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

Goal state before:
pp: case pos
G : Type u_1
inst✝ : Group G
hH : IsNilpotent G
⊢ nilpotencyClass (G ⧸ center G) = nilpotencyClass G - 1

pp: case neg
G : Type u_1
inst✝ : Group G
hH : ¬IsNilpotent G
⊢ nilpotencyClass (G ⧸ center G) = nilpotencyClass G - 1

Tactic:
;

Goal state after:
case pos
G : Type u_1
inst✝ : Group G
hH : IsNilpotent G
⊢ nilpotencyClass (G ⧸ center G) = nilpotencyClass G - 1

case neg
G : Type u_1
inst✝ : Group G
hH : ¬IsNilpotent G
⊢ nilpotencyClass (G ⧸ center G) = nilpotencyClass G - 1
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: case succ
G : Type u_1
inst✝ : Group G
i : ℕ
ih : derivedSeries G i ≤ ⊤.lowerCentralSeries i
⊢ derivedSeries G i ≤ ⊤

Tactic:
;

Goal state after:
case succ
G : Type u_1
inst✝ : Group G
i : ℕ
ih : derivedSeries G i ≤ ⊤.lowerCentralSeries i
⊢ derivedSeries G i ≤ ⊤
Module: Mathlib.GroupTheory.FreeGroup.NielsenSchreier

Goal state before:
pp: case e_5
G : Type u
inst✝¹ : Groupoid G
inst✝ : IsFreeGroupoid G
a b : G
p : a ⟶ b
X : Type u := ⋯
f : G → X := ⋯
F : G ⥤ CategoryTheory.SingleObj X := ⋯
⊢ F = (Functor.const G).obj ()

Tactic:
;

Goal state after:
case e_5
G : Type u
inst✝¹ : Groupoid G
inst✝ : IsFreeGroupoid G
a b : G
p : a ⟶ b
X : Type u := ⋯
f : G → X := ⋯
F : G ⥤ CategoryTheory.SingleObj X := ⋯
⊢ F = (Functor.const G).obj ()
Module: Mathlib.GroupTheory.FreeGroup.NielsenSchreier

Goal state before:
pp: G : Type u
inst✝¹ : Groupoid G
inst✝ : IsFreeGroupoid G
a b : G
⊢ Nonempty (a ⟶ b) → Nonempty (Path (symgen a) (symgen b))

Tactic:
rintro ⟨p⟩
  rw [← @WeaklyConnectedComponent.eq (Generators G), eq_comm, ← FreeGroup.of_injective.eq_iff, ←
    mul_inv_eq_one]
  let X := FreeGroup (WeaklyConnectedComponent <| Generators G)
  let f : G → X := fun g => FreeGroup.of (WeaklyConnectedComponent.mk g)
  let F : G ⥤ CategoryTheory.SingleObj.{u} (X : Type u) := SingleObj.differenceFunctor f
  change (F.map p) = ((@CategoryTheory.Functor.const G _ _ (SingleObj.category X)).obj ()).map p
  congr; ext
  rw [Functor.const_obj_map, id_as_one, differenceFunctor_map, @mul_inv_eq_one _ _ (f _)]
  apply congr_arg FreeGroup.of
  apply (WeaklyConnectedComponent.eq _ _).mpr
  exact ⟨Hom.toPath (Sum.inr (by assumption))⟩

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

Goal state before:
pp: case inl
M : Type u_3
α : Type u_4
N : Type u_5
β : Type u_6
inst✝³ : Monoid M
inst✝² : MulAction M α
inst✝¹ : Monoid N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
B : Set α
hB : B.Subsingleton
⊢ (⇑f '' B).Subsingleton

Tactic:
;

Goal state after:
case inl
M : Type u_3
α : Type u_4
N : Type u_5
β : Type u_6
inst✝³ : Monoid M
inst✝² : MulAction M α
inst✝¹ : Monoid N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
B : Set α
hB : B.Subsingleton
⊢ (⇑f '' B).Subsingleton
Module: Mathlib.GroupTheory.GroupAction.Blocks

Goal state before:
pp: case inr
M : Type u_3
α : Type u_4
N : Type u_5
β : Type u_6
inst✝³ : Monoid M
inst✝² : MulAction M α
inst✝¹ : Monoid N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
B : Set α
hB : B = univ
⊢ ⇑f '' B = univ

Tactic:
;

Goal state after:
case inr
M : Type u_3
α : Type u_4
N : Type u_5
β : Type u_6
inst✝³ : Monoid M
inst✝² : MulAction M α
inst✝¹ : Monoid N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
B : Set α
hB : B = univ
⊢ ⇑f '' B = univ
Module: Mathlib.GroupTheory.GroupAction.Blocks

Goal state before:
pp: case inr
M : Type u_3
α : Type u_4
N : Type u_5
β : Type u_6
inst✝³ : Monoid M
inst✝² : MulAction M α
inst✝¹ : Monoid N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
B : Set α
hB : B = univ
⊢ IsTrivialBlock (⇑f '' B)

Tactic:
· apply Or.intro_right; rw [hB]
    simp only [Set.image_univ, Set.range_eq_univ, hf]

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

Goal state before:
pp: case inl
M : Type u_3
α : Type u_4
N : Type u_5
β : Type u_6
inst✝³ : Monoid M
inst✝² : MulAction M α
inst✝¹ : Monoid N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Injective ⇑f
B : Set β
hB : B.Subsingleton
⊢ (⇑f ⁻¹' B).Subsingleton

Tactic:
;

Goal state after:
case inl
M : Type u_3
α : Type u_4
N : Type u_5
β : Type u_6
inst✝³ : Monoid M
inst✝² : MulAction M α
inst✝¹ : Monoid N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Injective ⇑f
B : Set β
hB : B.Subsingleton
⊢ (⇑f ⁻¹' B).Subsingleton
Module: Mathlib.GroupTheory.GroupAction.Blocks

Goal state before:
pp: case inr
M : Type u_3
α : Type u_4
N : Type u_5
β : Type u_6
inst✝³ : Monoid M
inst✝² : MulAction M α
inst✝¹ : Monoid N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Injective ⇑f
B : Set β
hB : B = univ
⊢ ⇑f ⁻¹' B = univ

Tactic:
;

Goal state after:
case inr
M : Type u_3
α : Type u_4
N : Type u_5
β : Type u_6
inst✝³ : Monoid M
inst✝² : MulAction M α
inst✝¹ : Monoid N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Injective ⇑f
B : Set β
hB : B = univ
⊢ ⇑f ⁻¹' B = univ
Module: Mathlib.GroupTheory.GroupAction.Blocks

Goal state before:
pp: case inr
M : Type u_3
α : Type u_4
N : Type u_5
β : Type u_6
inst✝³ : Monoid M
inst✝² : MulAction M α
inst✝¹ : Monoid N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Injective ⇑f
B : Set β
hB : B = univ
⊢ ⇑f ⁻¹' univ = univ

Tactic:
;

Goal state after:
case inr
M : Type u_3
α : Type u_4
N : Type u_5
β : Type u_6
inst✝³ : Monoid M
inst✝² : MulAction M α
inst✝¹ : Monoid N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Injective ⇑f
B : Set β
hB : B = univ
⊢ ⇑f ⁻¹' univ = univ
Module: Mathlib.GroupTheory.GroupAction.Blocks

Goal state before:
pp: case f
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
B : Set X
H : Subgroup G
hB : IsBlock (↥H) B
g : G
h' : ↥(Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) H)
h : G
hH : h ∈ H
hh : g * h * g⁻¹ = ↑h'
this : h' • g • B = g • h • B
a✝ : ⟨h, hH⟩ • B = B
⊢ g • h • B = g • B

Tactic:
;

Goal state after:
case f
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
B : Set X
H : Subgroup G
hB : IsBlock (↥H) B
g : G
h' : ↥(Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) H)
h : G
hH : h ∈ H
hh : g * h * g⁻¹ = ↑h'
this : h' • g • B = g • h • B
a✝ : ⟨h, hH⟩ • B = B
⊢ g • h • B = g • B
Module: Mathlib.GroupTheory.GroupAction.Blocks

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
B : Set X
H : Subgroup G
hB : IsBlock (↥H) B
g : G
h' : ↥(Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) H)
h : G
hH : h ∈ H
hh : g * h * g⁻¹ = ↑h'
this : ↑h' • g • B = g • h • B
⊢ h' • g • B = ↑h' • g • B

Tactic:
;

Goal state after:
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
B : Set X
H : Subgroup G
hB : IsBlock (↥H) B
g : G
h' : ↥(Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) H)
h : G
hH : h ∈ H
hh : g * h * g⁻¹ = ↑h'
this : ↑h' • g • B = g • h • B
⊢ h' • g • B = ↑h' • g • B
Module: Mathlib.GroupTheory.GroupAction.Blocks

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
B : Set X
hGX : IsPretransitive G X
hB : IsBlock G B
hBe : B.Nonempty
g : G
⊢ IsBlock G ((fun g ↦ g • B) g)

Tactic:
;

Goal state after:
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
B : Set X
hGX : IsPretransitive G X
hB : IsBlock G B
hBe : B.Nonempty
g : G
⊢ IsBlock G ((fun g ↦ g • B) g)
Module: Mathlib.GroupTheory.GroupAction.Blocks

Goal state before:
pp: case right
G : Type u_1
inst✝² : Group G
X : Type u_2
inst✝¹ : MulAction G X
N : Subgroup G
inst✝ : N.Normal
b : Set X
⊢ (b ∈ range fun a ↦ orbit (↥N) a) → IsBlock G b

Tactic:
;

Goal state after:
case right
G : Type u_1
inst✝² : Group G
X : Type u_2
inst✝¹ : MulAction G X
N : Subgroup G
inst✝ : N.Normal
b : Set X
⊢ (b ∈ range fun a ↦ orbit (↥N) a) → IsBlock G b
Module: Mathlib.GroupTheory.GroupAction.Blocks

Goal state before:
pp: G : Type u_1
inst✝² : Group G
X : Type u_2
inst✝¹ : MulAction G X
N : Subgroup G
inst✝ : N.Normal
⊢ IsBlockSystem G (range fun a ↦ orbit (↥N) a)

Tactic:
by
  constructor
  · apply IsPartition.of_orbits
  · intro b; rintro ⟨a, rfl⟩
    exact .orbit_of_normal a

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

Goal state before:
pp: M : Type u_1
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
inst✝¹ : Nontrivial M
is_perfect : commutator M = ⊤
inst✝ : IsQuasiPreprimitive M α
IwaS : IwasawaStructure M α
is_faithful : FaithfulSMul M α
⊢ IsSimpleGroup M

Tactic:
apply IsSimpleGroup.mk

Goal state after:
M : Type u_1
inst✝³ : Group M
α : Type u_2
inst✝² : MulAction M α
inst✝¹ : Nontrivial M
is_perfect : commutator M = ⊤
inst✝ : IsQuasiPreprimitive M α
IwaS : IwasawaStructure M α
is_faithful : FaithfulSMul M α
⊢ ∀ (H : Subgroup M), H.Normal → H = ⊥ ∨ H = ⊤
Module: Mathlib.GroupTheory.GroupAction.Blocks

Goal state before:
pp: case mp
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g ∈ stabilizer G (orbit (↥H) a)
b : ↥H
h : g • ↑b • a = a
⊢ g ∈ H

Tactic:
← mul_smul,

Goal state after:
case mp
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
g : G
hg : g ∈ stabilizer G (orbit (↥H) a)
b : ↥H
h : (g * ↑b) • a = a
⊢ g ∈ H
Module: Mathlib.GroupTheory.GroupAction.Blocks

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
B✝ : Set X
htGX : IsPretransitive G X
a : X
B : Set X
ha : a ∈ B
hB : IsBlock G B
⊢ ∀ {b : { B // a ∈ B ∧ IsBlock G B }},
    {
            toFun := fun x ↦
              match x with
              | ⟨B, ⋯⟩ => ⟨stabilizer G B, ⋯⟩,
            invFun := fun x ↦
              match x with
              | ⟨H, hH⟩ => ⟨orbit (↥H) a, ⋯⟩,
            left_inv := ⋯, right_inv := ⋯ }
          ⟨B, ⋯⟩ ≤
        {
            toFun := fun x ↦
              match x with
              | ⟨B, ⋯⟩ => ⟨stabilizer G B, ⋯⟩,
            invFun := fun x ↦
              match x with
              | ⟨H, hH⟩ => ⟨orbit (↥H) a, ⋯⟩,
            left_inv := ⋯, right_inv := ⋯ }
          b ↔
      ⟨B, ⋯⟩ ≤ b

Tactic:
;

Goal state after:
G : Type u_1
inst✝¹ : Group G
X : Type u_2
inst✝ : MulAction G X
B✝ : Set X
htGX : IsPretransitive G X
a : X
B : Set X
ha : a ∈ B
hB : IsBlock G B
⊢ ∀ {b : { B // a ∈ B ∧ IsBlock G B }},
    {
            toFun := fun x ↦
              match x with
              | ⟨B, ⋯⟩ => ⟨stabilizer G B, ⋯⟩,
            invFun := fun x ↦
              match x with
              | ⟨H, hH⟩ => ⟨orbit (↥H) a, ⋯⟩,
            left_inv := ⋯, right_inv := ⋯ }
          ⟨B, ⋯⟩ ≤
        {
            toFun := fun x ↦
              match x with
              | ⟨B, ⋯⟩ => ⟨stabilizer G B, ⋯⟩,
            invFun := fun x ↦
              match x with
              | ⟨H, hH⟩ => ⟨orbit (↥H) a, ⋯⟩,
            left_inv := ⋯, right_inv := ⋯ }
          b ↔
      ⟨B, ⋯⟩ ≤ b
Module: Mathlib.GroupTheory.GroupAction.Blocks

Goal state before:
pp: G : Type u_1
inst✝² : Group G
X : Type u_2
inst✝¹ : MulAction G X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
x : X
hx : x ∈ B
x✝ : ↥(stabilizer G B)
⊢ x✝ ∈ (stabilizer G x).subgroupOf (stabilizer G B) ↔ x✝ ∈ stabilizer (↥(stabilizer G B)) x

Tactic:
;

Goal state after:
G : Type u_1
inst✝² : Group G
X : Type u_2
inst✝¹ : MulAction G X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
x : X
hx : x ∈ B
x✝ : ↥(stabilizer G B)
⊢ x✝ ∈ (stabilizer G x).subgroupOf (stabilizer G B) ↔ x✝ ∈ stabilizer (↥(stabilizer G B)) x
Module: Mathlib.GroupTheory.GroupAction.Blocks

Goal state before:
pp: G : Type u_1
inst✝² : Group G
X : Type u_2
inst✝¹ : MulAction G X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
x : X
hx : x ∈ B
⊢ (stabilizer G x).index = Nat.card X

Tactic:
index_stabilizer_of_transitive

Goal state after:
G : Type u_1
inst✝² : Group G
X : Type u_2
inst✝¹ : MulAction G X
inst✝ : IsPretransitive G X
B : Set X
hB : IsBlock G B
x : X
hx : x ∈ B
⊢ Nat.card X = Nat.card X
Module: Mathlib.GroupTheory.GroupAction.Blocks

Goal state before:
pp: case inl
G : Type u_1
inst✝³ : Group G
X : Type u_2
inst✝² : MulAction G X
inst✝¹ : IsPretransitive G X
B : Set X
inst✝ : Finite X
hB : IsBlock G B
hB' : Nat.card X < 2 * (orbit G B).ncard
h : B = ∅
⊢ 0 < 2

Tactic:
;

Goal state after:
case inl
G : Type u_1
inst✝³ : Group G
X : Type u_2
inst✝² : MulAction G X
inst✝¹ : IsPretransitive G X
B : Set X
inst✝ : Finite X
hB : IsBlock G B
hB' : Nat.card X < 2 * (orbit G B).ncard
h : B = ∅
⊢ 0 < 2
Module: Mathlib.GroupTheory.GroupAction.Primitive

Goal state before:
pp: case inl
G : Type u_1
X : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H✝ : ∀ ⦃B : Set X⦄, a ∈ B → IsBlock G B → IsTrivialBlock B
H : (orbit G a).Subsingleton
⊢ False

Tactic:
;

Goal state after:
case inl
G : Type u_1
X : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H✝ : ∀ ⦃B : Set X⦄, a ∈ B → IsBlock G B → IsTrivialBlock B
H : (orbit G a).Subsingleton
⊢ False
Module: Mathlib.GroupTheory.GroupAction.Primitive

Goal state before:
pp: case inl
G : Type u_1
X : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H✝ : ∀ ⦃B : Set X⦄, a ∈ B → IsBlock G B → IsTrivialBlock B
H : orbit G a = {a}
g : G
⊢ g • a ∈ {a}

Tactic:
;

Goal state after:
case inl
G : Type u_1
X : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H✝ : ∀ ⦃B : Set X⦄, a ∈ B → IsBlock G B → IsTrivialBlock B
H : orbit G a = {a}
g : G
⊢ g • a ∈ {a}
Module: Mathlib.GroupTheory.GroupAction.Primitive

Goal state before:
pp: case inr
G : Type u_1
X : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H✝ : ∀ ⦃B : Set X⦄, a ∈ B → IsBlock G B → IsTrivialBlock B
H : orbit G a = Set.univ
x : X
⊢ ∃ g, g • a = x

Tactic:
;

Goal state after:
case inr
G : Type u_1
X : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H✝ : ∀ ⦃B : Set X⦄, a ∈ B → IsBlock G B → IsTrivialBlock B
H : orbit G a = Set.univ
x : X
⊢ ∃ g, g • a = x
Module: Mathlib.GroupTheory.GroupAction.Primitive

Goal state before:
pp: case inr
G : Type u_1
X : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H✝ : ∀ ⦃B : Set X⦄, a ∈ B → IsBlock G B → IsTrivialBlock B
H : orbit G a = Set.univ
x : X
⊢ x ∈ Set.univ

Tactic:
;

Goal state after:
case inr
G : Type u_1
X : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
ha : a ∉ fixedPoints G X
H✝ : ∀ ⦃B : Set X⦄, a ∈ B → IsBlock G B → IsTrivialBlock B
H : orbit G a = Set.univ
x : X
⊢ x ∈ Set.univ
Module: Mathlib.GroupTheory.GroupAction.Primitive

Goal state before:
pp: case mp
G : Type u_3
inst✝³ : Group G
X : Type u_4
inst✝² : MulAction G X
inst✝¹ : IsPretransitive G X
inst✝ : Nontrivial X
a : X
h : IsSimpleOrder (BlockMem G a)
⊢ IsPreprimitive G X

Tactic:
;

Goal state after:
case mp
G : Type u_3
inst✝³ : Group G
X : Type u_4
inst✝² : MulAction G X
inst✝¹ : IsPretransitive G X
inst✝ : Nontrivial X
a : X
h : IsSimpleOrder (BlockMem G a)
⊢ IsPreprimitive G X
Module: Mathlib.GroupTheory.GroupAction.Primitive

Goal state before:
pp: case mp.inl
G : Type u_3
inst✝³ : Group G
X : Type u_4
inst✝² : MulAction G X
inst✝¹ : IsPretransitive G X
inst✝ : Nontrivial X
a : X
h : IsSimpleOrder (BlockMem G a)
h_bot_or_top : ∀ (a_1 : BlockMem G a), a_1 = ⊥ ∨ a_1 = ⊤ := ⋯
B : Set X
haB : a ∈ B
hB : IsBlock G B
hB' : B = ↑⊥
⊢ B.Subsingleton

Tactic:
;

Goal state after:
case mp.inl
G : Type u_3
inst✝³ : Group G
X : Type u_4
inst✝² : MulAction G X
inst✝¹ : IsPretransitive G X
inst✝ : Nontrivial X
a : X
h : IsSimpleOrder (BlockMem G a)
h_bot_or_top : ∀ (a_1 : BlockMem G a), a_1 = ⊥ ∨ a_1 = ⊤ := ⋯
B : Set X
haB : a ∈ B
hB : IsBlock G B
hB' : B = ↑⊥
⊢ B.Subsingleton
Module: Mathlib.GroupTheory.GroupAction.Primitive

Goal state before:
pp: case mp.inl
G : Type u_3
inst✝³ : Group G
X : Type u_4
inst✝² : MulAction G X
inst✝¹ : IsPretransitive G X
inst✝ : Nontrivial X
a : X
h : IsSimpleOrder (BlockMem G a)
h_bot_or_top : ∀ (a_1 : BlockMem G a), a_1 = ⊥ ∨ a_1 = ⊤ := eq_bot_or_eq_top
B : Set X
haB : a ∈ B
hB : IsBlock G B
hB' : B = ↑⊥
⊢ (↑⊥).Subsingleton

Tactic:
;

Goal state after:
case mp.inl
G : Type u_3
inst✝³ : Group G
X : Type u_4
inst✝² : MulAction G X
inst✝¹ : IsPretransitive G X
inst✝ : Nontrivial X
a : X
h : IsSimpleOrder (BlockMem G a)
h_bot_or_top : ∀ (a_1 : BlockMem G a), a_1 = ⊥ ∨ a_1 = ⊤ := eq_bot_or_eq_top
B : Set X
haB : a ∈ B
hB : IsBlock G B
hB' : B = ↑⊥
⊢ (↑⊥).Subsingleton
Module: Mathlib.GroupTheory.GroupAction.Primitive

Goal state before:
pp: case mp.inr
G : Type u_3
inst✝³ : Group G
X : Type u_4
inst✝² : MulAction G X
inst✝¹ : IsPretransitive G X
inst✝ : Nontrivial X
a : X
h : IsSimpleOrder (BlockMem G a)
h_bot_or_top : ∀ (a_1 : BlockMem G a), a_1 = ⊥ ∨ a_1 = ⊤ := ⋯
B : Set X
haB : a ∈ B
hB : IsBlock G B
hB' : B = ↑⊤
⊢ B = Set.univ

Tactic:
;

Goal state after:
case mp.inr
G : Type u_3
inst✝³ : Group G
X : Type u_4
inst✝² : MulAction G X
inst✝¹ : IsPretransitive G X
inst✝ : Nontrivial X
a : X
h : IsSimpleOrder (BlockMem G a)
h_bot_or_top : ∀ (a_1 : BlockMem G a), a_1 = ⊥ ∨ a_1 = ⊤ := ⋯
B : Set X
haB : a ∈ B
hB : IsBlock G B
hB' : B = ↑⊤
⊢ B = Set.univ
Module: Mathlib.GroupTheory.GroupAction.Primitive

Goal state before:
pp: case mp.inr
G : Type u_3
inst✝³ : Group G
X : Type u_4
inst✝² : MulAction G X
inst✝¹ : IsPretransitive G X
inst✝ : Nontrivial X
a : X
h : IsSimpleOrder (BlockMem G a)
h_bot_or_top : ∀ (a_1 : BlockMem G a), a_1 = ⊥ ∨ a_1 = ⊤ := eq_bot_or_eq_top
B : Set X
haB : a ∈ B
hB : IsBlock G B
hB' : B = ↑⊤
⊢ ↑⊤ = Set.univ

Tactic:
;

Goal state after:
case mp.inr
G : Type u_3
inst✝³ : Group G
X : Type u_4
inst✝² : MulAction G X
inst✝¹ : IsPretransitive G X
inst✝ : Nontrivial X
a : X
h : IsSimpleOrder (BlockMem G a)
h_bot_or_top : ∀ (a_1 : BlockMem G a), a_1 = ⊥ ∨ a_1 = ⊤ := eq_bot_or_eq_top
B : Set X
haB : a ∈ B
hB : IsBlock G B
hB' : B = ↑⊤
⊢ ↑⊤ = Set.univ
Module: Mathlib.GroupTheory.GroupAction.Primitive

Goal state before:
pp: case mpr
G : Type u_3
inst✝³ : Group G
X : Type u_4
inst✝² : MulAction G X
inst✝¹ : IsPretransitive G X
inst✝ : Nontrivial X
a : X
hGX' : IsPreprimitive G X
⊢ IsSimpleOrder (BlockMem G a)

Tactic:
;

Goal state after:
case mpr
G : Type u_3
inst✝³ : Group G
X : Type u_4
inst✝² : MulAction G X
inst✝¹ : IsPretransitive G X
inst✝ : Nontrivial X
a : X
hGX' : IsPreprimitive G X
⊢ IsSimpleOrder (BlockMem G a)
Module: Mathlib.GroupTheory.SpecificGroups.Alternating

Goal state before:
pp: α : Type u_1
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : Subsingleton α
⊢ alternatingGroup α = ⊤

Tactic:
;

Goal state after:
α : Type u_1
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : Subsingleton α
⊢ alternatingGroup α = ⊤
Module: Mathlib.GroupTheory.SpecificGroups.Alternating

Goal state before:
pp: case inl
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
σ : Perm α
hσ✝ : σ ∈ alternatingGroup α
hσ : (↑⟨σ, hσ✝⟩).support.card + 2 ≤ card α
τ : Perm α
hτ : τ ∈ alternatingGroup α
hc : IsConj ↑⟨σ, hσ✝⟩ ↑⟨τ, hτ⟩
π : Perm α
hπ : π * σ * π⁻¹ = τ
h : sign π = 1
⊢ IsConj ⟨σ, hσ✝⟩ ⟨τ, hτ⟩

Tactic:
rw [isConj_iff]

Goal state after:
case inl
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
σ : Perm α
hσ✝ : σ ∈ alternatingGroup α
hσ : (↑⟨σ, hσ✝⟩).support.card + 2 ≤ card α
τ : Perm α
hτ : τ ∈ alternatingGroup α
hc : IsConj ↑⟨σ, hσ✝⟩ ↑⟨τ, hτ⟩
π : Perm α
hπ : π * σ * π⁻¹ = τ
h : sign π = 1
⊢ ∃ c, c * ⟨σ, hσ✝⟩ * c⁻¹ = ⟨τ, hτ⟩
Module: Mathlib.GroupTheory.GroupAction.Primitive

Goal state before:
pp: G : Type u_1
X : Type u_2
inst✝⁴ : Group G
inst✝³ : MulAction G X
M : Type u_3
inst✝² : Group M
α : Type u_4
inst✝¹ : MulAction M α
inst✝ : IsPreprimitive M α
N : Subgroup M
x✝ : N.Normal
a : α
ha : a ∉ fixedPoints (↥N) α
h : (orbit (↥N) a).Subsingleton
n : ↥N
this : orbit (↥N) a = {a}
⊢ n • a ∈ orbit (↥N) a

Tactic:
;

Goal state after:
G : Type u_1
X : Type u_2
inst✝⁴ : Group G
inst✝³ : MulAction G X
M : Type u_3
inst✝² : Group M
α : Type u_4
inst✝¹ : MulAction M α
inst✝ : IsPreprimitive M α
N : Subgroup M
x✝ : N.Normal
a : α
ha : a ∉ fixedPoints (↥N) α
h : (orbit (↥N) a).Subsingleton
n : ↥N
this : orbit (↥N) a = {a}
⊢ n • a ∈ orbit (↥N) a
Module: Mathlib.GroupTheory.GroupAction.Primitive

Goal state before:
pp: case inl
G : Type u_1
X : Type u_2
inst✝⁶ : Group G
inst✝⁵ : MulAction G X
H : Type u_3
Y : Type u_4
inst✝⁴ : Group H
inst✝³ : MulAction H Y
φ : G → H
f : X →ₑ[φ] Y
inst✝² : Finite Y
inst✝¹ : IsPretransitive H Y
inst✝ : IsPreprimitive G X
hf' : Nat.card Y < 2 * (Set.range ⇑f).ncard
B : Set Y
hB : IsBlock H B
hB' : B = ∅
⊢ IsTrivialBlock B

pp: case inr
G : Type u_1
X : Type u_2
inst✝⁶ : Group G
inst✝⁵ : MulAction G X
H : Type u_3
Y : Type u_4
inst✝⁴ : Group H
inst✝³ : MulAction H Y
φ : G → H
f : X →ₑ[φ] Y
inst✝² : Finite Y
inst✝¹ : IsPretransitive H Y
inst✝ : IsPreprimitive G X
hf' : Nat.card Y < 2 * (Set.range ⇑f).ncard
B : Set Y
hB : IsBlock H B
hB' : B.Nonempty
⊢ IsTrivialBlock B

Tactic:
;

Goal state after:
case inl
G : Type u_1
X : Type u_2
inst✝⁶ : Group G
inst✝⁵ : MulAction G X
H : Type u_3
Y : Type u_4
inst✝⁴ : Group H
inst✝³ : MulAction H Y
φ : G → H
f : X →ₑ[φ] Y
inst✝² : Finite Y
inst✝¹ : IsPretransitive H Y
inst✝ : IsPreprimitive G X
hf' : Nat.card Y < 2 * (Set.range ⇑f).ncard
B : Set Y
hB : IsBlock H B
hB' : B = ∅
⊢ IsTrivialBlock B

case inr
G : Type u_1
X : Type u_2
inst✝⁶ : Group G
inst✝⁵ : MulAction G X
H : Type u_3
Y : Type u_4
inst✝⁴ : Group H
inst✝³ : MulAction H Y
φ : G → H
f : X →ₑ[φ] Y
inst✝² : Finite Y
inst✝¹ : IsPretransitive H Y
inst✝ : IsPreprimitive G X
hf' : Nat.card Y < 2 * (Set.range ⇑f).ncard
B : Set Y
hB : IsBlock H B
hB' : B.Nonempty
⊢ IsTrivialBlock B
Module: Mathlib.GroupTheory.GroupAction.Primitive

Goal state before:
pp: G : Type u_1
X : Type u_2
inst✝⁶ : Group G
inst✝⁵ : MulAction G X
H : Type u_3
Y : Type u_4
inst✝⁴ : Group H
inst✝³ : MulAction H Y
φ : G → H
f : X →ₑ[φ] Y
inst✝² : Finite Y
inst✝¹ : IsPretransitive H Y
inst✝ : IsPreprimitive G X
hf' : Nat.card Y < 2 * (Set.range ⇑f).ncard
B : Set Y
hB : IsBlock H B
hB' : B.Nonempty
hB_ne_top : ¬B = Set.univ
⊢ B.ncard * (Set.range ⇑f).ncard ≤ B.ncard * (orbit H B).ncard

Tactic:
apply Nat.mul_le_mul_left

Goal state after:
G : Type u_1
X : Type u_2
inst✝⁶ : Group G
inst✝⁵ : MulAction G X
H : Type u_3
Y : Type u_4
inst✝⁴ : Group H
inst✝³ : MulAction H Y
φ : G → H
f : X →ₑ[φ] Y
inst✝² : Finite Y
inst✝¹ : IsPretransitive H Y
inst✝ : IsPreprimitive G X
hf' : Nat.card Y < 2 * (Set.range ⇑f).ncard
B : Set Y
hB : IsBlock H B
hB' : B.Nonempty
hB_ne_top : ¬B = Set.univ
⊢ (Set.range ⇑f).ncard ≤ (orbit H B).ncard
Module: Mathlib.GroupTheory.GroupAction.Primitive

Goal state before:
pp: G : Type u_1
X : Type u_2
inst✝² : Group G
inst✝¹ : MulAction G X
inst✝ : IsPreprimitive G X
A : Set X
hfA : A.Finite
hA : A.Nonempty
hA' : A ≠ Set.univ
a b : X
h : a ≠ b
B : Set X := ⋂ g, ⋂ (_ : a ∈ g • A), g • A
this : B = {a}
⊢ b ∉ {a}

Tactic:
;

Goal state after:
G : Type u_1
X : Type u_2
inst✝² : Group G
inst✝¹ : MulAction G X
inst✝ : IsPreprimitive G X
A : Set X
hfA : A.Finite
hA : A.Nonempty
hA' : A ≠ Set.univ
a b : X
h : a ≠ b
B : Set X := ⋂ g, ⋂ (_ : a ∈ g • A), g • A
this : B = {a}
⊢ b ∉ {a}
Module: Mathlib.GroupTheory.GroupAction.Primitive

Goal state before:
pp: G : Type u_1
X : Type u_2
inst✝² : Group G
inst✝¹ : MulAction G X
inst✝ : IsPreprimitive G X
A : Set X
hfA : A.Finite
hA : A.Nonempty
hA' : A ≠ Set.univ
a b : X
h : a ≠ b
B : Set X := ⋂ g, ⋂ (_ : a ∈ g • A), g • A
this : B = {a}
⊢ ¬b = a

Tactic:
;

Goal state after:
G : Type u_1
X : Type u_2
inst✝² : Group G
inst✝¹ : MulAction G X
inst✝ : IsPreprimitive G X
A : Set X
hfA : A.Finite
hA : A.Nonempty
hA' : A ≠ Set.univ
a b : X
h : a ≠ b
B : Set X := ⋂ g, ⋂ (_ : a ∈ g • A), g • A
this : B = {a}
⊢ ¬b = a
Module: Mathlib.GroupTheory.GroupAction.Primitive

Goal state before:
pp: case inl
G : Type u_1
X : Type u_2
inst✝² : Group G
inst✝¹ : MulAction G X
inst✝ : IsPreprimitive G X
A : Set X
hfA : A.Finite
hA : A.Nonempty
hA' : A ≠ Set.univ
a b : X
h : a ≠ b
B : Set X := ⋂ g, ⋂ (_ : a ∈ g • A), g • A
hyp : (⋂ k, ⋂ (_ : a ∈ k • A), k • A).Subsingleton
⊢ ∀ (i : G), a ∈ ⋂ (_ : a ∈ i • A), i • A

Tactic:
;

Goal state after:
case inl
G : Type u_1
X : Type u_2
inst✝² : Group G
inst✝¹ : MulAction G X
inst✝ : IsPreprimitive G X
A : Set X
hfA : A.Finite
hA : A.Nonempty
hA' : A ≠ Set.univ
a b : X
h : a ≠ b
B : Set X := ⋂ g, ⋂ (_ : a ∈ g • A), g • A
hyp : (⋂ k, ⋂ (_ : a ∈ k • A), k • A).Subsingleton
⊢ ∀ (i : G), a ∈ ⋂ (_ : a ∈ i • A), i • A