mathlib-initiative/mathlib-tactics huggingface dataset:
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
N : Type u_3
β : Type u_4
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
hφ : Function.Surjective φ
f : α →ₑ[φ] β
hf : Function.Bijective ⇑f
n : ℕ
H : IsMultiplyPreprimitive N β n
s : Set α
hs : s.encard + 1 = ↑n
t : Set β := ⇑f '' s
x✝ : ↥(fixingSubgroup M s)
g : M
hg : ∀ y ∈ s, g • y = y
y : β
hy : y ∈ t
this : ∃ x ∈ s, y = f x
⊢ φ g • y = 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✝² : Group M
inst✝¹ : MulAction M α
s : Set α
inst✝ : Finite α
hs : IsPreprimitive ↥(fixingSubgroup M s) ↥(ofFixingSubgroup M s)
g : M
ha : s ∪ g • s ≠ ⊤
⊢ IsPreprimitive ↥(fixingSubgroup M (s ∩ g • s)) ↥(ofFixingSubgroup M (s ∩ g • 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
hn2 : 2 * (n + 1) ≥ Nat.card α
a : α
ha : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
⊢ a ∉ s ∪ g • 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
N : Type u_3
β : Type u_4
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
hφ : Function.Surjective φ
f : α →ₑ[φ] β
hf : Function.Bijective ⇑f
n : ℕ
H : IsMultiplyPreprimitive N β n
s : Set α
hs : s.encard + 1 = ↑n
t : Set β := ⇑f '' s
x✝ : ↥(fixingSubgroup M s)
g : M
hg : g ∈ fixingSubgroup M s
⊢ φ g ∈ fixingSubgroup N t

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
α : Type u_2
inst✝ : MulAction G α
s : Set α
⊢ fixingSubgroup G s ≤ stabilizer G 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
N : Type u_3
β : Type u_4
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
hφ : Function.Surjective φ
f : α →ₑ[φ] β
hf : Function.Bijective ⇑f
n : ℕ
H : IsMultiplyPreprimitive N β n
s : Set α
hs : s.encard + 1 = ↑n
t : Set β := ⇑f '' s
ψ : ↥(fixingSubgroup M s) → ↥(fixingSubgroup N t) :=
  fun x ↦
    match x with
    | ⟨g, hg⟩ => ⟨φ g, ⋯⟩
x : ↥(ofFixingSubgroup M s)
⊢ f ↑x ∈ ofFixingSubgroup N t

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
N : Type u_3
β : Type u_4
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
hφ : Function.Surjective φ
f : α →ₑ[φ] β
hf : Function.Bijective ⇑f
n : ℕ
H : IsMultiplyPreprimitive N β n
s : Set α
hs : s.encard + 1 = ↑n
t : Set β := ⇑f '' s
ψ : ↥(fixingSubgroup M s) → ↥(fixingSubgroup N t) :=
  fun x ↦
    match x with
    | ⟨g, hg⟩ => ⟨φ g, ⋯⟩
m : ↥(fixingSubgroup M s)
x : ↥(ofFixingSubgroup M s)
⊢ ⟨f ↑(m • x), ⋯⟩ = ψ m • ⟨f ↑x, ⋯⟩

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝² : Group G
A : Type u_2
inst✝¹ : MulAction G A
H : Subgroup G
inst✝ : H.Normal
g : G
hg : g ∈ H
⊢ g ∈ toEndHom.ker

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝² : Group G
A : Type u_2
inst✝¹ : MulAction G A
H : Subgroup G
inst✝ : H.Normal
g : G ⧸ H
a : ↑(fixedPoints (↥H) A)
⊢ Quotient.out g • a = g • a

Tactic:
by

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

Goal state before:
pp: α : Type v
M : Type u
inst✝¹ : Monoid M
inst✝ : MulAction M α
m : M
a : α
fixed : m • a = a
⊢ m ^ 1 • a = a

Tactic:
by

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

Goal state before:
pp: α : Type v
G : Type u
inst✝¹ : Group G
inst✝ : MulAction G α
g : G
a : α
⊢ period g⁻¹ a = period g a

Tactic:
by

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

Goal state before:
pp: α : Type v
M : Type u
inst✝¹ : Monoid M
inst✝ : MulAction M α
m : M
a : α
⊢ period m a ∣ orderOf m

Tactic:
by

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

Goal state before:
pp: α : Type v
M : Type u
inst✝¹ : Monoid M
inst✝ : MulAction M α
m : M
a : α
⊢ period m a ∣ Monoid.exponent M

Tactic:
by

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

Goal state before:
pp: α : Type v
M : Type u
inst✝¹ : Monoid M
inst✝ : MulAction M α
exp_pos : 0 < Monoid.exponent M
m : M
⊢ BddAbove (Set.range fun a ↦ period m a)

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Perm.MaximalSubgroups

Goal state before:
pp: M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
s : Set α
hs : Function.Surjective toPerm
⊢ IsPreprimitive ↥(stabilizer M s) ↑s

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Perm.MaximalSubgroups

Goal state before:
pp: M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
s : Set α
hs : ∀ a ∈ s, ∀ b ∈ s, ∃ g, g • a = b
hs' : ∀ a ∈ sᶜ, ∀ b ∈ sᶜ, ∃ g, g • a = b
hM : stabilizer M s ≠ ⊤
this : ∃ a b g, a ∈ s ∧ b ∈ sᶜ ∧ g • a = b
⊢ IsPretransitive M α

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hmt : IsMultiplyPretransitive (↥G) α (Nat.card α - 2)
⊢ alternatingGroup α ≤ G

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Perm.MaximalSubgroups

Goal state before:
pp: M : Type u_1
α : Type u_2
inst✝¹ : Group M
inst✝ : MulAction M α
s : Set α
hs : ∀ a ∈ s, ∀ b ∈ s, ∃ g, g • a = b
hs' : ∀ a ∈ sᶜ, ∀ b ∈ sᶜ, ∃ g, g • a = b
hM : stabilizer M s ≠ ⊤
⊢ IsPretransitive M α

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h : 3 ≤ Nat.card α
this : IsMultiplyPretransitive (↥(alternatingGroup α)) α (Nat.card α - 2) := isMultiplyPretransitive α
⊢ 2 ≤ 3

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h : 3 ≤ Nat.card α
⊢ IsPretransitive (↥(alternatingGroup α)) α

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
N : Type u_3
β : Type u_4
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
hφ : Function.Surjective φ
f : α →ₑ[φ] β
hf : Function.Bijective ⇑f
n : ℕ
H : IsMultiplyPreprimitive N β n
s : Set α
hs : s.encard + 1 = ↑n
t : Set β := ⇑f '' s
ψ : ↥(fixingSubgroup M s) → ↥(fixingSubgroup N t) :=
  fun x ↦
    match x with
    | ⟨g, hg⟩ => ⟨φ g, ⋯⟩
g : ↥(ofFixingSubgroup M s) →ₑ[ψ] ↥(ofFixingSubgroup N t) := { toFun := fun x ↦ ⟨f ↑x, ⋯⟩, map_smul' := ⋯ }
k : M
hk : φ k ∈ fixingSubgroup N t
this : k ∈ fixingSubgroup M s
⊢ ∃ a, ψ a = ⟨φ k, hk⟩

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Perm.MaximalSubgroups

Goal state before:
pp: α : Type u_2
s : Set α
inst✝ : DecidablePred fun x ↦ x ∈ s
g : Perm ↑s
⊢ ofSubtype g ∈ stabilizer (Perm α) s

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Perm.MaximalSubgroups

Goal state before:
pp: α : Type u_2
inst✝ : DecidableEq α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
this : swap a b • s ⊆ s
⊢ swap a b ∈ stabilizer (Perm α) s

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
h2 : 2 < Nat.card α
h4 : 3 < Nat.card α
this : IsPreprimitive (↥(alternatingGroup α)) α
⊢ IsTrivialBlock B

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Perm.MaximalSubgroups

Goal state before:
pp: α : Type u_2
inst✝ : DecidableEq α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
x : α
hx : x ∈ s
⊢ (fun x ↦ swap a b • x) x ∈ s

Tactic:
by_cases h : x ∈ ({a, b} : Set α)

Goal state after:
case pos
α : Type u_2
inst✝ : DecidableEq α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
x : α
hx : x ∈ s
h : x ∈ {a, b}
⊢ (fun x ↦ swap a b • x) x ∈ s

case neg
α : Type u_2
inst✝ : DecidableEq α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
x : α
hx : x ∈ s
h : x ∉ {a, b}
⊢ (fun x ↦ swap a b • x) x ∈ s
Module: Mathlib.GroupTheory.GroupAction.MultipleTransitivity

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
B : Set α
hB : IsBlock (↥(alternatingGroup α)) B
⊢ IsTrivialBlock B

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Perm.MaximalSubgroups

Goal state before:
pp: α : Type u_2
inst✝ : DecidableEq α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
⊢ swap a b ∈ stabilizer (Perm α) s

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Perm.MaximalSubgroups

Goal state before:
pp: α : Type u_2
s : Set α
⊢ ∀ a ∈ s, ∀ b ∈ s, ∃ g ∈ stabilizer (Perm α) s, g • a = b

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Perm.MaximalSubgroups

Goal state before:
pp: α : Type u_2
s : Set α
g : Perm ↑s
⊢ ∃ a, toPerm a = g

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Perm.MaximalSubgroups

Goal state before:
pp: α : Type u_2
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∉ s
h : stabilizer (Perm α) s = ⊤
⊢ swap a b ∈ stabilizer (Perm α) s

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.SpecificGroups.Alternating.MaximalSubgroups

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : 2 < s.ncard
⊢ ∃ g ∈ stabilizer (Perm α) s, g.IsThreeCycle

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.SpecificGroups.Alternating.MaximalSubgroups

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hα : 4 < Nat.card α
hs : 2 ≥ s.ncard
this : s.ncard + sᶜ.ncard = Nat.card α
⊢ 2 < sᶜ.ncard

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.SpecificGroups.Alternating.MaximalSubgroups

Goal state before:
pp: case inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hα : 4 < Nat.card α
hs : 2 ≥ s.ncard
⊢ ∃ g ∈ stabilizer (Perm α) s, g.IsThreeCycle

Tactic:
· have := ncard_add_ncard_compl s
    rw [← stabilizer_compl]
    exact exists_mem_stabilizer_isThreeCycle_of_two_lt_ncard (by grind)

Goal state after:
Module: Mathlib.GroupTheory.SpecificGroups.Alternating.MaximalSubgroups

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hα : 4 < Nat.card α
⊢ ∃ g ∈ stabilizer (Perm α) s, g.IsThreeCycle

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Perm.MaximalSubgroups

Goal state before:
pp: α : Type u_2
s : Set α
hs : s.Nonempty
hsc : sᶜ.Nonempty
⊢ stabilizer (Perm α) s ≠ ⊤

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.SpecificGroups.Alternating.MaximalSubgroups

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h4 : 4 < Nat.card α
G : Subgroup (Perm α)
hG' : IsPreprimitive (↥G) α
s : Set α
hG : stabilizer (Perm α) s ⊓ alternatingGroup α ≤ G
⊢ alternatingGroup α ≤ G

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
hn2 : 2 * (n + 1) ≥ Nat.card α
a : α
ha✝ : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
ha : a ∉ s ∪ g • s
h : s ∪ g • s = Set.univ
⊢ a ∈ s ∪ g • 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
N : Type u_3
β : Type u_4
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
hφ : Function.Surjective φ
f : α →ₑ[φ] β
hf : Function.Bijective ⇑f
n : ℕ
H : IsMultiplyPreprimitive N β n
s : Set α
hs : s.encard + 1 = ↑n
t : Set β := ⇑f '' s
ψ : ↥(fixingSubgroup M s) → ↥(fixingSubgroup N t) :=
  fun x ↦
    match x with
    | ⟨g, hg⟩ => ⟨φ g, ⋯⟩
g : ↥(ofFixingSubgroup M s) →ₑ[ψ] ↥(ofFixingSubgroup N t) := { toFun := fun x ↦ ⟨f ↑x, ⋯⟩, map_smul' := ⋯ }
x : α
hx : x ∈ ofFixingSubgroup M s
y : α
hy : y ∈ ofFixingSubgroup M s
h : g ⟨x, hx⟩ = g ⟨y, hy⟩
this : f x = f y
⊢ ⟨x, hx⟩ = ⟨y, hy⟩

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Perm.MaximalSubgroups

Goal state before:
pp: α : Type u_2
inst✝ : DecidableEq α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
⊢ ∀ (t : Set α), 1 < t.encard → ∃ g, g.IsSwap ∧ g ∈ stabilizer (Perm α) t

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
hn2 : 2 * (n + 1) ≥ Nat.card α
a : α
ha✝ : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
ha : a ∉ s ∪ g • s
⊢ t.ncard = t.ncard - 1 + 1

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.SpecificGroups.Alternating.MaximalSubgroups

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : sᶜ.Nontrivial
⊢ ∃ k, k.IsSwap ∧ _root_.Disjoint s ↑k.support

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.SpecificGroups.Alternating.MaximalSubgroups

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : sᶜ.Nontrivial
k : Perm α
hk_swap : k.IsSwap
hk_support : _root_.Disjoint s ↑k.support
⊢ k • s = s

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Perm.MaximalSubgroups

Goal state before:
pp: α : Type u_2
inst✝ : DecidableEq α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < t.encard → ∃ g, g.IsSwap ∧ g ∈ stabilizer (Perm α) t
h1' : s.encard ≤ 1
h1c' : sᶜ.encard ≤ 1
⊢ 1 + 1 = 2

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
N : Type u_3
β : Type u_4
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
hφ : Function.Surjective φ
f : α →ₑ[φ] β
hf : Function.Bijective ⇑f
n : ℕ
H : IsMultiplyPreprimitive N β n
s : Set α
hs : s.encard + 1 = ↑n
t : Set β := ⇑f '' s
ψ : ↥(fixingSubgroup M s) → ↥(fixingSubgroup N t) :=
  fun x ↦
    match x with
    | ⟨g, hg⟩ => ⟨φ g, ⋯⟩
g : ↥(ofFixingSubgroup M s) →ₑ[ψ] ↥(ofFixingSubgroup N t) := { toFun := fun x ↦ ⟨f ↑x, ⋯⟩, map_smul' := ⋯ }
y : α
hx : f y ∈ ofFixingSubgroup N t
this : y ∈ ofFixingSubgroup M s
⊢ ∃ a, g a = ⟨f y, hx⟩

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
hn2 : 2 * (n + 1) ≥ Nat.card α
a : α
ha✝ : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
ha : a ∉ s ∪ g • s
htm : t.ncard = t.ncard - 1 + 1
h : s ⊆ t
this : s = g • s
⊢ b ∈ g • s

Tactic:
simpa only [Set.smul_set_compl, Set.mem_compl_iff, Set.not_notMem] using hgb

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
hn2 : 2 * (n + 1) ≥ Nat.card α
a : α
ha✝ : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
ha : a ∉ s ∪ g • s
htm : t.ncard = t.ncard - 1 + 1
h : s ⊆ t
this : s = g • s
⊢ b ∈ 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
N : Type u_3
β : Type u_4
inst✝¹ : Group N
inst✝ : MulAction N β
φ : M → N
hφ : Function.Surjective φ
f : α →ₑ[φ] β
hf : Function.Bijective ⇑f
n : ℕ
⊢ IsMultiplyPreprimitive M α n ↔ IsMultiplyPreprimitive N β n

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Perm.MaximalSubgroups

Goal state before:
pp: α : Type u_2
inst✝ : DecidableEq α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < t.encard → ∃ g, g.IsSwap ∧ g ∈ stabilizer (Perm α) t
h1' : s.encard ≤ 1
h1c' : sᶜ.encard ≤ 1
⊢ univ.encard = 2

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Perm.MaximalSubgroups

Goal state before:
pp: α : Type u_2
inst✝ : DecidableEq α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < t.encard → ∃ g, g.IsSwap ∧ g ∈ stabilizer (Perm α) t
h1' : s.encard ≤ 1
h1c' : sᶜ.encard ≤ 1
hα : univ.encard = 2
⊢ Finite α

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
hn2 : 2 * (n + 1) ≥ Nat.card α
a : α
ha✝ : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
ha : a ∉ s ∪ g • s
htm : t.ncard = t.ncard - 1 + 1
⊢ t.ncard - 1 < n

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.SpecificGroups.Alternating.MaximalSubgroups

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : sᶜ.Nontrivial
k : Perm α
hk_swap : k.IsSwap
hk_support : _root_.Disjoint s ↑k.support
hks : k • s = s
g : Perm ↑s
hsg : sign g = -1
x : ↑s
this : k ↑x = ↑x
⊢ ↑((toPerm ⟨⟨Perm.ofSubtype g * k, ⋯⟩, ⋯⟩) x) = ↑(g x)

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Perm.MaximalSubgroups

Goal state before:
pp: α : Type u_2
inst✝ : DecidableEq α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < t.encard → ∃ g, g.IsSwap ∧ g ∈ stabilizer (Perm α) t
h1' : s.encard ≤ 1
h1c' : sᶜ.encard ≤ 1
hα : univ.encard = 2
x✝ : Finite α
⊢ Nat.card α = 2

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
hn2 : 2 * (n + 1) ≥ Nat.card α
a : α
ha✝ : a ∈ sᶜ
b : α
hb : b ∈ sᶜ
hab : a ≠ b
g : G
hga : a ∈ g • sᶜ
hgb : b ∉ g • sᶜ
t : Set α := s ∩ g • s
ha : a ∉ s ∪ g • s
htm : t.ncard = t.ncard - 1 + 1
hmn : t.ncard - 1 < n
htm' : t.ncard - 1 + 2 < Nat.card α
hsgs_ne_top : s ∪ g • s ≠ ⊤
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.Perm.MaximalSubgroups

Goal state before:
pp: α : Type u_2
inst✝ : DecidableEq α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < t.encard → ∃ g, g.IsSwap ∧ g ∈ stabilizer (Perm α) t
h1' : s.encard ≤ 1
h1c' : sᶜ.encard ≤ 1
hα✝ : univ.encard = 2
x✝ : Finite α
hα : Nat.card α = 2
⊢ Fact (Nat.Prime (Nat.card (Perm α)))

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.SpecificGroups.Alternating.MaximalSubgroups

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
hs : sᶜ.Nontrivial
⊢ Function.Surjective toPerm

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.SpecificGroups.Alternating.MaximalSubgroups

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∈ sᶜ
c : α
hc : c ∈ sᶜ
hbc : b ≠ c
this : ∃ g, g ∉ stabilizer (↥(alternatingGroup α)) s
⊢ stabilizer (↥(alternatingGroup α)) s ≠ ⊤

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.SpecificGroups.Alternating.MaximalSubgroups

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∈ sᶜ
c : α
hc : c ∈ sᶜ
hbc : b ≠ c
⊢ swap a b * swap a c ∈ alternatingGroup α

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Perm.MaximalSubgroups

Goal state before:
pp: α : Type u_2
inst✝ : DecidableEq α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
⊢ ∃ g, g.IsSwap ∧ g ∈ G

Tactic:
by

Goal state after: