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

Goal state before:
pp: case zero
G : Type u_1
inst✝¹ : Group G
S : Subgroup G
K : Type u_2
inst✝ : Group K
f : G →* K
⊢ map f (S.lowerCentralSeries 0) = (map f S).lowerCentralSeries 0

Tactic:
simp

Goal state after:
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: case refine_1
G : Type u_1
inst✝ : Group G
hG : Group.IsNilpotent G
n : ℕ
H : ℕ → Subgroup G
hH : IsDescendingCentralSeries H
hn : H n = ⊥
⊢ ⊤.lowerCentralSeries n = ⊥

Tactic:
rw [← le_bot_iff, ← hn]

Goal state after:
case refine_1
G : Type u_1
inst✝ : Group G
hG : Group.IsNilpotent G
n : ℕ
H : ℕ → Subgroup G
hH : IsDescendingCentralSeries H
hn : H n = ⊥
⊢ ⊤.lowerCentralSeries n ≤ H n
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: case zero
G : Type u_1
inst✝¹ : Group G
H : Type u_2
inst✝ : Group H
f : G →* H
h : Function.Surjective ⇑f
⊢ Subgroup.map f (upperCentralSeries G 0) ≤ upperCentralSeries H 0

Tactic:
simp

Goal state after:
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: G : Type u_1
inst✝ : Group G
S : Subgroup G
n : ℕ
h : S.lowerCentralSeries n ≤ center G
⊢ ⁅S.lowerCentralSeries n, S⁆ = ⊥

Tactic:
commutator_def,

Goal state after:
G : Type u_1
inst✝ : Group G
S : Subgroup G
n : ℕ
h : S.lowerCentralSeries n ≤ center G
⊢ closure {g | ∃ g₁ ∈ S.lowerCentralSeries n, ∃ g₂ ∈ S, ⁅g₁, g₂⁆ = g} = ⊥
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: case a
G : Type u_1
inst✝ : Group G
hH : IsNilpotent G
n : ℕ
hn : nilpotencyClass G = n + 1
⊢ n ≤ nilpotencyClass (G ⧸ center G)

Tactic:
apply le_of_add_le_add_right

Goal state after:
case bc
G : Type u_1
inst✝ : Group G
hH : IsNilpotent G
n : ℕ
hn : nilpotencyClass G = n + 1
⊢ n + ?a.a✝ ≤ nilpotencyClass (G ⧸ center G) + ?a.a✝

case a.a
G : Type u_1
inst✝ : Group G
hH : IsNilpotent G
n : ℕ
hn : nilpotencyClass G = n + 1
⊢ ℕ
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: case succ
G : Type u_1
inst✝¹ : Group G
hH : IsNilpotent G
inst✝ : Nontrivial G
n✝ : ℕ
h : nilpotencyClass G = n✝ + 1
⊢ n✝ + 1 = n✝ + 1 - 1 + 1

Tactic:
simp

Goal state after:
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: case zero
G : Type u_1
inst✝ : Group G
⊢ derivedSeries G 0 ≤ ⊤.lowerCentralSeries 0

Tactic:
simp

Goal state after:
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:
simp

Goal state after:
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: case neg
G : Type u_1
inst✝ : Group G
a b : ℕ
ab : a < b
hn : upperCentralSeries G a = upperCentralSeries G b
hG : ¬Group.IsNilpotent G
⊢ 0 ≤ a

Tactic:
apply Nat.zero_le

Goal state after:
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: G : Type u_1
inst✝² : Group G
inst✝¹ : Nontrivial G
inst✝ : IsNilpotent G
⊢ ¬upperCentralSeries G 0 = ⊤

Tactic:
simp

Goal state after:
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: case zero
G₁ : Type u_2
G₂ : Type u_3
inst✝¹ : Group G₁
inst✝ : Group G₂
S₁ : Subgroup G₁
S₂ : Subgroup G₂
⊢ (S₁.prod S₂).lowerCentralSeries 0 = (S₁.lowerCentralSeries 0).prod (S₂.lowerCentralSeries 0)

Tactic:
simp

Goal state after:
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: case zero
η : Type u_2
Gs : η → Type u_3
inst✝ : (i : η) → Group (Gs i)
Ss : (i : η) → Subgroup (Gs i)
⊢ (pi Set.univ Ss).lowerCentralSeries 0 ≤ pi Set.univ fun i ↦ (Ss i).lowerCentralSeries 0

Tactic:
simp

Goal state after:
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: case zero
η : Type u_2
Gs : η → Type u_3
inst✝¹ : (i : η) → Group (Gs i)
inst✝ : Finite η
Ss : (i : η) → Subgroup (Gs i)
⊢ (pi Set.univ Ss).lowerCentralSeries 0 = pi Set.univ fun i ↦ (Ss i).lowerCentralSeries 0

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.VectorBundle.Riemannian

Goal state before:
pp: B : Type u_1
inst✝⁷ : TopologicalSpace B
F : Type u_2
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
E : B → Type u_3
inst✝⁴ : TopologicalSpace (TotalSpace F E)
inst✝³ : (x : B) → NormedAddCommGroup (E x)
inst✝² : (x : B) → InnerProductSpace ℝ (E x)
inst✝¹ : FiberBundle F E
inst✝ : VectorBundle ℝ F E
h : IsContinuousRiemannianBundle F E
x : B
r : ℝ
hr : 1 < r
r' : ℝ
hr' : 1 < r'
r'r : r' < r
h'x : x ∈ (trivializationAt F E x).baseSet
G : E x ≃L[ℝ] F := Trivialization.continuousLinearEquivAt ℝ (trivializationAt F E x) x h'x
C : ℝ := ‖↑G‖ ^ 2
δ : ℝ
δpos : 0 < δ
h'δ : 1 + δ * C < r' ^ 2
g : (x : B) → E x →L[ℝ] E x →L[ℝ] ℝ
g_cont :
  Continuous[inst✝⁷, topologicalSpaceTotalSpace (RingHom.id ℝ) F E (F →L[ℝ] ℝ) fun x ↦ E x →L[ℝ] ℝ] fun x ↦
    TotalSpace.mk' (F →L[ℝ] F →L[ℝ] ℝ) x (g x)
hg : ∀ (x : B) (v w : E x), ⟪v, w⟫ = ((g x) v) w
g' : B → F →L[ℝ] F →L[ℝ] ℝ := fun y ↦ inCoordinates F E (F →L[ℝ] ℝ) (fun x ↦ E x →L[ℝ] ℝ) x y x y (g y)
hg' : ContinuousAt g' x
this✝ : ∀ᶠ (y : B) in 𝓝 x, dist (g' y) (g' x) < δ
y : B
hy : dist (g' y) (g' x) < δ
h'y : y ∈ (trivializationAt F E x).baseSet
this : ‖g' y - g' x‖ ≤ δ
v : E x
w : F := (Trivialization.continuousLinearMapAt ℝ (trivializationAt F E x) x) v
hgx : ((g x) v) v = ((g' x) w) w
hgy : ((g y) ((trivializationAt F E x).symm y w)) ((trivializationAt F E x).symm y w) = ((g' y) w) w
⊢ δ * ‖w‖ ^ 2 + ((g' x) w) w ≤ δ * ‖↑G (G.symm w)‖ ^ 2 + ((g' x) w) w

Tactic:
simp

Goal state after:
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
x✝ : G
⊢ x✝ • Set.univ = Set.univ

Tactic:
simp

Goal state after:
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
a : X
g : G
⊢ orbit (↥N) (g • a) = orbit (↥N) a ∨ Disjoint (orbit (↥N) (g • a)) (orbit (↥N) a)

Tactic:
apply orbit.eq_or_disjoint

Goal state after:
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:
simp

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

Goal state before:
pp: case e_a
G : Type u_1
inst✝¹ : Group G
α : Type u_2
inst✝ : MulAction G α
a : α
⊢ Cardinal.toENat (Cardinal.mk ↑{a}) = 1

Tactic:
simp

Goal state after:
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
⊢ orbit (↥N) a = {a}

Tactic:
ext b

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
b : α
⊢ b ∈ orbit (↥N) a ↔ b ∈ {a}
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
hfin : Fintype ↑(Set.range fun g ↦ g • B)
⊢ ↑(Set.range fun g ↦ g • B).ncard ≤ (Set.range fun m ↦ m • B).ncard

Tactic:
Nat.cast_id

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
hfin : Fintype ↑(Set.range fun g ↦ g • B)
⊢ (Set.range fun g ↦ g • B).ncard ≤ (Set.range fun m ↦ m • B).ncard
Module: Mathlib.SetTheory.Cardinal.Embedding

Goal state before:
pp: α : Type u_1
m n : ℕ
inst✝ : Finite α
hmn : m ≤ n
hn : n ≤ Nat.card α
⊢ Surjective fun x ↦ (castLEEmb hmn).trans x

Tactic:
obtain ⟨k, rfl⟩ := Nat.exists_eq_add_of_le hmn
  exact Fin.Embedding.restrictSurjective_of_add_le_natCard hn

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

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
h2 : Nat.card α ≤ 2
a✝ : Nontrivial α
hα' : Nat.card α = 2
⊢ 2 ≠ 0

Tactic:
simp

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

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
n : ℕ
⊢ 3 ≤ Nat.card (Fin (n + 3))

Tactic:
simp

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

Goal state before:
pp: g : Perm (Fin 5)
ha : g ∈ alternatingGroup (Fin 5)
h1 : g ≠ 1
h2 : g.cycleType = Multiset.replicate g.cycleType.card 2
h : g.cycleType.card * 2 ≤ card (Fin 5)
⊢ 0 < 2

Tactic:
simp

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

Goal state before:
pp: case «2»
g : Perm (Fin 5)
h1 : g ≠ 1
h_1 : g.cycleType.card = 2
ha : Even 2
h2 : g.cycleType = Multiset.replicate 2 2
h✝ : 2 * 2 ≤ card (Fin 5)
h : 2 ≤ 3
h04 : 0 ≠ 4
h13 : 1 ≠ 3
⊢ {2} + {2} = Multiset.replicate 2 2

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.VectorBundle.Riemannian

Goal state before:
pp: B : Type u_1
inst✝⁷ : TopologicalSpace B
F : Type u_2
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
E : B → Type u_3
inst✝⁴ : TopologicalSpace (TotalSpace F E)
inst✝³ : (x : B) → NormedAddCommGroup (E x)
inst✝² : (x : B) → InnerProductSpace ℝ (E x)
inst✝¹ : FiberBundle F E
inst✝ : VectorBundle ℝ F E
h : IsContinuousRiemannianBundle F E
x y : B
hy :
  ‖Trivialization.symmL ℝ (trivializationAt F E x) y ∘SL
        Trivialization.continuousLinearMapAt ℝ (trivializationAt F E x) x‖ <
    2
A :
  Trivialization.continuousLinearMapAt ℝ (trivializationAt F E x) x ∘SL
      Trivialization.symmL ℝ (trivializationAt F E x) x =
    ContinuousLinearMap.id ℝ F
⊢ Trivialization.symmL ℝ (trivializationAt F E x) y =
    Trivialization.symmL ℝ (trivializationAt F E x) y ∘SL ContinuousLinearMap.id ℝ F

Tactic:
simp

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

Goal state before:
pp: case mp
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
⊢ IsMultiplyPretransitive G α 2 → ∀ {a b c d : α}, a ≠ b → c ≠ d → ∃ g, g • a = c ∧ g • b = d

Tactic:
intro _ a b c d h h'

Goal state after:
case mp
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
a✝ : IsMultiplyPretransitive G α 2
a b c d : α
h : a ≠ b
h' : c ≠ d
⊢ ∃ g, g • a = c ∧ g • b = d
Module: Mathlib.GroupTheory.SpecificGroups.Alternating

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Finset α
⊢ ⟨Perm.ofSubtype ↑1, ⋯⟩ = 1

Tactic:
simp

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

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Finset α
⊢ ∀ (x y : ↥(alternatingGroup ↥s)), ⟨Perm.ofSubtype ↑(x * y), ⋯⟩ = ⟨Perm.ofSubtype ↑x, ⋯⟩ * ⟨Perm.ofSubtype ↑y, ⋯⟩

Tactic:
simp

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

Goal state before:
pp: case mpr
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
H : ∀ {a b c d : α}, a ≠ b → c ≠ d → ∃ g, g • a = c ∧ g • b = d
⊢ ∀ (x y : Fin 2 ↪ α), ∃ g, g • x = y

Tactic:
intro j

Goal state after:
case mpr
G : Type u_1
α : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
H : ∀ {a b c d : α}, a ≠ b → c ≠ d → ∃ g, g • a = c ∧ g • b = d
j : Fin 2 ↪ α
⊢ ∀ (y : Fin 2 ↪ α), ∃ g, g • j = y
Module: Mathlib.GroupTheory.SpecificGroups.Alternating

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Finset α
k : ↥(alternatingGroup α)
⊢ ↑(↑k).support ⊆ {x | x ∈ s} ↔ (↑k).support ⊆ s

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.VectorBundle.Riemannian

Goal state before:
pp: case inl
B✝ : Type u_1
inst✝¹⁸ : TopologicalSpace B✝
F✝ : Type u_2
inst✝¹⁷ : NormedAddCommGroup F✝
inst✝¹⁶ : NormedSpace ℝ F✝
E✝ : B✝ → Type u_3
inst✝¹⁵ : TopologicalSpace (TotalSpace F✝ E✝)
inst✝¹⁴ : (x : B✝) → NormedAddCommGroup (E✝ x)
inst✝¹³ : (x : B✝) → InnerProductSpace ℝ (E✝ x)
inst✝¹² : FiberBundle F✝ E✝
inst✝¹¹ : VectorBundle ℝ F✝ E✝
B : Type u_4
inst✝¹⁰ : TopologicalSpace B
F : Type u_5
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace ℝ F
E : B → Type u_6
inst✝⁷ : TopologicalSpace (TotalSpace F E)
inst✝⁶ : (b : B) → TopologicalSpace (E b)
inst✝⁵ : (b : B) → AddCommGroup (E b)
inst✝⁴ : (b : B) → Module ℝ (E b)
inst✝³ : ∀ (b : B), IsTopologicalAddGroup (E b)
inst✝² : ∀ (b : B), ContinuousConstSMul ℝ (E b)
inst✝¹ : FiberBundle F E
inst✝ : VectorBundle ℝ F E
g : RiemannianMetric E
b : B
⊢ 0 ≤ RCLike.re (((g.inner b) 0) 0)

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.VectorBundle.Riemannian

Goal state before:
pp: B✝ : Type u_1
inst✝¹⁸ : TopologicalSpace B✝
F✝ : Type u_2
inst✝¹⁷ : NormedAddCommGroup F✝
inst✝¹⁶ : NormedSpace ℝ F✝
E✝ : B✝ → Type u_3
inst✝¹⁵ : TopologicalSpace (TotalSpace F✝ E✝)
inst✝¹⁴ : (x : B✝) → NormedAddCommGroup (E✝ x)
inst✝¹³ : (x : B✝) → InnerProductSpace ℝ (E✝ x)
inst✝¹² : FiberBundle F✝ E✝
inst✝¹¹ : VectorBundle ℝ F✝ E✝
B : Type u_4
inst✝¹⁰ : TopologicalSpace B
F : Type u_5
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace ℝ F
E : B → Type u_6
inst✝⁷ : TopologicalSpace (TotalSpace F E)
inst✝⁶ : (b : B) → TopologicalSpace (E b)
inst✝⁵ : (b : B) → AddCommGroup (E b)
inst✝⁴ : (b : B) → Module ℝ (E b)
inst✝³ : ∀ (b : B), IsTopologicalAddGroup (E b)
inst✝² : ∀ (b : B), ContinuousConstSMul ℝ (E b)
inst✝¹ : FiberBundle F E
inst✝ : VectorBundle ℝ F E
g : RiemannianMetric E
b : B
v w x : E b
⊢ ((g.inner b) (v + w)) x = ((g.inner b) v) x + ((g.inner b) w) x

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.VectorBundle.Riemannian

Goal state before:
pp: B✝ : Type u_1
inst✝¹⁸ : TopologicalSpace B✝
F✝ : Type u_2
inst✝¹⁷ : NormedAddCommGroup F✝
inst✝¹⁶ : NormedSpace ℝ F✝
E✝ : B✝ → Type u_3
inst✝¹⁵ : TopologicalSpace (TotalSpace F✝ E✝)
inst✝¹⁴ : (x : B✝) → NormedAddCommGroup (E✝ x)
inst✝¹³ : (x : B✝) → InnerProductSpace ℝ (E✝ x)
inst✝¹² : FiberBundle F✝ E✝
inst✝¹¹ : VectorBundle ℝ F✝ E✝
B : Type u_4
inst✝¹⁰ : TopologicalSpace B
F : Type u_5
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace ℝ F
E : B → Type u_6
inst✝⁷ : TopologicalSpace (TotalSpace F E)
inst✝⁶ : (b : B) → TopologicalSpace (E b)
inst✝⁵ : (b : B) → AddCommGroup (E b)
inst✝⁴ : (b : B) → Module ℝ (E b)
inst✝³ : ∀ (b : B), IsTopologicalAddGroup (E b)
inst✝² : ∀ (b : B), ContinuousConstSMul ℝ (E b)
inst✝¹ : FiberBundle F E
inst✝ : VectorBundle ℝ F E
g : RiemannianMetric E
b : B
c v : E b
⊢ ∀ (r : ℝ), ((g.inner b) (r • c)) v = (starRingEnd ℝ) r * ((g.inner b) c) v

Tactic:
simp

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

Goal state before:
pp: G : Type u_1
α : Type u_2
inst✝² : Group G
inst✝¹ : MulAction G α
m n : ℕ
Hn : IsMultiplyPretransitive G α n
s : Set α
inst✝ : Finite ↑s
hmn : s.ncard + m = n
x y : Fin m ↪ ↥(ofFixingSubgroup G s)
this : IsMultiplyPretransitive G α (s.ncard + m)
Hs : Nonempty (Fin s.ncard ≃ ↑s)
x' : Fin (s.ncard + m) ↪ α := append x
hx : x' = append x
y' : Fin (s.ncard + m) ↪ α := append y
hy : y' = append y
g : G
hg : g • x' = y'
a : ↑s
i : Fin s.ncard := (Classical.choice Hs).symm a
ha : (Classical.choice Hs) i = a
⊢ g • ↑a = ↑a

Tactic:
rw [← ha]

Goal state after:
G : Type u_1
α : Type u_2
inst✝² : Group G
inst✝¹ : MulAction G α
m n : ℕ
Hn : IsMultiplyPretransitive G α n
s : Set α
inst✝ : Finite ↑s
hmn : s.ncard + m = n
x y : Fin m ↪ ↥(ofFixingSubgroup G s)
this : IsMultiplyPretransitive G α (s.ncard + m)
Hs : Nonempty (Fin s.ncard ≃ ↑s)
x' : Fin (s.ncard + m) ↪ α := append x
hx : x' = append x
y' : Fin (s.ncard + m) ↪ α := append y
hy : y' = append y
g : G
hg : g • x' = y'
a : ↑s
i : Fin s.ncard := (Classical.choice Hs).symm a
ha : (Classical.choice Hs) i = a
⊢ g • ↑((Classical.choice Hs) i) = ↑((Classical.choice Hs) i)
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:
apply Nat.zero_le

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:
simp

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:
simp

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:
simp

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:
simp

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
⊢ (insert a (Subtype.val '' s)).encard + 1 = ↑n.succ

Tactic:
rw [Set.encard_insert_of_notMem, Subtype.coe_injective.encard_image, hs, Nat.cast_succ]

Goal state after:
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
⊢ a ∉ Subtype.val '' s
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:
simp

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:
simp

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
a : α
x : ↥(ofFixingSubgroup M {a})
⊢ ↑x ∈ ofStabilizer M a

Tactic:
simp

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

Goal state before:
pp: case inr.left
M : Type u_1
α : Type u_2
inst✝² : Group M
inst✝¹ : MulAction M α
n : ℕ
hα : ↑n.succ ≤ ENat.card α
inst✝ : IsMultiplyPretransitive M α n.succ
hn : n > 0
⊢ IsMultiplyPretransitive M α n

Tactic:
exact isMultiplyPretransitive_of_le' (Nat.le_succ n) hα

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

Goal state before:
pp: case left
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
⊢ IsMultiplyPretransitive M α n

pp: case right
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}, s.encard + 1 = ↑n → IsPreprimitive ↥(fixingSubgroup M s) ↥(ofFixingSubgroup M s)

Tactic:
· exact (IsPretransitive.of_embedding_congr hφ hf).mpr H.isMultiplyPretransitive

Goal state after:
case right
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}, s.encard + 1 = ↑n → IsPreprimitive ↥(fixingSubgroup M s) ↥(ofFixingSubgroup M s)
Module: Mathlib.GroupTheory.Perm.MaximalSubgroups

Goal state before:
pp: case h
α : Type u_2
s : Set α
inst✝ : DecidablePred fun x ↦ x ∈ s
g : Perm ↑s
g' : α
a : g' ∈ s
⊢ ↑(g⁻¹ ⟨g', a⟩) ∈ s ∧ (ofSubtype g) ↑(g⁻¹ ⟨g', a⟩) = g'

Tactic:
simp

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:
classical
  obtain ⟨a, ha⟩ := hs
  obtain ⟨b, hb⟩ := hsc
  intro h
  rw [Set.mem_compl_iff] at hb; apply hb
  have hg : swap a b ∈ stabilizer (Perm α) s := by simp_all
  rw [mem_stabilizer_iff] at hg
  rw [← hg, Set.mem_smul_set]
  aesop

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

Goal state before:
pp: case right
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 : 1 < sᶜ.encard
g : Perm α
hg : g.IsSwap
hg' : g ∈ stabilizer (Perm α) s
⊢ g ∈ G

Tactic:
exact hG.le hg'

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 B : Set α
G : Subgroup ↥(alternatingGroup α)
hs : s.Nontrivial
hB_ss_sc : B ⊂ sᶜ
hG : stabilizer (↥(alternatingGroup α)) s ≤ G
hB : IsBlock (↥G) B
g : Perm ↑sᶜ
⊢ ∃ a, toPerm a = g

Tactic:
obtain ⟨⟨k, hk⟩, rfl⟩ := alternatingGroup.stabilizer.surjective_toPerm (by rwa [compl_compl]) g

Goal state after:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s B : Set α
G : Subgroup ↥(alternatingGroup α)
hs : s.Nontrivial
hB_ss_sc : B ⊂ sᶜ
hG : stabilizer (↥(alternatingGroup α)) s ≤ G
hB : IsBlock (↥G) B
k : ↥(alternatingGroup α)
hk : k ∈ stabilizer (↥(alternatingGroup α)) sᶜ
⊢ ∃ a, toPerm a = toPerm ⟨k, hk⟩
Module: Mathlib.GroupTheory.Perm.MaximalSubgroups

Goal state before:
pp: M : Type u_1
α : Type u_2
inst✝³ : Group M
inst✝² : MulAction M α
s : Set α
inst✝¹ : Finite α
inst✝ : IsMultiplyPretransitive M α (s.ncard + 1)
G : Subgroup M
hG : stabilizer M s ≤ G
B : Set α
hBs : ¬B ⊆ s
hBsc : ¬B ⊆ sᶜ
hB : IsBlock (↥G) B
a : α
ha : a ∈ B
ha' : a ∈ s
b : α
hb : b ∈ B
hb' : b ∈ sᶜ
x : α
hx' : x ∈ sᶜ
⊢ ∃ k, k • b = x

Tactic:
suffices h : IsPretransitive (fixingSubgroup M s) (ofFixingSubgroup M s) by
      obtain ⟨k, hk⟩ := h.exists_smul_eq (⟨b, hb'⟩ : ofFixingSubgroup M s) ⟨x, hx'⟩
      rw [← Subtype.coe_inj, val_smul] at hk
      exact ⟨k, hk⟩

Goal state after:
M : Type u_1
α : Type u_2
inst✝³ : Group M
inst✝² : MulAction M α
s : Set α
inst✝¹ : Finite α
inst✝ : IsMultiplyPretransitive M α (s.ncard + 1)
G : Subgroup M
hG : stabilizer M s ≤ G
B : Set α
hBs : ¬B ⊆ s
hBsc : ¬B ⊆ sᶜ
hB : IsBlock (↥G) B
a : α
ha : a ∈ B
ha' : a ∈ s
b : α
hb : b ∈ B
hb' : b ∈ sᶜ
x : α
hx' : x ∈ sᶜ
⊢ IsPretransitive ↥(fixingSubgroup M s) ↥(ofFixingSubgroup M s)
Module: Mathlib.GroupTheory.GroupAction.Jordan

Goal state before:
pp: α : Type u_1
K : Type u_2
inst✝¹ : Group K
inst✝ : MulAction K α
hα : Nat.card α = 2
hK : fixedPoints K α ≠ _root_.Set.univ
n : ℕ
x✝ : Infinite α
⊢ False

Tactic:
simp [Nat.card_eq_zero_of_infinite] at hα

Goal state after: