mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.GroupTheory.GroupAction.MultipleTransitivity

Goal state before:
pp: k : ℕ
hrec :
  ∀ {G : Type u_1} [inst : Group G] {α : Type u_2} [inst_1 : MulAction G α] [Finite α],
    IsMultiplyPretransitive G α k →
      ∀ {s : Set α}, s.ncard = k → (fixingSubgroup G s).index * (Nat.card α - k)! = (Nat.card α)!
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : MulAction G α
inst✝ : Finite α
Hk : IsMultiplyPretransitive G α (k + 1)
s : Set α
hs : s.ncard = k + 1
⊢ IsMultiplyPretransitive G α 1

Tactic:
]

Goal state after:
k : ℕ
hrec :
  ∀ {G : Type u_1} [inst : Group G] {α : Type u_2} [inst_1 : MulAction G α] [Finite α],
    IsMultiplyPretransitive G α k →
      ∀ {s : Set α}, s.ncard = k → (fixingSubgroup G s).index * (Nat.card α - k)! = (Nat.card α)!
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : MulAction G α
inst✝ : Finite α
Hk : IsMultiplyPretransitive G α (k + 1)
s : Set α
hs : s.ncard = k + 1
⊢ IsMultiplyPretransitive G α 1
Module: Mathlib.GroupTheory.GroupAction.MultipleTransitivity

Goal state before:
pp: case hmn
k : ℕ
hrec :
  ∀ {G : Type u_1} [inst : Group G] {α : Type u_2} [inst_1 : MulAction G α] [Finite α],
    IsMultiplyPretransitive G α k →
      ∀ {s : Set α}, s.ncard = k → (fixingSubgroup G s).index * (Nat.card α - k)! = (Nat.card α)!
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : MulAction G α
inst✝ : Finite α
Hk : IsMultiplyPretransitive G α (k + 1)
s : Set α
hs : s.ncard = k + 1
⊢ 0 ≤ k

Tactic:
]

Goal state after:
case hmn
k : ℕ
hrec :
  ∀ {G : Type u_1} [inst : Group G] {α : Type u_2} [inst_1 : MulAction G α] [Finite α],
    IsMultiplyPretransitive G α k →
      ∀ {s : Set α}, s.ncard = k → (fixingSubgroup G s).index * (Nat.card α - k)! = (Nat.card α)!
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : MulAction G α
inst✝ : Finite α
Hk : IsMultiplyPretransitive G α (k + 1)
s : Set α
hs : s.ncard = k + 1
⊢ 0 ≤ k
Module: Mathlib.GroupTheory.GroupAction.MultipleTransitivity

Goal state before:
pp: case hα
k : ℕ
hrec :
  ∀ {G : Type u_1} [inst : Group G] {α : Type u_2} [inst_1 : MulAction G α] [Finite α],
    IsMultiplyPretransitive G α k →
      ∀ {s : Set α}, s.ncard = k → (fixingSubgroup G s).index * (Nat.card α - k)! = (Nat.card α)!
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : MulAction G α
inst✝ : Finite α
Hk : IsMultiplyPretransitive G α (k + 1)
s : Set α
hs : s.ncard = k + 1
⊢ s.ncard ≤ univ.ncard

Tactic:
]

Goal state after:
case hα
k : ℕ
hrec :
  ∀ {G : Type u_1} [inst : Group G] {α : Type u_2} [inst_1 : MulAction G α] [Finite α],
    IsMultiplyPretransitive G α k →
      ∀ {s : Set α}, s.ncard = k → (fixingSubgroup G s).index * (Nat.card α - k)! = (Nat.card α)!
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : MulAction G α
inst✝ : Finite α
Hk : IsMultiplyPretransitive G α (k + 1)
s : Set α
hs : s.ncard = k + 1
⊢ s.ncard ≤ univ.ncard
Module: Mathlib.GroupTheory.GroupAction.MultipleTransitivity

Goal state before:
pp: k : ℕ
hrec :
  ∀ {G : Type u_1} [inst : Group G] {α : Type u_2} [inst_1 : MulAction G α] [Finite α],
    IsMultiplyPretransitive G α k →
      ∀ {s : Set α}, s.ncard = k → (fixingSubgroup G s).index * (Nat.card α - k)! = (Nat.card α)!
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : MulAction G α
inst✝ : Finite α
Hk : IsMultiplyPretransitive G α (k + 1)
s : Set α
hs : s.ncard = k + 1
hGX : IsPretransitive G α
⊢ 0 < k + 1

Tactic:
]

Goal state after:
k : ℕ
hrec :
  ∀ {G : Type u_1} [inst : Group G] {α : Type u_2} [inst_1 : MulAction G α] [Finite α],
    IsMultiplyPretransitive G α k →
      ∀ {s : Set α}, s.ncard = k → (fixingSubgroup G s).index * (Nat.card α - k)! = (Nat.card α)!
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : MulAction G α
inst✝ : Finite α
Hk : IsMultiplyPretransitive G α (k + 1)
s : Set α
hs : s.ncard = k + 1
hGX : IsPretransitive G α
⊢ 0 < k + 1
Module: Mathlib.GroupTheory.GroupAction.MultipleTransitivity

Goal state before:
pp: case succ
k : ℕ
hrec :
  ∀ {G : Type u_1} [inst : Group G] {α : Type u_2} [inst_1 : MulAction G α] [Finite α],
    IsMultiplyPretransitive G α k →
      ∀ {s : Set α}, s.ncard = k → (fixingSubgroup G s).index * (Nat.card α - k)! = (Nat.card α)!
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : MulAction G α
inst✝ : Finite α
Hk : IsMultiplyPretransitive G α (k + 1)
s : Set α
hs : s.ncard = k + 1
hGX : IsPretransitive G α
a : α
has : a ∈ s
⊢ (fixingSubgroup G s).index * (Nat.card α - (k + 1))! = (Nat.card α)!

Tactic:
let t : Set (SubMulAction.ofStabilizer G a) := Subtype.val ⁻¹' s

Goal state after:
case succ
k : ℕ
hrec :
  ∀ {G : Type u_1} [inst : Group G] {α : Type u_2} [inst_1 : MulAction G α] [Finite α],
    IsMultiplyPretransitive G α k →
      ∀ {s : Set α}, s.ncard = k → (fixingSubgroup G s).index * (Nat.card α - k)! = (Nat.card α)!
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : MulAction G α
inst✝ : Finite α
Hk : IsMultiplyPretransitive G α (k + 1)
s : Set α
hs : s.ncard = k + 1
hGX : IsPretransitive G α
a : α
has : a ∈ s
t : Set ↥(ofStabilizer G a) := Subtype.val ⁻¹' s
⊢ (fixingSubgroup G s).index * (Nat.card α - (k + 1))! = (Nat.card α)!
Module: Mathlib.GroupTheory.GroupAction.MultipleTransitivity

Goal state before:
pp: k : ℕ
hrec :
  ∀ {G : Type u_1} [inst : Group G] {α : Type u_2} [inst_1 : MulAction G α] [Finite α],
    IsMultiplyPretransitive G α k →
      ∀ {s : Set α}, s.ncard = k → (fixingSubgroup G s).index * (Nat.card α - k)! = (Nat.card α)!
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : MulAction G α
inst✝ : Finite α
Hk : IsMultiplyPretransitive G α (k + 1)
s : Set α
hs : s.ncard = k + 1
hGX : IsPretransitive G α
a : α
has : a ∈ s
t : Set ↥(ofStabilizer G a) := Subtype.val ⁻¹' s
⊢ s ∩ range Subtype.val = s \ {a}

Tactic:
]

Goal state after:
k : ℕ
hrec :
  ∀ {G : Type u_1} [inst : Group G] {α : Type u_2} [inst_1 : MulAction G α] [Finite α],
    IsMultiplyPretransitive G α k →
      ∀ {s : Set α}, s.ncard = k → (fixingSubgroup G s).index * (Nat.card α - k)! = (Nat.card α)!
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : MulAction G α
inst✝ : Finite α
Hk : IsMultiplyPretransitive G α (k + 1)
s : Set α
hs : s.ncard = k + 1
hGX : IsPretransitive G α
a : α
has : a ∈ s
t : Set ↥(ofStabilizer G a) := Subtype.val ⁻¹' s
⊢ s ∩ range Subtype.val = s \ {a}
Module: Mathlib.GroupTheory.GroupAction.MultipleTransitivity

Goal state before:
pp: k : ℕ
hrec :
  ∀ {G : Type u_1} [inst : Group G] {α : Type u_2} [inst_1 : MulAction G α] [Finite α],
    IsMultiplyPretransitive G α k →
      ∀ {s : Set α}, s.ncard = k → (fixingSubgroup G s).index * (Nat.card α - k)! = (Nat.card α)!
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : MulAction G α
inst✝ : Finite α
Hk : IsMultiplyPretransitive G α (k + 1)
s : Set α
hs : s.ncard = k + 1
hGX : IsPretransitive G α
a : α
has : a ∈ s
t : Set ↥(ofStabilizer G a) := Subtype.val ⁻¹' s
⊢ {x | x ∈ ofStabilizer G a} ∩ s = {a}ᶜ ∩ s

Tactic:
]

Goal state after:
k : ℕ
hrec :
  ∀ {G : Type u_1} [inst : Group G] {α : Type u_2} [inst_1 : MulAction G α] [Finite α],
    IsMultiplyPretransitive G α k →
      ∀ {s : Set α}, s.ncard = k → (fixingSubgroup G s).index * (Nat.card α - k)! = (Nat.card α)!
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : MulAction G α
inst✝ : Finite α
Hk : IsMultiplyPretransitive G α (k + 1)
s : Set α
hs : s.ncard = k + 1
hGX : IsPretransitive G α
a : α
has : a ∈ s
t : Set ↥(ofStabilizer G a) := Subtype.val ⁻¹' s
⊢ {x | x ∈ ofStabilizer G a} ∩ s = {a}ᶜ ∩ s
Module: Mathlib.GroupTheory.GroupAction.MultipleTransitivity

Goal state before:
pp: k : ℕ
hrec :
  ∀ {G : Type u_1} [inst : Group G] {α : Type u_2} [inst_1 : MulAction G α] [Finite α],
    IsMultiplyPretransitive G α k →
      ∀ {s : Set α}, s.ncard = k → (fixingSubgroup G s).index * (Nat.card α - k)! = (Nat.card α)!
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : MulAction G α
inst✝ : Finite α
Hk : IsMultiplyPretransitive G α (k + 1)
s : Set α
hs : s.ncard = k + 1
hGX : IsPretransitive G α
a : α
has : a ∈ s
t : Set ↥(ofStabilizer G a) := Subtype.val ⁻¹' s
hat : Subtype.val '' t = s \ {a}
⊢ s = s

Tactic:
]

Goal state after:
Module: Mathlib.GroupTheory.GroupAction.MultipleTransitivity

Goal state before:
pp: case succ
k : ℕ
hrec :
  ∀ {G : Type u_1} [inst : Group G] {α : Type u_2} [inst_1 : MulAction G α] [Finite α],
    IsMultiplyPretransitive G α k →
      ∀ {s : Set α}, s.ncard = k → (fixingSubgroup G s).index * (Nat.card α - k)! = (Nat.card α)!
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : MulAction G α
inst✝ : Finite α
Hk : IsMultiplyPretransitive G α (k + 1)
s : Set α
hs : s.ncard = k + 1
hGX : IsPretransitive G α
a : α
has : a ∈ s
t : Set ↥(ofStabilizer G a) := Subtype.val ⁻¹' s
hat : Subtype.val '' t = s \ {a}
hat' : s = insert a (Subtype.val '' t)
hfs : fixingSubgroup G s = Subgroup.map (stabilizer G a).subtype (fixingSubgroup (↥(stabilizer G a)) t)
⊢ (fixingSubgroup G s).index * (Nat.card α - (k + 1))! = (Nat.card α)!

Tactic:
]

Goal state after:
case succ
k : ℕ
hrec :
  ∀ {G : Type u_1} [inst : Group G] {α : Type u_2} [inst_1 : MulAction G α] [Finite α],
    IsMultiplyPretransitive G α k →
      ∀ {s : Set α}, s.ncard = k → (fixingSubgroup G s).index * (Nat.card α - k)! = (Nat.card α)!
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : MulAction G α
inst✝ : Finite α
Hk : IsMultiplyPretransitive G α (k + 1)
s : Set α
hs : s.ncard = k + 1
hGX : IsPretransitive G α
a : α
has : a ∈ s
t : Set ↥(ofStabilizer G a) := Subtype.val ⁻¹' s
hat : Subtype.val '' t = s \ {a}
hat' : s = insert a (Subtype.val '' t)
hfs : fixingSubgroup G s = Subgroup.map (stabilizer G a).subtype (fixingSubgroup (↥(stabilizer G a)) t)
⊢ (fixingSubgroup G s).index * (Nat.card α - (k + 1))! = (Nat.card α)!
Module: Mathlib.GroupTheory.GroupAction.Jordan

Goal state before:
pp: G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
hsn' : 2 < ENat.card α
hG' : IsMultiplyPretransitive G α 2
a : α
⊢ IsMultiplyPretransitive G α 1

Tactic:
]

Goal state after:
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
hsn' : 2 < ENat.card α
hG' : IsMultiplyPretransitive G α 2
a : α
⊢ IsMultiplyPretransitive G α 1
Module: Mathlib.GroupTheory.GroupAction.Jordan

Goal state before:
pp: G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
hsn' : 2 < ENat.card α
hG' : IsMultiplyPretransitive G α 2
a : α
this : IsPretransitive G α
⊢ 1 < ENat.card α

Tactic:
]

Goal state after:
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
hsn' : 2 < ENat.card α
hG' : IsMultiplyPretransitive G α 2
a : α
this : IsPretransitive G α
⊢ 1 < ENat.card α
Module: Mathlib.GroupTheory.GroupAction.MultipleTransitivity

Goal state before:
pp: case succ
k : ℕ
hrec :
  ∀ {G : Type u_1} [inst : Group G] {α : Type u_2} [inst_1 : MulAction G α] [Finite α],
    IsMultiplyPretransitive G α k →
      ∀ {s : Set α}, s.ncard = k → (fixingSubgroup G s).index * (Nat.card α - k)! = (Nat.card α)!
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : MulAction G α
inst✝ : Finite α
Hk : IsMultiplyPretransitive G α (k + 1)
s : Set α
hs : s.ncard = k + 1
hGX : IsPretransitive G α
a : α
has : a ∈ s
t : Set ↥(ofStabilizer G a) := Subtype.val ⁻¹' s
hat : Subtype.val '' t = s \ {a}
hat' : s = insert a (Subtype.val '' t)
hfs : fixingSubgroup G s = Subgroup.map (stabilizer G a).subtype (fixingSubgroup (↥(stabilizer G a)) t)
⊢ (fixingSubgroup (↥(stabilizer G a)) t ⊔ ⊥).index * (stabilizer G a).subtype.range.index * (Nat.card α - (k + 1))! =
    (Nat.card α)!

Tactic:
]

Goal state after:
case succ
k : ℕ
hrec :
  ∀ {G : Type u_1} [inst : Group G] {α : Type u_2} [inst_1 : MulAction G α] [Finite α],
    IsMultiplyPretransitive G α k →
      ∀ {s : Set α}, s.ncard = k → (fixingSubgroup G s).index * (Nat.card α - k)! = (Nat.card α)!
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : MulAction G α
inst✝ : Finite α
Hk : IsMultiplyPretransitive G α (k + 1)
s : Set α
hs : s.ncard = k + 1
hGX : IsPretransitive G α
a : α
has : a ∈ s
t : Set ↥(ofStabilizer G a) := Subtype.val ⁻¹' s
hat : Subtype.val '' t = s \ {a}
hat' : s = insert a (Subtype.val '' t)
hfs : fixingSubgroup G s = Subgroup.map (stabilizer G a).subtype (fixingSubgroup (↥(stabilizer G a)) t)
⊢ (fixingSubgroup (↥(stabilizer G a)) t ⊔ ⊥).index * (stabilizer G a).subtype.range.index * (Nat.card α - (k + 1))! =
    (Nat.card α)!
Module: Mathlib.GroupTheory.GroupAction.Jordan

Goal state before:
pp: G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
hsn' : 2 < ENat.card α
hG' : IsMultiplyPretransitive G α 2
a : α
this✝ : IsPretransitive G α
this : Nontrivial α
⊢ IsPreprimitive G α

Tactic:
]

Goal state after:
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
hsn' : 2 < ENat.card α
hG' : IsMultiplyPretransitive G α 2
a : α
this✝ : IsPretransitive G α
this : Nontrivial α
⊢ IsPreprimitive G α
Module: Mathlib.GroupTheory.GroupAction.Jordan

Goal state before:
pp: G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
hsn' : 2 < ENat.card α
hG' : IsMultiplyPretransitive G α 2
a : α
this✝ : IsPretransitive G α
this : Nontrivial α
⊢ IsPreprimitive G α

Tactic:
exact isPreprimitive_of_is_two_pretransitive hG'

Goal state after:
Module: Mathlib.GroupTheory.GroupAction.Jordan

Goal state before:
pp: G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
hsn' : 2 < ENat.card α
hG' : IsMultiplyPretransitive G α 2
a : α
this✝ : IsPretransitive G α
this : Nontrivial α
hGa : IsCoatom (stabilizer G a)
hyp : normalClosure ↑(stabilizer G a) ≤ stabilizer G a
⊢ 1 < ENat.card ↥(ofStabilizer G a)

Tactic:
]

Goal state after:
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
hsn' : 2 < ENat.card α
hG' : IsMultiplyPretransitive G α 2
a : α
this✝ : IsPretransitive G α
this : Nontrivial α
hGa : IsCoatom (stabilizer G a)
hyp : normalClosure ↑(stabilizer G a) ≤ stabilizer G a
⊢ 1 < ENat.card ↥(ofStabilizer G a)
Module: Mathlib.GroupTheory.GroupAction.Jordan

Goal state before:
pp: case bc
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
hsn' : 2 < ENat.card α
hG' : IsMultiplyPretransitive G α 2
a : α
this✝ : IsPretransitive G α
this : Nontrivial α
hGa : IsCoatom (stabilizer G a)
hyp : normalClosure ↑(stabilizer G a) ≤ stabilizer G a
⊢ 1 + 1 < ENat.card α

Tactic:
]

Goal state after:
case bc
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
hsn' : 2 < ENat.card α
hG' : IsMultiplyPretransitive G α 2
a : α
this✝ : IsPretransitive G α
this : Nontrivial α
hGa : IsCoatom (stabilizer G a)
hyp : normalClosure ↑(stabilizer G a) ≤ stabilizer G a
⊢ 1 + 1 < ENat.card α
Module: Mathlib.GroupTheory.GroupAction.Jordan

Goal state before:
pp: case right
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
hsn' : 2 < ENat.card α
hG' : IsMultiplyPretransitive G α 2
a : α
this✝¹ : IsPretransitive G α
this✝ : Nontrivial α
hGa : IsCoatom (stabilizer G a)
hyp : normalClosure ↑(stabilizer G a) ≤ stabilizer G a
this : ∃ x y, x ≠ y
⊢ False

Tactic:
]

Goal state after:
case right
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
hsn' : 2 < ENat.card α
hG' : IsMultiplyPretransitive G α 2
a : α
this✝¹ : IsPretransitive G α
this✝ : Nontrivial α
hGa : IsCoatom (stabilizer G a)
hyp : normalClosure ↑(stabilizer G a) ≤ stabilizer G a
this : ∃ x y, x ≠ y
⊢ False
Module: Mathlib.GroupTheory.GroupAction.Jordan

Goal state before:
pp: G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
hsn' : 2 < ENat.card α
hG' : IsMultiplyPretransitive G α 2
a : α
this✝ : IsPretransitive G α
this : Nontrivial α
hGa : IsCoatom (stabilizer G a)
hyp : normalClosure ↑(stabilizer G a) ≤ stabilizer G a
b c : ↥(ofStabilizer G a)
hbc : b ≠ c
⊢ IsMultiplyPretransitive (↥(stabilizer G a)) (↥(ofStabilizer G a)) 1

Tactic:
]

Goal state after:
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
hsn' : 2 < ENat.card α
hG' : IsMultiplyPretransitive G α 2
a : α
this✝ : IsPretransitive G α
this : Nontrivial α
hGa : IsCoatom (stabilizer G a)
hyp : normalClosure ↑(stabilizer G a) ≤ stabilizer G a
b c : ↥(ofStabilizer G a)
hbc : b ≠ c
⊢ IsMultiplyPretransitive (↥(stabilizer G a)) (↥(ofStabilizer G a)) 1
Module: Mathlib.GroupTheory.GroupAction.Jordan

Goal state before:
pp: G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
hsn' : 2 < ENat.card α
hG' : IsMultiplyPretransitive G α 2
a : α
this✝ : IsPretransitive G α
this : Nontrivial α
hGa : IsCoatom (stabilizer G a)
hyp : normalClosure ↑(stabilizer G a) ≤ stabilizer G a
b c : ↥(ofStabilizer G a)
hbc : b ≠ c
⊢ IsMultiplyPretransitive G α (Nat.succ 1)

Tactic:
]

Goal state after:
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
hsn' : 2 < ENat.card α
hG' : IsMultiplyPretransitive G α 2
a : α
this✝ : IsPretransitive G α
this : Nontrivial α
hGa : IsCoatom (stabilizer G a)
hyp : normalClosure ↑(stabilizer G a) ≤ stabilizer G a
b c : ↥(ofStabilizer G a)
hbc : b ≠ c
⊢ IsMultiplyPretransitive G α (Nat.succ 1)
Module: Mathlib.GroupTheory.GroupAction.MultiplePrimitivity

Goal state before:
pp: case mp
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
H1 : IsMultiplyPreprimitive M α 1
⊢ IsPreprimitive ↥(fixingSubgroup M ∅) ↥(ofFixingSubgroup M ∅)

Tactic:
]

Goal state after:
case mp
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
H1 : IsMultiplyPreprimitive M α 1
⊢ IsPreprimitive ↥(fixingSubgroup M ∅) ↥(ofFixingSubgroup M ∅)
Module: Mathlib.GroupTheory.GroupAction.Jordan

Goal state before:
pp: case right
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
hsn' : 2 < ENat.card α
hG' : IsMultiplyPretransitive G α 2
a : α
this✝¹ : IsPretransitive G α
this✝ : Nontrivial α
hGa : IsCoatom (stabilizer G a)
hyp : normalClosure ↑(stabilizer G a) ≤ stabilizer G a
b c : ↥(ofStabilizer G a)
hbc : b ≠ c
this : IsPretransitive ↥(stabilizer G a) ↥(ofStabilizer G a)
g : G
hg : g ∈ stabilizer G a
hgbc : ↑(⟨g, hg⟩ • b) = ↑c
⊢ ↑b = ↑c

Tactic:
]

Goal state after:
case right
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
hsn' : 2 < ENat.card α
hG' : IsMultiplyPretransitive G α 2
a : α
this✝¹ : IsPretransitive G α
this✝ : Nontrivial α
hGa : IsCoatom (stabilizer G a)
hyp : normalClosure ↑(stabilizer G a) ≤ stabilizer G a
b c : ↥(ofStabilizer G a)
hbc : b ≠ c
this : IsPretransitive ↥(stabilizer G a) ↥(ofStabilizer G a)
g : G
hg : g ∈ stabilizer G a
hgbc : ↑(⟨g, hg⟩ • b) = ↑c
⊢ ↑b = ↑c
Module: Mathlib.GroupTheory.GroupAction.MultiplePrimitivity

Goal state before:
pp: case mpr
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
h : IsPreprimitive M α
⊢ IsMultiplyPretransitive M α 1 ∧
    ∀ {s : Set α}, s.encard + 1 = ↑1 → IsPreprimitive ↥(fixingSubgroup M s) ↥(ofFixingSubgroup M s)

Tactic:
]

Goal state after:
case mpr
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
h : IsPreprimitive M α
⊢ IsMultiplyPretransitive M α 1 ∧
    ∀ {s : Set α}, s.encard + 1 = ↑1 → IsPreprimitive ↥(fixingSubgroup M s) ↥(ofFixingSubgroup M s)
Module: Mathlib.GroupTheory.GroupAction.MultipleTransitivity

Goal state before:
pp: k : ℕ
hrec :
  ∀ {G : Type u_1} [inst : Group G] {α : Type u_2} [inst_1 : MulAction G α] [Finite α],
    IsMultiplyPretransitive G α k →
      ∀ {s : Set α}, s.ncard = k → (fixingSubgroup G s).index * (Nat.card α - k)! = (Nat.card α)!
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : MulAction G α
inst✝ : Finite α
Hk : IsMultiplyPretransitive G α (k + 1)
s : Set α
hs : s.ncard = k + 1
hGX : IsPretransitive G α
a : α
has : a ∈ s
t : Set ↥(ofStabilizer G a) := Subtype.val ⁻¹' s
hat : Subtype.val '' t = s \ {a}
hat' : s = insert a (Subtype.val '' t)
hfs : fixingSubgroup G s = Subgroup.map (stabilizer G a).subtype (fixingSubgroup (↥(stabilizer G a)) t)
⊢ (insert a (Subtype.val '' t)).ncard = t.ncard + 1

Tactic:
]

Goal state after:
k : ℕ
hrec :
  ∀ {G : Type u_1} [inst : Group G] {α : Type u_2} [inst_1 : MulAction G α] [Finite α],
    IsMultiplyPretransitive G α k →
      ∀ {s : Set α}, s.ncard = k → (fixingSubgroup G s).index * (Nat.card α - k)! = (Nat.card α)!
G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : MulAction G α
inst✝ : Finite α
Hk : IsMultiplyPretransitive G α (k + 1)
s : Set α
hs : s.ncard = k + 1
hGX : IsPretransitive G α
a : α
has : a ∈ s
t : Set ↥(ofStabilizer G a) := Subtype.val ⁻¹' s
hat : Subtype.val '' t = s \ {a}
hat' : s = insert a (Subtype.val '' t)
hfs : fixingSubgroup G s = Subgroup.map (stabilizer G a).subtype (fixingSubgroup (↥(stabilizer G a)) t)
⊢ (insert a (Subtype.val '' t)).ncard = t.ncard + 1
Module: Mathlib.GroupTheory.GroupAction.SubMulAction.OfFixingSubgroup

Goal state before:
pp: M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
s : Set α
x✝ : ↥(fixingSubgroup M s)
x : α
c : M
hc✝ : c ∈ fixingSubgroup M s
hc : c⁻¹ ∈ fixingSubgroup M s
⊢ x ∈ sᶜ → ⟨c, hc✝⟩ • x ∈ sᶜ

Tactic:
]

Goal state after:
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
s : Set α
x✝ : ↥(fixingSubgroup M s)
x : α
c : M
hc✝ : c ∈ fixingSubgroup M s
hc : c⁻¹ ∈ fixingSubgroup M s
⊢ x ∈ sᶜ → ⟨c, hc✝⟩ • x ∈ sᶜ
Module: Mathlib.GroupTheory.GroupAction.Jordan

Goal state before:
pp: case right
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
hsn' : 2 < ENat.card α
hG' : IsMultiplyPretransitive G α 2
a : α
this✝¹ : IsPretransitive G α
this✝ : Nontrivial α
hGa : IsCoatom (stabilizer G a)
hyp : normalClosure ↑(stabilizer G a) ≤ stabilizer G a
b c : ↥(ofStabilizer G a)
hbc : b ≠ c
this : IsPretransitive ↥(stabilizer G a) ↥(ofStabilizer G a)
g : G
hg : g ∈ stabilizer G a
hgbc : ↑(⟨g, hg⟩ • b) = ↑c
h : G
hinvab : h⁻¹ • a = ↑b
⊢ ↑b = ↑c

Tactic:
]

Goal state after:
case right
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
hsn' : 2 < ENat.card α
hG' : IsMultiplyPretransitive G α 2
a : α
this✝¹ : IsPretransitive G α
this✝ : Nontrivial α
hGa : IsCoatom (stabilizer G a)
hyp : normalClosure ↑(stabilizer G a) ≤ stabilizer G a
b c : ↥(ofStabilizer G a)
hbc : b ≠ c
this : IsPretransitive ↥(stabilizer G a) ↥(ofStabilizer G a)
g : G
hg : g ∈ stabilizer G a
hgbc : ↑(⟨g, hg⟩ • b) = ↑c
h : G
hinvab : h⁻¹ • a = ↑b
⊢ ↑b = ↑c
Module: Mathlib.GroupTheory.GroupAction.SubMulAction.OfFixingSubgroup

Goal state before:
pp: M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
s : Set α
x✝ : ↥(fixingSubgroup M s)
x : α
c : M
hc✝ : c ∈ fixingSubgroup M s
hc : c⁻¹ ∈ fixingSubgroup M s
hcx : ⟨c, hc✝⟩ • x ∈ s
⊢ 1 • x ∈ s

Tactic:
← inv_mul_cancel c,

Goal state after:
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
s : Set α
x✝ : ↥(fixingSubgroup M s)
x : α
c : M
hc✝ : c ∈ fixingSubgroup M s
hc : c⁻¹ ∈ fixingSubgroup M s
hcx : ⟨c, hc✝⟩ • x ∈ s
⊢ (c⁻¹ * c) • x ∈ s
Module: Mathlib.GroupTheory.GroupAction.MultiplePrimitivity

Goal state before:
pp: case inl
M : Type u_1
α : Type u_2
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : IsPretransitive M α
n : ℕ
a : α
inst✝ : IsMultiplyPreprimitive M α n.succ
h0 : n = 0
⊢ IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(ofStabilizer M a)) n

Tactic:
]

Goal state after:
case inl
M : Type u_1
α : Type u_2
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : IsPretransitive M α
n : ℕ
a : α
inst✝ : IsMultiplyPreprimitive M α n.succ
h0 : n = 0
⊢ IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(ofStabilizer M a)) n
Module: Mathlib.GroupTheory.GroupAction.SubMulAction.OfFixingSubgroup

Goal state before:
pp: M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
s : Set α
x✝ : ↥(fixingSubgroup M s)
x : α
c : M
hc✝ : c ∈ fixingSubgroup M s
hc : c⁻¹ ∈ fixingSubgroup M s
hcx : ⟨c, hc✝⟩ • x ∈ s
⊢ c • x ∈ s

Tactic:
]

Goal state after:
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
s : Set α
x✝ : ↥(fixingSubgroup M s)
x : α
c : M
hc✝ : c ∈ fixingSubgroup M s
hc : c⁻¹ ∈ fixingSubgroup M s
hcx : ⟨c, hc✝⟩ • x ∈ s
⊢ c • x ∈ s
Module: Mathlib.GroupTheory.GroupAction.MultiplePrimitivity

Goal state before:
pp: case inl
M : Type u_1
α : Type u_2
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : IsPretransitive M α
n : ℕ
a : α
inst✝ : IsMultiplyPreprimitive M α n.succ
h0 : n = 0
⊢ IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(ofStabilizer M a)) 0

Tactic:
]

Goal state after:
case inl
M : Type u_1
α : Type u_2
inst✝³ : Group M
inst✝² : MulAction M α
inst✝¹ : IsPretransitive M α
n : ℕ
a : α
inst✝ : IsMultiplyPreprimitive M α n.succ
h0 : n = 0
⊢ IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(ofStabilizer M a)) 0