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rubin-lean4/Rubin/RigidStabilizerBasis.lean

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/-
This file describes [`RigidStabilizerBasis`], which are all non-trivial subgroups of `G` constructed
by finite intersections of `RigidStabilizer G (RegularSupport α g)`.
-/
import Mathlib.Topology.Basic
import Mathlib.Topology.Homeomorph
import Rubin.RegularSupport
import Rubin.RigidStabilizer
namespace Rubin
variable {G α : Type _}
variable [Group G]
variable [MulAction G α]
variable [TopologicalSpace α]
/--
Finite intersections of rigid stabilizers of regular supports
(which are equivalent to the rigid stabilizers of finite intersections of regular supports).
-/
def RigidStabilizerInter₀ {G: Type _} (α : Type _) [Group G] [MulAction G α] [TopologicalSpace α]
(S : Finset G) : Subgroup G :=
⨅ (g ∈ S), RigidStabilizer G (RegularSupport α g)
theorem RigidStabilizerInter₀.eq_sInter (S : Finset G) :
RigidStabilizerInter₀ α S = RigidStabilizer G (⋂ g ∈ S, (RegularSupport α g)) :=
by
have img_eq : ⋂ g ∈ S, RegularSupport α g = ⋂₀ ((fun g : G => RegularSupport α g) '' S) :=
by simp only [Set.sInter_image, Finset.mem_coe]
rw [img_eq]
rw [rigidStabilizer_sInter]
unfold RigidStabilizerInter₀
ext x
repeat rw [Subgroup.mem_iInf]
constructor
· intro H T
rw [Subgroup.mem_iInf]
intro T_in_img
simp at T_in_img
let ⟨g, ⟨g_in_S, T_eq⟩⟩ := T_in_img
specialize H g
rw [Subgroup.mem_iInf] at H
rw [<-T_eq]
apply H; assumption
· intro H g
rw [Subgroup.mem_iInf]
intro g_in_S
specialize H (RegularSupport α g)
rw [Subgroup.mem_iInf] at H
simp at H
apply H g <;> tauto
/--
A predecessor structure to [`RigidStabilizerBasis`], where equality is defined on the choice of
group elements who regular support's rigid stabilizer get intersected upon.
--/
structure RigidStabilizerBasis₀ (G α : Type _) [Group G] [MulAction G α] [TopologicalSpace α] where
seed : Finset G
val_ne_bot : RigidStabilizerInter₀ α seed ≠ ⊥
def RigidStabilizerBasis₀.val (B : RigidStabilizerBasis₀ G α) : Subgroup G :=
RigidStabilizerInter₀ α B.seed
theorem RigidStabilizerBasis₀.val_def (B : RigidStabilizerBasis₀ G α) : B.val = RigidStabilizerInter₀ α B.seed := rfl
/--
The set of all non-trivial subgroups of `G` constructed
by finite intersections of `RigidStabilizer G (RegularSupport α g)`.
--/
def RigidStabilizerBasis (G α : Type _) [Group G] [MulAction G α] [TopologicalSpace α] : Set (Subgroup G) :=
{ H.val | H : RigidStabilizerBasis₀ G α }
theorem RigidStabilizerBasis.mem_iff (H : Subgroup G) :
H ∈ RigidStabilizerBasis G α ↔ ∃ B : RigidStabilizerBasis₀ G α, B.val = H := by rfl
theorem RigidStabilizerBasis.mem_iff' (H : Subgroup G)
(H_ne_bot : H ≠ ⊥) :
H ∈ RigidStabilizerBasis G α ↔ ∃ seed : Finset G, RigidStabilizerInter₀ α seed = H :=
by
rw [mem_iff]
constructor
· intro ⟨B, B_eq⟩
use B.seed
rw [RigidStabilizerBasis₀.val_def] at B_eq
exact B_eq
· intro ⟨seed, seed_eq⟩
let B := RigidStabilizerInter₀ α seed
have val_ne_bot : B ≠ ⊥ := by
unfold_let
rw [seed_eq]
exact H_ne_bot
use ⟨seed, val_ne_bot⟩
rw [<-seed_eq]
rfl
def RigidStabilizerBasis.asSets (G α : Type _) [Group G] [MulAction G α] [TopologicalSpace α] : Set (Set G) :=
{ (H.val : Set G) | H : RigidStabilizerBasis₀ G α }
theorem RigidStabilizerBasis.mem_asSets_iff (S : Set G) :
S ∈ RigidStabilizerBasis.asSets G α ↔ ∃ H ∈ RigidStabilizerBasis G α, H.carrier = S :=
by
unfold asSets RigidStabilizerBasis
simp
rfl
theorem RigidStabilizerBasis.mem_iff_asSets (H : Subgroup G) :
H ∈ RigidStabilizerBasis G α ↔ (H : Set G) ∈ RigidStabilizerBasis.asSets G α :=
by
unfold asSets RigidStabilizerBasis
simp
variable [ContinuousConstSMul G α]
lemma RigidStabilizerBasis.smulImage_mem₀ {H : Subgroup G} (H_in_ristBasis : H ∈ RigidStabilizerBasis G α)
(f : G) : ((fun g => f * g * f⁻¹) '' H.carrier) ∈ RigidStabilizerBasis.asSets G α :=
by
have G_decidable : DecidableEq G := Classical.decEq _
rw [mem_iff] at H_in_ristBasis
let ⟨seed, H_eq⟩ := H_in_ristBasis
rw [mem_asSets_iff]
let new_seed := Finset.image (fun g => f * g * f⁻¹) seed.seed
have new_seed_ne_bot : RigidStabilizerInter₀ α new_seed ≠ ⊥ := by
rw [RigidStabilizerInter₀.eq_sInter]
unfold_let
simp [<-regularSupport_smulImage]
rw [<-ne_eq]
rw [<-smulImage_iInter_fin]
have val_ne_bot := seed.val_ne_bot
rw [Subgroup.ne_bot_iff_exists_ne_one] at val_ne_bot
let ⟨⟨g, g_in_ristInter⟩, g_ne_one⟩ := val_ne_bot
simp at g_ne_one
have fgf_in_ristInter₂ : f * g * f⁻¹ ∈ RigidStabilizer G (f •'' ⋂ x ∈ seed.seed, RegularSupport α x) := by
rw [rigidStabilizer_smulImage]
group
rw [RigidStabilizerInter₀.eq_sInter] at g_in_ristInter
exact g_in_ristInter
have fgf_ne_one : f * g * f⁻¹ ≠ 1 := by
intro h₁
have h₂ := congr_arg (fun x => x * f) h₁
group at h₂
have h₃ := congr_arg (fun x => f⁻¹ * x) h₂
group at h₃
exact g_ne_one h₃
rw [Subgroup.ne_bot_iff_exists_ne_one]
use ⟨f * g * f⁻¹, fgf_in_ristInter₂⟩
simp
rw [<-ne_eq]
exact fgf_ne_one
use RigidStabilizerInter₀ α new_seed
apply And.intro
exact ⟨⟨new_seed, new_seed_ne_bot⟩, rfl⟩
rw [RigidStabilizerInter₀.eq_sInter]
unfold_let
simp [<-regularSupport_smulImage]
rw [<-smulImage_iInter_fin]
ext x
simp only [Subsemigroup.mem_carrier, Submonoid.mem_toSubsemigroup,
Subgroup.mem_toSubmonoid, Set.mem_image]
rw [rigidStabilizer_smulImage]
rw [<-H_eq, RigidStabilizerBasis₀.val_def, RigidStabilizerInter₀.eq_sInter]
constructor
· intro fxf_mem
use f⁻¹ * x * f
constructor
· exact fxf_mem
· group
· intro ⟨y, ⟨y_in_H, y_conj⟩⟩
rw [<-y_conj]
group
exact y_in_H
def RigidStabilizerBasisMul (G α : Type _) [Group G] [MulAction G α] [TopologicalSpace α]
[ContinuousConstSMul G α] (f : G) (H : Subgroup G) : Subgroup G
where
carrier := (fun g => f * g * f⁻¹) '' H.carrier
mul_mem' := by
intro a b a_mem b_mem
simp at a_mem
simp at b_mem
let ⟨a', a'_in_H, a'_eq⟩ := a_mem
let ⟨b', b'_in_H, b'_eq⟩ := b_mem
use a' * b'
constructor
· apply Subsemigroup.mul_mem' <;> simp <;> assumption
· simp
rw [<-a'_eq, <-b'_eq]
group
one_mem' := by
simp
use 1
constructor
exact Subgroup.one_mem H
group
inv_mem' := by
simp
intro g g_in_H
use g⁻¹
constructor
exact Subgroup.inv_mem H g_in_H
rw [mul_assoc]
theorem RigidStabilizerBasisMul_mem (f : G) {H : Subgroup G} (H_in_basis : H ∈ RigidStabilizerBasis G α)
: RigidStabilizerBasisMul G α f H ∈ RigidStabilizerBasis G α :=
by
rw [RigidStabilizerBasis.mem_iff_asSets]
unfold RigidStabilizerBasisMul
simp
apply RigidStabilizerBasis.smulImage_mem₀
assumption
instance rigidStabilizerBasis_smul : SMul G (RigidStabilizerBasis G α) where
smul := fun g H => ⟨
RigidStabilizerBasisMul G α g H.val,
RigidStabilizerBasisMul_mem g H.prop
theorem RigidStabilizerBasis.smul_eq (g : G) {H: Subgroup G} (H_in_basis : H ∈ RigidStabilizerBasis G α) :
g • (⟨H, H_in_basis⟩ : RigidStabilizerBasis G α) = ⟨
RigidStabilizerBasisMul G α g H,
RigidStabilizerBasisMul_mem g H_in_basis
⟩ := rfl
theorem RigidStabilizerBasis.mem_smul (f g : G) {H: Subgroup G} (H_in_basis : H ∈ RigidStabilizerBasis G α):
f ∈ (g • (⟨H, H_in_basis⟩ : RigidStabilizerBasis G α)).val ↔ g⁻¹ * f * g ∈ H :=
by
rw [RigidStabilizerBasis.smul_eq]
simp
rw [<-Subgroup.mem_carrier]
unfold RigidStabilizerBasisMul
simp
constructor
· intro ⟨x, x_in_H, f_eq⟩
rw [<-f_eq]
group
exact x_in_H
· intro gfg_in_H
use g⁻¹ * f * g
constructor
assumption
group
instance rigidStabilizerBasis_mulAction : MulAction G (RigidStabilizerBasis G α) where
one_smul := by
intro ⟨H, H_in_ristBasis⟩
ext x
rw [RigidStabilizerBasis.mem_smul]
group
mul_smul := by
intro f g ⟨B, B_in_ristBasis⟩
ext x
repeat rw [RigidStabilizerBasis.mem_smul]
group
end Rubin