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

Goal state before:
pp: 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)
n : ℕ
⊢ (pi Set.univ Ss).lowerCentralSeries n = pi Set.univ fun i ↦ (Ss i).lowerCentralSeries n

Tactic:
by

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

Goal state before:
pp: G : Type u_1
X : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G X
a : X
⊢ IsPretransitive G X ↔ orbit G a = Set.univ

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: η : Type u_2
Gs : η → Type u_3
inst✝¹ : (i : η) → Group (Gs i)
inst✝ : Finite η
n : ℕ
⊢ ⊤.lowerCentralSeries n = pi Set.univ fun i ↦ ⊤.lowerCentralSeries n

Tactic:
by

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

Goal state before:
pp: M : Type u_3
N : Type u_4
α : Type u_5
β : Type u_6
inst✝³ : Monoid M
inst✝² : Monoid N
inst✝¹ : MulAction M α
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
h : IsPretransitive M α
x' : α
g : M
⊢ φ g • f x' = f (g • x')

Tactic:
by

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

Goal state before:
pp: M : Type u_3
N : Type u_4
α : Type u_5
β : Type u_6
inst✝³ : Monoid M
inst✝² : Monoid N
inst✝¹ : MulAction M α
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
h : IsPretransitive M α
⊢ IsPretransitive N β

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: G : Type u_1
inst✝⁴ : Group G
N : Subgroup G
inst✝³ : N.Normal
η : Type u_2
Gs : η → Type u_3
inst✝² : (i : η) → Group (Gs i)
inst✝¹ : Finite η
inst✝ : ∀ (i : η), IsNilpotent (Gs i)
⊢ IsNilpotent ((i : η) → Gs i)

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: η : Type u_2
Gs : η → Type u_3
inst✝² : (i : η) → Group (Gs i)
inst✝¹ : Fintype η
inst✝ : ∀ (i : η), IsNilpotent (Gs i)
⊢ nilpotencyClass ((i : η) → Gs i) = Finset.univ.sup fun i ↦ nilpotencyClass (Gs i)

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
N : Subgroup G
inst✝ : N.Normal
h : Group.IsNilpotent G
⊢ Group.IsSolvable G

Tactic:
by

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

Goal state before:
pp: M : Type u_3
N : Type u_4
α : Type u_5
β : Type u_6
inst✝³ : Monoid M
inst✝² : Monoid N
inst✝¹ : MulAction M α
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hφ : Function.Surjective φ
hf : Function.Bijective ⇑f
⊢ IsPretransitive M α ↔ IsPretransitive N β

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : IsNilpotent G
ih : ∀ (H : Subgroup (G ⧸ center G)), normalizer ↑H = H → H = ⊤
H : Subgroup G
hH : normalizer ↑H = H
hch : center G ≤ H
⊢ (mk' (center G)).ker ≤ H

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : IsNilpotent G
ih : ∀ (H : Subgroup (G ⧸ center G)), normalizer ↑H = H → H = ⊤
H : Subgroup G
hH : normalizer ↑H = H
hch : center G ≤ H
hkh : (mk' (center G)).ker ≤ H
hsur : Function.Surjective ⇑(mk' (center G))
H' : Subgroup (G ⧸ center G) := Subgroup.map (mk' (center G)) H
⊢ normalizer ↑H' = H'

Tactic:
by

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
a b : X
⊢ orbit G a = orbit G b ∨ Disjoint (orbit G a) (orbit G b)

Tactic:
by

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
⊢ (range fun x ↦ orbit G x).PairwiseDisjoint id

Tactic:
by

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
⊢ Setoid.IsPartition (range fun a ↦ orbit G a)

Tactic:
by

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

Goal state before:
pp: 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 : IsTrivialBlock B
⊢ IsTrivialBlock (⇑f '' B)

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: G : Type u_1
inst✝ : Group G
h : IsNilpotent G
⊢ NormalizerCondition G

Tactic:
by

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

Goal state before:
pp: 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 : IsTrivialBlock B
⊢ IsTrivialBlock (⇑f ⁻¹' B)

Tactic:
by

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

Goal state before:
pp: M : Type u_3
α : Type u_4
inst✝¹ : Group M
inst✝ : MulAction M α
B : Set α
hB : IsTrivialBlock B
g : M
⊢ IsTrivialBlock (g • B)

Tactic:
by

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:
by

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

Goal state before:
pp: M : Type u_3
α : Type u_4
inst✝¹ : Group M
inst✝ : MulAction M α
B : Set α
g : M
⊢ IsTrivialBlock (g • B) ↔ IsTrivialBlock B

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: G✝ : Type u_1
inst✝² : Finite G✝
p : ℕ
hp : Fact (Nat.Prime p)
G : Type u_1
inst✝¹ : Finite G
inst✝ : Nontrivial G
ih : ∀ (β : Type u_1) [Finite β], Nat.card β < Nat.card G → ∀ [hG : Group β], IsPGroup p β → Group.IsNilpotent β
hG : Group G
h : IsPGroup p G
⊢ Nat.card (G ⧸ center G) < Nat.card G

Tactic:
by

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

Goal state before:
pp: G : Type u_1
X : Type u_2
inst✝ : SMul G X
B : Set X
⊢ IsBlock G B ↔ ∀ ⦃g₁ g₂ : G⦄, (g₁ • B ∩ g₂ • B).Nonempty → g₁ • B = g₂ • B

Tactic:
by

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

Goal state before:
pp: G : Type u_1
X : Type u_2
inst✝ : SMul G X
B : Set X
g₁ g₂ : G
hB : IsBlock G B
hg : g₁ • B ⊆ g₂ • B
hg' : g₁ • B ≠ g₂ • B
⊢ B = ∅

Tactic:
by

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

Goal state before:
pp: G : Type u_1
X : Type u_2
inst✝ : SMul G X
B : Set X
g₁ g₂ : G
hB : IsBlock G B
hg : g₁ • B ⊆ g₂ • B
⊢ g₁ • B = g₂ • B

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: G : Type u_1
hG : Group G
inst✝ : Finite G
p : ℕ
hp : Fact (Nat.Prime p)
h : IsPGroup p G
⊢ Group.IsNilpotent G

Tactic:
by

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

Goal state before:
pp: G : Type u_1
X : Type u_2
inst✝ : SMul G X
B : Set X
s : Set G
g : G
hB : IsBlock G B
hgs : ¬g • B ⊆ s • B
⊢ Disjoint (g • B) (s • B)

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: G : Type u_1
hG : Group G
inst✝ : Finite G
e : ((p : ↥(Nat.card G).primeFactors) → (P : Sylow (↑p) G) → ↥↑P) ≃* G
ps : Finset ℕ := (Nat.card G).primeFactors
⊢ ∀ (p : ↥ps) (P : Sylow (↑p) G), IsNilpotent ↥↑P

Tactic:
by

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

Goal state before:
pp: G : Type u_1
X : Type u_2
inst✝ : SMul G X
B : Set X
hfB : IsFixedBlock G B
⊢ IsBlock G B

Tactic:
by

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

Goal state before:
pp: G : Type u_1
X : Type u_2
inst✝ : SMul G X
⊢ IsBlock G ∅

Tactic:
by

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

Goal state before:
pp: G : Type u_1
X : Type u_2
inst✝ : SMul G X
a : X
⊢ IsBlock G {a}

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: G : Type u_1
hG : Group G
inst✝ : Finite G
e : ((p : ↥(Nat.card G).primeFactors) → (P : Sylow (↑p) G) → ↥↑P) ≃* G
⊢ IsNilpotent G

Tactic:
by

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

Goal state before:
pp: M : Type u_1
X : Type u_2
inst✝¹ : Monoid M
inst✝ : MulAction M X
B : Set X
s : Set M
hB : IsBlock M B
hs : ¬B ⊆ s • B
⊢ ¬1 • B ⊆ s • B

Tactic:
by

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

Goal state before:
pp: M : Type u_1
X : Type u_2
inst✝¹ : Monoid M
inst✝ : MulAction M X
B : Set X
s : Set M
hB : IsBlock M B
hs : ¬B ⊆ s • B
⊢ Disjoint B (s • B)

Tactic:
by

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
B : Set X
hB : IsBlock G B
g : G
⊢ g • B ≠ B → Disjoint (g • B) B

Tactic:
by

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
B : Set X
⊢ IsBlock G B ↔ ∀ ⦃g : G⦄, g • B ≠ B → Disjoint (g • B) B

Tactic:
by

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
B : Set X
⊢ IsBlock G B ↔ ∀ ⦃g : G⦄, (g • B ∩ B).Nonempty → g • B = B

Tactic:
by

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
B : Set X
⊢ IsBlock G B ↔ ∀ ⦃g : G⦄ ⦃a : X⦄, a ∈ B → g • a ∈ B → g • B = B

Tactic:
by

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:
by

Goal state after:
Module: Mathlib.GroupTheory.Nilpotent

Goal state before:
pp: G : Type u_1
hG : Group G
inst✝ : Finite G
⊢ [IsNilpotent G, NormalizerCondition G, ∀ (H : Subgroup G), IsCoatom H → H.Normal,
      ∀ (p : ℕ), Fact (Nat.Prime p) → ∀ (P : Sylow p G), (↑P).Normal,
      Nonempty (((p : ↥(Nat.card G).primeFactors) → (P : Sylow (↑p) G) → ↥↑P) ≃* G)].TFAE

Tactic:
by

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
B₁ B₂ : Set X
h₁ : IsBlock G B₁
h₂ : IsBlock G B₂
⊢ IsBlock G (B₁ ∩ B₂)

Tactic:
by

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
ι : Sort u_3
B : ι → Set X
hB : ∀ (i : ι), IsBlock G (B i)
⊢ IsBlock G (⋂ i, B i)

Tactic:
by

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
B : Set X
hB : IsTrivialBlock B
⊢ IsBlock G B

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝³ : Group G
X : Type u_2
inst✝² : MulAction G X
B : Set X
H : Type u_3
Y : Type u_4
inst✝¹ : Group H
inst✝ : MulAction H Y
φ : H → G
j : Y →ₑ[φ] X
hB : IsBlock G B
⊢ IsBlock H (⇑j ⁻¹' B)

Tactic:
by

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
B : Set X
H : Type u_3
Y : Type u_4
inst✝ : SMul H Y
φ : G → H
j : X →ₑ[φ] Y
hφ : Function.Surjective φ
hj : Function.Injective ⇑j
hB : IsBlock G B
⊢ IsBlock H (⇑j '' B)

Tactic:
by

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
C : SubMulAction G X
B : Set ↥C
⊢ IsBlock G (Subtype.val '' B) ↔ IsBlock G B

Tactic:
by

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 * (‖↑G‖ * ‖G.symm w‖) ^ 2 + ((g' x) w) w = δ * C * ‖G.symm w‖ ^ 2 + ((g' x) w) w

Tactic:
by

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
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 = g • B ∨ Disjoint (h' • g • B) (g • B)

Tactic:
by

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
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 = g • h • B

Tactic:
by

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
B : Set X
H : Subgroup G
hB : IsBlock (↥H) B
g : G
⊢ IsBlock (↥(Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) H)) (g • B)

Tactic:
by

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
B : Set X
g : G
hB : IsBlock G B
⊢ IsBlock G (g • B)

Tactic:
by

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
B : Set X
hGX : IsPretransitive G X
hB : IsBlock G B
hBe : B.Nonempty
⊢ IsBlockSystem G (range fun g ↦ g • B)

Tactic:
by

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
N : Subgroup G
nN : N.Normal
a : X
g : G
⊢ g • orbit (↥N) a = orbit (↥N) (g • a)

Tactic:
by

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
⊢ IsBlock G (orbit (↥N) a)

Tactic:
by

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
⊢ IsBlockSystem G (range fun a ↦ orbit (↥N) a)

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝⁵ : Group G
S : Type u_3
H : Type u_4
inst✝⁴ : Group H
inst✝³ : SetLike S H
inst✝² : SubgroupClass S H
s : S
inst✝¹ : MulAction G H
inst✝ : IsScalarTower G H H
⊢ IsBlock G ↑s

Tactic:
by

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 * C * ‖G.symm w‖ ^ 2 + ((g' x) w) w = δ * C * ((g x) (G.symm w)) (G.symm w) + ((g' x) w) w

Tactic:
by

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

Goal state before:
pp: G : Type u_1
inst✝⁵ : Group G
S : Type u_3
H : Type u_4
inst✝⁴ : Group H
inst✝³ : SetLike S H
inst✝² : SubgroupClass S H
s : S
inst✝¹ : MulAction G H
inst✝ : IsScalarTower G Hᵐᵒᵖ H
⊢ IsBlock G ↑s

Tactic:
by

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

Goal state before:
pp: M : Type u_1
inst✝² : Group M
α : Type u_2
inst✝¹ : MulAction M α
IwaS : IwasawaStructure M α
inst✝ : IsQuasiPreprimitive M α
N : Subgroup M
nN : N.Normal
hNX : fixedPoints (↥N) α ≠ Set.univ
is_transN : IsPretransitive (↥N) α
ntα : Nontrivial α
a : α
⊢ ⊤ ≤ N ⊔ IwaS.T a

Tactic:
← IwaS.is_generator,

Goal state after:
M : Type u_1
inst✝² : Group M
α : Type u_2
inst✝¹ : MulAction M α
IwaS : IwasawaStructure M α
inst✝ : IsQuasiPreprimitive M α
N : Subgroup M
nN : N.Normal
hNX : fixedPoints (↥N) α ≠ Set.univ
is_transN : IsPretransitive (↥N) α
ntα : Nontrivial α
a : α
⊢ iSup IwaS.T ≤ N ⊔ IwaS.T a
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
H : Subgroup G
a : X
hH : stabilizer G a ≤ H
g : G
h₁ h₂ : ↥H
h : (fun x ↦ g • x) ((fun m ↦ m • a) h₁) = (fun m ↦ m • a) h₂
this : g ∈ H
⊢ g • orbit (↥H) a = orbit (↥H) a

Tactic:
by

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
H : Subgroup G
a : X
hH : stabilizer G a ≤ H
⊢ IsBlock G (orbit (↥H) a)

Tactic:
by

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

Goal state before:
pp: M : Type u_1
inst✝² : Group M
α : Type u_2
inst✝¹ : MulAction M α
IwaS : IwasawaStructure M α
inst✝ : IsQuasiPreprimitive M α
N : Subgroup M
nN : N.Normal
hNX : fixedPoints (↥N) α ≠ Set.univ
⊢ commutator M ≤ N

Tactic:
by

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
B : Set X
hB : IsBlock G B
a : X
ha : a ∈ B
g : G
hg : g ∈ stabilizer G a
⊢ a ∈ g • B

Tactic:
by

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:
by

Goal state after:
Module: Mathlib.GroupTheory.IndexNormal

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : H.index = 1
⊢ H.Normal

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.IndexNormal

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : H.index = 2
x : G
hxH : x ∈ H
g : G
⊢ g * x * g⁻¹ ∈ H

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.IndexNormal

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H : Subgroup G
hHp : H.index = (Nat.card G).minFac
hG0 : ¬Nat.card G = 0
hG1 : ¬Nat.card G = 1
this : H.normalCore.relIndex H = 1
⊢ H.Normal

Tactic:
by

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

Goal state before:
pp: case mpr
G : Type u_1
inst✝² : Group G
X : Type u_2
inst✝¹ : MulAction G X
B : Set X
inst✝ : IsPretransitive G X
hB : IsBlock G B
a : X
ha : a ∈ B
x : X
⊢ x ∈ B → x ∈ orbit (↥(stabilizer G B)) a

Tactic:
· intro hx
    obtain ⟨k, rfl⟩ := exists_smul_eq G a x
    exact ⟨⟨k, hB.smul_eq_of_mem ha hx⟩, rfl⟩

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
B : Set X
inst✝ : IsPretransitive G X
hB : IsBlock G B
a : X
ha : a ∈ B
⊢ orbit (↥(stabilizer G B)) a = B

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.IndexNormal

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H : Subgroup G
hHp : H.index = (Nat.card G).minFac
hG0 : ¬Nat.card G = 0
hG1 : ¬Nat.card G = 1
this : Finite G
index_ne_zero : H.index ≠ 0
hp : Nat.Prime H.index
⊢ H.normalCore.index ∣ H.index !

Tactic:
by

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
a : X
H : Subgroup G
hH : stabilizer G a ≤ H
⊢ stabilizer G (orbit (↥H) a) = H

Tactic:
by

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
B✝ : Set X
htGX : IsPretransitive G X
a : X
B : Set X
ha : a ∈ B
hB : IsBlock G B
B' : Set X
ha' : a ∈ B'
hB' : IsBlock G B'
hBB' : ⟨stabilizer G B, ⋯⟩ ≤ ⟨stabilizer G B', ⋯⟩
k : G
hb : k • a ∈ B
this : k ∈ stabilizer G B'
⊢ k • a ∈ B'

Tactic:
by

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
B : Set X
htGX : IsPretransitive G X
a : X
⊢ ∀ {a_1 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 := ⋯ }
          a_1 ≤
        {
            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 ↔
      a_1 ≤ b

Tactic:
by

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
B : Set X
a : X
⊢ ∀ (a_1 : BlockMem G a), a_1 ≤ ⟨univ, ⋯⟩

Tactic:
by

Goal state after: