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