k) D) a))^[p ^ m + 1] b = 0

Tactic:
by rw [hm2 (p ^ m + 1) (Nat.le_add_right _ _), Pi.zero_apply]

Goal state after:
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
m : ℕ
hm2 : ∀ (n : ℕ), p ^ m ≤ n → (⇑((ad (↥k) D) a))^[n] = 0
h_exist : ∃ n, 0 < n ∧ (⇑((ad (↥k) D) a))^[n + 1] b = 0
t : ℕ := Nat.find h_exist - 1
ht : t = Nat.find h_exist - 1
h_pos : t ≤ 0
h_find : Nat.find h_exist = 1
⊢ (⇑((ad (↥k) D) a))^[1] b ≠ 0

Tactic:
]

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
m : ℕ
hm2 : ∀ (n : ℕ), p ^ m ≤ n → (⇑((ad (↥k) D) a))^[n] = 0
h_exist : ∃ n, 0 < n ∧ (⇑((ad (↥k) D) a))^[n + 1] b = 0
t : ℕ := Nat.find h_exist - 1
ht : t = Nat.find h_exist - 1
h_pos : t ≤ 0
h_find : Nat.find h_exist = 1
⊢ (⇑((ad (↥k) D) a))^[1] b ≠ 0
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
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
hcovers : ∅ = Set.univ
hempty : IsEmpty κ
⊢ False

Tactic:
]

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
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
hcovers : ∅ = Set.univ
hempty : IsEmpty κ
⊢ False
Module: Mathlib.FieldTheory.JacobsonNoether

Goal state before:
pp: case neg
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
m : ℕ
hm2 : ∀ (n : ℕ), p ^ m ≤ n → (⇑((ad (↥k) D) a))^[n] = 0
h_exist : ∃ n, 0 < n ∧ (⇑((ad (↥k) D) a))^[n + 1] b = 0
t : ℕ := Nat.find h_exist - 1
ht : t = Nat.find h_exist - 1
h_pos : Nat.find h_exist ≤ 1
⊢ Nat.find h_exist = 1

Tactic:
]

Goal state after:
case neg
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
m : ℕ
hm2 : ∀ (n : ℕ), p ^ m ≤ n → (⇑((ad (↥k) D) a))^[n] = 0
h_exist : ∃ n, 0 < n ∧ (⇑((ad (↥k) D) a))^[n + 1] b = 0
t : ℕ := Nat.find h_exist - 1
ht : t = Nat.find h_exist - 1
h_pos : Nat.find h_exist ≤ 1
⊢ Nat.find h_exist = 1
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
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
hcovers : ⋃ k ∈ Finset.univ, f k • ↑(K k) = Set.univ
k : κ
hcovers' : ⋃ i ∈ {x | K x = K k}, f i • ↑(K i) = Set.univ
i : κ
hi : i ∈ {x | K x = K k}
⊢ f i • ↑(K k) = ?m.746 i hi

Tactic:
]

Goal state after:
Module: Mathlib.GroupTheory.CosetCover

Goal state before:
pp: case pos
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
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
hcovers : ⋃ k ∈ Finset.univ, f k • ↑(K k) = Set.univ
k : κ
hcovers' : ⋃ i ∈ {x | K x = K k}, f i • ↑(K k) = Set.univ
⊢ ∃ i ∈ s, H i ≠ H j ∧ (H i).FiniteIndex

Tactic:
]

Goal state after:
case pos
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
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
hcovers : ⋃ k ∈ Finset.univ, f k • ↑(K k) = Set.univ
k : κ
hcovers' : ⋃ i ∈ {x | K x = K k}, f i • ↑(K k) = Set.univ
⊢ ∃ i ∈ s, H i ≠ H j ∧ (H i).FiniteIndex
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 := ⋯ }
⊢ (mulLeft (↥k) a) c - (mulRight (↥k) a) c = 0

Tactic:
]

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 := ⋯ }
⊢ (mulLeft (↥k) a) c - (mulRight (↥k) a) c = 0
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 := ⋯ }
this : ((ad (↥k) D) a) c = 0
⊢ (mulLeft (↥k) a) c - (mulRight (↥k) a) c = ((ad (↥k) D) a) c

Tactic:
]

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 := ⋯ }
this : ((ad (↥k) D) a) c = 0
⊢ (mulLeft (↥k) a) c - (mulRight (↥k) a) c = ((ad (↥k) D) a) c
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
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 ↦
    match x 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
hcovers : ⋃ k ∈ Finset.univ, f k • ↑(K k) = Set.univ
k : κ
hcovers' : ¬⋃ i ∈ {x | K x = K k}, f i • ↑(K i) = Set.univ
x : Subgroup G
⊢ (∃ y, K y = x) → x ∈ (Finset.image H s).erase (H j)

Tactic:
]

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
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 ↦
    match x 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
hcovers : ⋃ k ∈ Finset.univ, f k • ↑(K k) = Set.univ
k : κ
hcovers' : ¬⋃ i ∈ {x | K x = K k}, f i • ↑(K i) = Set.univ
x : Subgroup G
⊢ (∃ y, K y = x) → x ∈ (Finset.image H s).erase (H j)
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 := ⋯ }
⊢ 0 = 0

Tactic:
]

Goal state after:
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 : C x ∈ I
⊢ x = 0

Tactic:
]

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.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 : C x ∈ I
hx0 : ¬x = 0
this : 1 ∈ I
⊢ 1 ∈ I

Tactic:
]

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
hx0 : ¬x = 0
this : 1 ∈ I
⊢ 1 ∈ I
Module: Mathlib.RingTheory.MvPolynomial

Goal state before:
pp: σ : Type u
K : Type v
inst✝¹ : CommRing K
inst✝ : Nontrivial K
⊢ lift.{v, u} #(σ →₀ ℕ) = lift.{v, u} #(σ →₀ ℕ)

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.MvPolynomial

Goal state before:
pp: K : Type v
inst✝¹ : CommRing K
inst✝ : Nontrivial K
σ : Type v
⊢ lift.{v, v} #(σ →₀ ℕ) = lift.{v, v} #(σ →₀ ℕ)

Tactic:
]

Goal state after:
Module: Mathlib.FieldTheory.MvRatFunc.Rank

Goal state before:
pp: case refine_1
σ : Type u
F : Type v
inst✝¹ : Field F
inst✝ : Nonempty σ
R : Type (max v u) := MvPolynomial σ F
K : Type (max u v) := FractionRing R
⊢ max (max (lift.{u, v} #F) (lift.{v, u} #σ)) ℵ₀ ≤ max (max (lift.{u, v} #F) (lift.{v, u} #σ)) ℵ₀

Tactic:
]

Goal state after:
Module: Mathlib.FieldTheory.MvRatFunc.Rank

Goal state before:
pp: case refine_2
σ : Type u
F : Type v
inst✝¹ : Field F
inst✝ : Nonempty σ
R : Type (max v u) := MvPolynomial σ F
K : Type (max u v) := FractionRing R
hinj : Function.Injective ⇑(algebraMap R K)
h1 : lift.{v, u} #σ ≤ Module.rank F K ∧ ℵ₀ ≤ Module.rank F K
⊢ max (max (lift.{u, v} #F) (lift.{v, u} #σ)) ℵ₀ ≤ Module.rank F (FractionRing (MvPolynomial σ F))

Tactic:
]

Goal state after:
case refine_2
σ : Type u
F : Type v
inst✝¹ : Field F
inst✝ : Nonempty σ
R : Type (max v u) := MvPolynomial σ F
K : Type (max u v) := FractionRing R
hinj : Function.Injective ⇑(algebraMap R K)
h1 : lift.{v, u} #σ ≤ Module.rank F K ∧ ℵ₀ ≤ Module.rank F K
⊢ max (max (lift.{u, v} #F) (lift.{v, u} #σ)) ℵ₀ ≤ Module.rank F (FractionRing (MvPolynomial σ F))
Module: Mathlib.FieldTheory.MvRatFunc.Rank

Goal state before:
pp: case refine_2
σ : Type u
F : Type v
inst✝¹ : Field F
inst✝ : Nonempty σ
R : Type (max v u) := MvPolynomial σ F
K : Type (max u v) := FractionRing R
hinj : Function.Injective ⇑(algebraMap R K)
h1 : lift.{v, u} #σ ≤ Module.rank F K ∧ ℵ₀ ≤ Module.rank F K
i : σ
hx : Transcendental F ((algebraMap R K) (MvPolynomial.X i))
h2 : lift.{u, v} #F ≤ Module.rank F K
⊢ max (max (lift.{u, v} #F) (lift.{v, u} #σ)) ℵ₀ ≤ Module.rank F (FractionRing (MvPolynomial σ F))

Tactic:
]

Goal state after:
case refine_2
σ : Type u
F : Type v
inst✝¹ : Field F
inst✝ : Nonempty σ
R : Type (max v u) := MvPolynomial σ F
K : Type (max u v) := FractionRing R
hinj : Function.Injective ⇑(algebraMap R K)
h1 : lift.{v, u} #σ ≤ Module.rank F K ∧ ℵ₀ ≤ Module.rank F K
i : σ
hx : Transcendental F ((algebraMap R K) (MvPolynomial.X i))
h2 : lift.{u, v} #F ≤ Module.rank F K
⊢ max (max (lift.{u, v} #F) (lift.{v, u} #σ)) ℵ₀ ≤ Module.rank F (FractionRing (MvPolynomial σ F))