mathlib-initiative/mathlib-tactics huggingface dataset:
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
⊢ Nontrivial ↥(ofStabilizer G a)

Tactic:
by

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
b c : ↥(ofStabilizer G a)
hbc : b ≠ c
⊢ IsPretransitive ↥(stabilizer G a) ↥(ofStabilizer G a)

Tactic:
by

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

Goal state before:
pp: M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
s✝ : Set α
hs : s✝.encard + 1 = ↑0
⊢ IsPreprimitive ↥(fixingSubgroup M s✝) ↥(ofFixingSubgroup M s✝)

Tactic:
by

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

Goal state before:
pp: M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
H1 : IsMultiplyPreprimitive M α 1
⊢ ∅.encard + 1 = ↑1

Tactic:
by

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

Goal state before:
pp: M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
⊢ IsMultiplyPreprimitive M α 1 ↔ IsPreprimitive M α

Tactic:
by

Goal state after:
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
⊢ x ∈ sᶜ → ⟨c, hc⟩ • x ∈ sᶜ

Tactic:
by

Goal state after:
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
t : Set ↥(ofFixingSubgroup M s)
⊢ Disjoint s (Subtype.val '' t)

Tactic:
by

Goal state after:
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
⊢ Injective ⇑(ofFixingSubgroup_equivariantMap M s)

Tactic:
by

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
⊢ normalClosure ↑(stabilizer G a) = ⊤

Tactic:
by

Goal state after:
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
⊢ Bijective ⇑(ofFixingSubgroup_equivariantMap M ∅)

Tactic:
by

Goal state after:
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
g : M
⊢ g ∈ fixingSubgroup M ∅

Tactic:
by

Goal state after:
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
a : α
s : Set α
m : M
⊢ m ∈ fixingSubgroup M (insert a s) ↔ m • a = a ∧ m ∈ fixingSubgroup M s

Tactic:
by

Goal state after:
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
a : α
s : Set ↥(ofStabilizer M a)
⊢ fixingSubgroup M (insert a ((fun x ↦ ↑x) '' s)) =
    Subgroup.map (stabilizer M a).subtype (fixingSubgroup (↥(stabilizer M a)) s)

Tactic:
by

Goal state after:
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
a : α
s : Set ↥(ofStabilizer M a)
x : α
⊢ x ∈ ofFixingSubgroup M (insert a ((fun x ↦ ↑x) '' s)) ↔
    ∃ (hx : x ∈ ofStabilizer M a), ⟨x, hx⟩ ∈ ofFixingSubgroup (↥(stabilizer M a)) s

Tactic:
by

Goal state after:
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
a : α
s : Set ↥(ofStabilizer M a)
m : ↥(fixingSubgroup M (insert a (Subtype.val '' s)))
x✝ : ↑s
x : ↥(ofStabilizer M a)
hx : x ∈ s
⊢ ⟨↑m, ⋯⟩ • ↑⟨x, hx⟩ = ↑⟨x, hx⟩

Tactic:
by

Goal state after:
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
a : α
s : Set ↥(ofStabilizer M a)
m : ↥(fixingSubgroup (↥(stabilizer M a)) s)
⊢ ↑↑m ∈ fixingSubgroup M (insert a (Subtype.val '' s))

Tactic:
by

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

Goal state before:
pp: n : ℕ
hrec :
  ∀ m < n,
    ∀ {G : Type u_1} {α : Type u_2} [inst : Group G] [inst_1 : MulAction G α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          s.ncard = m + 1 →
            m + 2 < Nat.card α →
              (IsPretransitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2) ∧
                (IsPreprimitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s) → IsMultiplyPreprimitive G α 2)
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
hG : IsPreprimitive G α
s : Set α
hsn : s.ncard = n + 1
hsn' : n + 2 < Nat.card α
this : Finite α
⊢ s ≠ Set.univ

Tactic:
by

Goal state after:
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
a : α
s : Set ↥(ofStabilizer M a)
x✝ : ↥(fixingSubgroup M (insert a (Subtype.val '' s)))
⊢ (fun m ↦ ⟨↑↑m, ⋯⟩) ((fun m ↦ ⟨⟨↑m, ⋯⟩, ⋯⟩) x✝) = x✝

Tactic:
by

Goal state after:
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)
this : a ∉ Subtype.val '' t
⊢ (insert a (Subtype.val '' t)).ncard = t.ncard + 1

Tactic:
by

Goal state after:
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
a : α
s : Set ↥(ofStabilizer M a)
x✝ : ↥(fixingSubgroup (↥(stabilizer M a)) s)
⊢ (fun m ↦ ⟨⟨↑m, ⋯⟩, ⋯⟩) ((fun m ↦ ⟨↑↑m, ⋯⟩) x✝) = x✝

Tactic:
by

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

Goal state before:
pp: 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
h1 : n ≥ 1
s : Set ↥(ofStabilizer M a)
hs : s.encard + 1 = ↑n
⊢ IsPreprimitive ↥(fixingSubgroup M (insert a (Subtype.val '' s))) ↥(ofFixingSubgroup M (insert a (Subtype.val '' s)))

Tactic:
by

Goal state after:
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
a : α
s : Set ↥(ofStabilizer M a)
x✝¹ x✝ : ↥(fixingSubgroup M (insert a (Subtype.val '' s)))
⊢ ⟨⟨↑(x✝¹ * x✝), ⋯⟩, ⋯⟩ = ⟨⟨↑x✝¹, ⋯⟩, ⋯⟩ * ⟨⟨↑x✝, ⋯⟩, ⋯⟩

Tactic:
by

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

Goal state before:
pp: n : ℕ
hrec :
  ∀ m < n,
    ∀ {G : Type u_1} {α : Type u_2} [inst : Group G] [inst_1 : MulAction G α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          s.ncard = m + 1 →
            m + 2 < Nat.card α →
              (IsPretransitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2) ∧
                (IsPreprimitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s) → IsMultiplyPreprimitive G α 2)
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
hG : IsPreprimitive G α
s : Set α
hsn : s.ncard = n + 1
hsn' : n + 2 < Nat.card α
this : Finite α
hs_ne_univ : s ≠ Set.univ
⊢ s.Nonempty

Tactic:
by

Goal state after:
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
a : α
s : Set ↥(ofStabilizer M a)
x : ↥(ofFixingSubgroup M (insert a (Subtype.val '' s)))
⊢ ↥(ofFixingSubgroup (↥(stabilizer M a)) s)

Tactic:
by

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

Goal state before:
pp: 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
⊢ IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(ofStabilizer M a)) n

Tactic:
by

Goal state after:
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
a : α
s : Set ↥(ofStabilizer M a)
⊢ Bijective ⇑(ofFixingSubgroup_insert_map a s)

Tactic:
by

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

Goal state before:
pp: n : ℕ
hrec :
  ∀ m < n,
    ∀ {G : Type u_1} {α : Type u_2} [inst : Group G] [inst_1 : MulAction G α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          s.ncard = m + 1 →
            m + 2 < Nat.card α →
              (IsPretransitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2) ∧
                (IsPreprimitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s) → IsMultiplyPreprimitive G α 2)
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
hG : IsPreprimitive G α
s : Set α
hsn : s.ncard = n + 1
hsn' : n + 2 < Nat.card α
this : Finite α
hs_ne_univ : s ≠ Set.univ
hs_nonempty : s.Nonempty
hn : n + 1 < 2
⊢ n = 0

Tactic:
by

Goal state after:
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)
⊢ t.ncard = k

Tactic:
by

Goal state after:
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 t : Set α
g : M
hg : g • t = s
k : M
hk : k ∈ fixingSubgroup M t
⊢ (MulAut.conj g) k ∈ fixingSubgroup M s

Tactic:
by

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

Goal state before:
pp: M : Type u_1
α : Type u_2
inst✝² : Group M
inst✝¹ : MulAction M α
inst✝ : IsPretransitive M α
n : ℕ
hn : 1 ≤ n
a : α
H : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(ofStabilizer M a)) n
s : Set α
hs : s.encard + 1 = ↑n.succ
⊢ ∃ b, b ∈ s

Tactic:
by

Goal state after:
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 t : Set α
g : M
hg : g • t = s
⊢ Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) (fixingSubgroup M t) = fixingSubgroup M s

Tactic:
by

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

Goal state before:
pp: n : ℕ
hrec :
  ∀ m < n,
    ∀ {G : Type u_1} {α : Type u_2} [inst : Group G] [inst_1 : MulAction G α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          s.ncard = m + 1 →
            m + 2 < Nat.card α →
              (IsPretransitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2) ∧
                (IsPreprimitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s) → IsMultiplyPreprimitive G α 2)
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
hG : IsPreprimitive G α
s : Set α
hsn' : n + 2 < Nat.card α
this✝ : Finite α
hs_ne_univ : s ≠ Set.univ
hs_nonempty : s.Nonempty
hn✝ : n + 1 < 2
hn : n = 0
a : α
hsa : s = {a}
this : IsPretransitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2
⊢ (IsPretransitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2) ∧
    (IsPreprimitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s) → IsMultiplyPreprimitive G α 2)

Tactic:
by

Goal state after:
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✝ s t : Set α
g : M
hg : g • t = s
x✝ : ↥(ofFixingSubgroup M t)
x : α
hx : x ∈ ofFixingSubgroup M t
⊢ g • x ∈ ofFixingSubgroup M s

Tactic:
intro hgxt; apply hx
      rw [← hg] at hgxt
      exact Set.smul_mem_smul_set_iff.mp hgxt

Goal state after:
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✝ s t : Set α
g : M
hg : g • t = s
x✝ : ↥(ofFixingSubgroup M t)
x : α
hx : x ∈ ofFixingSubgroup M t
⊢ g • x ∈ ofFixingSubgroup M s

Tactic:
by

Goal state after:
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✝ s t : Set α
g : M
hg : g • t = s
x✝¹ : ↥(fixingSubgroup M t)
x✝ : ↥(ofFixingSubgroup M t)
m : M
hm : m ∈ fixingSubgroup M t
x : α
hx : x ∈ ofFixingSubgroup M t
⊢ (match ⟨m, hm⟩ • ⟨x, hx⟩ with
    | ⟨x, hx⟩ => ⟨g • x, ⋯⟩) =
    (fixingSubgroupEquivFixingSubgroup hg) ⟨m, hm⟩ •
      match ⟨x, hx⟩ with
      | ⟨x, hx⟩ => ⟨g • x, ⋯⟩

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.GroupAction.SubMulAction.OfFixingSubgroup

Goal state before:
pp: case left
M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
s t : Set α
g : M
hst : g • s = t
x y : ↥(ofFixingSubgroup M s)
hxy : (conjMap_ofFixingSubgroup hst) x = (conjMap_ofFixingSubgroup hst) y
⊢ x = y

Tactic:
simpa [← SetLike.coe_eq_coe] using hxy

Goal state after:
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)
htcard : t.ncard = k
this : (fixingSubgroup (↥(stabilizer G a)) t).index * (Nat.card α - 1 - k)! = (Nat.card α - 1)!
⊢ (fixingSubgroup (↥(stabilizer G a)) t).index * (stabilizer G a).index * (Nat.card α - (k + 1))! = (Nat.card α)!

Tactic:
by

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

Goal state before:
pp: M : Type u_1
α : Type u_2
inst✝² : Group M
inst✝¹ : MulAction M α
inst✝ : IsPretransitive M α
n : ℕ
hn : 1 ≤ n
a : α
H : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(ofStabilizer M a)) n
s : Set α
hs : s.encard + 1 = ↑n.succ
b : α
hb : b ∈ s
g : M
hg : g • b = a
s' : Set α := g • s
hs' : s' = g • s
t : Set ↥(ofStabilizer M a) := Subtype.val ⁻¹' s'
x : α
hxs : x ∈ s'
hxa : ¬x = a
⊢ ⟨x, hxa⟩ ∈ t

Tactic:
by

Goal state after:
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 t : Set α
g : M
hst : g • s = t
⊢ Bijective ⇑(conjMap_ofFixingSubgroup hst)

Tactic:
by

Goal state after:
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 t : Set α
g : M
⊢ g ∈ fixingSubgroup M (s ∪ t) ↔ g ∈ fixingSubgroup M s ∧ g ∈ fixingSubgroup M t

Tactic:
by

Goal state after:
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✝ s t : Set α
m : ↥(fixingSubgroup M (s ∪ t))
⊢ ⟨↑m, ⋯⟩ ∈ fixingSubgroup (↥(fixingSubgroup M s)) (Subtype.val ⁻¹' t)

Tactic:
by

Goal state after:
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✝ s t : Set α
⊢ ⟨⟨↑1, ⋯⟩, ⋯⟩ = 1

Tactic:
by

Goal state after:
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✝ s t : Set α
x✝¹ x✝ : ↥(fixingSubgroup M (s ∪ t))
⊢ ⟨⟨↑(x✝¹ * x✝), ⋯⟩, ⋯⟩ = ⟨⟨↑x✝¹, ⋯⟩, ⋯⟩ * ⟨⟨↑x✝, ⋯⟩, ⋯⟩

Tactic:
by

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

Goal state before:
pp: case mpr.inr
M : Type u_1
α : Type u_2
inst✝² : Group M
inst✝¹ : MulAction M α
inst✝ : IsPretransitive M α
n : ℕ
hn : 1 ≤ n
a : α
H : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(ofStabilizer M a)) n
s : Set α
hs : s.encard + 1 = ↑n.succ
b : α
hb : b ∈ s
g : M
hg : g • b = a
s' : Set α := g • s
hs' : s' = g • s
t : Set ↥(ofStabilizer M a) := Subtype.val ⁻¹' s'
y : ↥(ofStabilizer M a)
hy : y ∈ t
⊢ ↑y ∈ s'

Tactic:
simpa only using! hy

Goal state after:
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✝ s t : Set α
m : ↥(fixingSubgroup M (s ∪ t))
⊢ ↑m ∈ fixingSubgroup M s

Tactic:
by

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

Goal state before:
pp: M : Type u_1
α : Type u_2
inst✝² : Group M
inst✝¹ : MulAction M α
inst✝ : IsPretransitive M α
n : ℕ
hn : 1 ≤ n
a : α
H : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(ofStabilizer M a)) n
s : Set α
hs : s.encard + 1 = ↑n.succ
b : α
hb : b ∈ s
g : M
hg : g • b = a
s' : Set α := g • s
hs' : s' = g • s
t : Set ↥(ofStabilizer M a) := Subtype.val ⁻¹' s'
⊢ s' = insert a (Subtype.val '' t)

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : MulAction G α
inst✝ : Finite α
k : ℕ
Hk : IsMultiplyPretransitive G α k
s : Set α
hs : s.ncard = k
⊢ (fixingSubgroup G s).index * (Nat.card α - k)! = (Nat.card α)!

Tactic:
by

Goal state after:
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✝ s t : Set α
x : ↥(ofFixingSubgroup M (s ∪ t))
hx : ⟨↑x, ⋯⟩ ∈ Subtype.val ⁻¹' t
⊢ ↑x ∈ s ∪ t

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝² : Group G
α : Type u_2
inst✝¹ : MulAction G α
inst✝ : Finite α
s : Set α
hMk : IsMultiplyPretransitive G α s.ncard
⊢ (fixingSubgroup G s).index = (Nat.card α).choose s.ncard * s.ncard !

Tactic:
by

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

Goal state before:
pp: ι : Type u_2
inst✝ : Finite ι
β : Type u_3
x y : ι ↪ β
σ : Perm β
hσ : ∀ (a : ι), σ (x a) = y a
i : ι
⊢ (σ • x) i = y i

Tactic:
by

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

Goal state before:
pp: ι : Type u_2
inst✝ : Finite ι
β : Type u_3
x y : ι ↪ β
⊢ ∃ σ, σ • x = y

Tactic:
by

Goal state after:
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✝ s t : Set α
x✝¹ : ↥(fixingSubgroup M (s ∪ t))
x✝ : ↥(ofFixingSubgroup M (s ∪ t))
m : M
hm : m ∈ fixingSubgroup M (s ∪ t)
x : α
hx : x ∈ ofFixingSubgroup M (s ∪ t)
⊢ ⟨⟨↑(⟨m, hm⟩ • ⟨x, hx⟩), ⋯⟩, ⋯⟩ = ⟨⟨↑⟨m, hm⟩, ⋯⟩, ⋯⟩ • ⟨⟨↑⟨x, hx⟩, ⋯⟩, ⋯⟩

Tactic:
by

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

Goal state before:
pp: α : Type u_1
n : ℕ
⊢ IsMultiplyPretransitive (Perm α) α n

Tactic:
by

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

Goal state before:
pp: case mpr.right
M : Type u_1
α : Type u_2
inst✝² : Group M
inst✝¹ : MulAction M α
inst✝ : IsPretransitive M α
n : ℕ
hn : 1 ≤ n
a : α
H : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(ofStabilizer M a)) n
s : Set α
hs : s.encard + 1 = ↑n.succ
b : α
hb : b ∈ s
g : M
hg : g • b = a
s' : Set α := ⋯
hs' : s' = g • s
t : Set ↥(ofStabilizer M a) := ⋯
hst : s' = insert a (Subtype.val '' t)
⊢ (fun x ↦ x + 1) (t.encard + 1) = (fun x ↦ x + 1) ↑n

Tactic:
simp only

Goal state after:
case mpr.right
M : Type u_1
α : Type u_2
inst✝² : Group M
inst✝¹ : MulAction M α
inst✝ : IsPretransitive M α
n : ℕ
hn : 1 ≤ n
a : α
H : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(ofStabilizer M a)) n
s : Set α
hs : s.encard + 1 = ↑n.succ
b : α
hb : b ∈ s
g : M
hg : g • b = a
s' : Set α := g • s
hs' : s' = g • s
t : Set ↥(ofStabilizer M a) := Subtype.val ⁻¹' s'
hst : s' = insert a (Subtype.val '' t)
⊢ t.encard + 1 + 1 = ↑n + 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 t : Set α
a : α
ha : a ∈ ofFixingSubgroup M s
ha' : ⟨a, ha⟩ ∈ ofFixingSubgroup (↥(fixingSubgroup M s)) (Subtype.val ⁻¹' t)
this : a ∈ ofFixingSubgroup M (s ∪ t)
⊢ ∃ a_1, (map_ofFixingSubgroupUnion M s t) a_1 = ⟨⟨a, ha⟩, ha'⟩

Tactic:
by

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

Goal state before:
pp: n : ℕ
hrec :
  ∀ m < n,
    ∀ {G : Type u_1} {α : Type u_2} [inst : Group G] [inst_1 : MulAction G α],
      IsPreprimitive G α →
        ∀ {s : Set α},
          s.ncard = m + 1 →
            m + 2 < Nat.card α →
              (IsPretransitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s) → IsMultiplyPretransitive G α 2) ∧
                (IsPreprimitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s) → IsMultiplyPreprimitive G α 2)
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
hG : IsPreprimitive G α
s : Set α
hsn : s.ncard = n + 1
hsn' : n + 2 < Nat.card α
this : Finite α
hs_ne_univ : s ≠ Set.univ
hs_nonempty : s.Nonempty
hn : n + 1 ≥ 2
hn1 : 2 * (n + 1) < Nat.card α
⊢ 1 < s.ncard

Tactic:
by

Goal state after:
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 t : Set α
⊢ Bijective ⇑(map_ofFixingSubgroupUnion M s t)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝ : Finite α
G : Subgroup (Perm α)
this✝ : Fintype α
hmt : IsMultiplyPretransitive (↥G) α (Fintype.card α - 1)
j : Fin (Fintype.card α - 1) ↪ Fin (Fintype.card α) := castLEEmb ⋯
k : Perm α
a✝ : k ∈ ⊤
x : Fin (Fintype.card α) ↪ α := (Fintype.equivFinOfCardEq ⋯).symm.toEmbedding
x' : Fin (Fintype.card α - 1) ↪ α := j.trans x
g : ↥G
hg' : g • x' = k • x'
this : k = ↑g
⊢ k ∈ G

Tactic:
by

Goal state after:
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 t : Set α
hst : t ⊆ s
⊢ Injective ⇑(ofFixingSubgroup_of_inclusion M hst)

Tactic:
by

Goal state after: