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rubin-lean4/Rubin/RegularSupport.lean

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import Mathlib.Data.Finset.Basic
import Mathlib.GroupTheory.Commutator
import Mathlib.GroupTheory.GroupAction.Basic
import Mathlib.Topology.Basic
import Mathlib.Topology.Separation
import Rubin.Tactic
import Rubin.Support
import Rubin.Topology
import Rubin.InteriorClosure
import Rubin.RigidStabilizer
namespace Rubin
section RegularSupport_def
variable {G : Type _}
variable (α : Type _)
variable [Group G]
variable [MulAction G α]
variable [TopologicalSpace α]
-- The "regular support" is the `Support` of `g : G` in `α` (aka the elements in `α` which are moved by `g`),
-- transformed with the interior closure.
def RegularSupport (g: G) : Set α :=
InteriorClosure (Support α g)
#align regular_support Rubin.RegularSupport
theorem RegularSupport.def {G : Type _} (α : Type _)
[Group G] [MulAction G α] [TopologicalSpace α]
(g: G) : RegularSupport α g = interior (closure (Support α g)) :=
by
simp [RegularSupport]
theorem regularSupport_open [MulAction G α] (g : G) : IsOpen (RegularSupport α g) := by
unfold RegularSupport
simp
theorem regularSupport_regular [MulAction G α] (g : G) : Regular (RegularSupport α g) := by
apply interiorClosure_regular
#align regular_regular_support Rubin.regularSupport_regular
end RegularSupport_def
section RegularSupport_continuous
variable {G α : Type _}
variable [Group G]
variable [TopologicalSpace α]
variable [MulAction G α]
variable [ContinuousMulAction G α]
theorem support_subset_regularSupport [T2Space α] {g : G} :
Support α g ⊆ RegularSupport α g :=
by
apply interiorClosure_subset
apply support_open (α := α) (g := g)
#align support_in_regular_support Rubin.support_subset_regularSupport
theorem regularSupport_subset {g : G} {U : Set α} :
Regular U → Support α g ⊆ U → RegularSupport α g ⊆ U :=
by
intro U_reg h
rw [<-U_reg]
apply interiorClosure_mono
exact h
theorem regularSupport_subset_closure_support {g : G} :
RegularSupport α g ⊆ closure (Support α g) :=
by
unfold RegularSupport
simp
exact interior_subset
theorem regularSupport_subset_of_rigidStabilizer {g : G} {U : Set α} (U_reg : Regular U) :
g ∈ RigidStabilizer G U → RegularSupport α g ⊆ U :=
by
intro g_in_rist
apply regularSupport_subset
· assumption
· apply rigidStabilizer_support.mp
assumption
theorem regularSupport_subset_iff_rigidStabilizer [T2Space α] {g : G} {U : Set α} (U_reg : Regular U) :
g ∈ RigidStabilizer G U ↔ RegularSupport α g ⊆ U :=
by
constructor
· apply regularSupport_subset_of_rigidStabilizer U_reg
· intro rs_subset
rw [rigidStabilizer_support]
apply subset_trans
apply support_subset_regularSupport
exact rs_subset
#align mem_regular_support Rubin.regularSupport_subset_of_rigidStabilizer
theorem regularSupport_smulImage {f g : G} :
f •'' (RegularSupport α g) = RegularSupport α (f * g * f⁻¹) :=
by
unfold RegularSupport
rw [support_conjugate]
rw [interiorClosure_smulImage]
theorem rigidStabilizer_iInter_regularSupport (F : Set G) :
G•[⋂ (g ∈ F), RegularSupport α g] = (⨅ (g ∈ F), G•[RegularSupport α g]) :=
by
let S := { RegularSupport α g | g ∈ F }
have h₁ : ⋂ (g ∈ F), RegularSupport α g = ⋂₀ S := by
ext x
simp
have h₂ : ⨅ (g ∈ F), G•[RegularSupport α g] = ⨅ (s ∈ S), G•[s] := by
ext x
rw [<-sInf_image]
simp
rw [Subgroup.mem_iInf]
simp only [Subgroup.mem_iInf, and_imp, forall_apply_eq_imp_iff₂]
rw [h₁, h₂]
rw [rigidStabilizer_sInter]
theorem rigidStabilizer_iInter_regularSupport' (F : Finset G) :
G•[⋂ (g ∈ F), RegularSupport α g] = (⨅ (g ∈ F), G•[RegularSupport α g]) :=
by
have h₁ : G•[⋂ (g ∈ F), RegularSupport α g] = G•[⋂ (g ∈ (F : Set G)), RegularSupport α g] := by simp
have h₂ : ⨅ (g ∈ F), G•[RegularSupport α g] = ⨅ (g ∈ (F : Set G)), G•[RegularSupport α g] := by simp
rw [h₁, h₂, rigidStabilizer_iInter_regularSupport]
end RegularSupport_continuous
end Rubin