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/-
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This files defines `RegularSupportBasis`, which is a basis of the topological space α,
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made up of finite intersections of `RegularSupport α g` for `g : G`.
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-/
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import Mathlib.Topology.Basic
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import Mathlib.Topology.Homeomorph
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import Mathlib.Topology.Algebra.ConstMulAction
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import Rubin.LocallyDense
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import Rubin.Topology
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import Rubin.Support
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import Rubin.RegularSupport
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import Rubin.HomeoGroup
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namespace Rubin
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/--
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Maps a "seed" of homeorphisms in α to the intersection of their regular support in α.
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Note that the condition that the resulting set is non-empty is introduced later in `RegularSupportBasis₀`
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--/
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def RegularSupport.FiniteInter {G : Type _} [Group G] (α : Type _) [TopologicalSpace α] [MulAction G α] (S : Finset G): Set α :=
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⋂ (g ∈ S), RegularSupport α g
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def RegularSupportBasis (G α : Type _) [Group G] [TopologicalSpace α] [MulAction G α] : Set (Set α) :=
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{ S : Set α | S.Nonempty ∧ ∃ (seed : Finset G), S = RegularSupport.FiniteInter α seed }
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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 [TopologicalSpace α]
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variable [MulAction G α]
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theorem RegularSupport.FiniteInter_sInter (S : Finset G) :
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RegularSupport.FiniteInter α S = ⋂₀ ((fun (g : G) => RegularSupport α g) '' S) :=
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by
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rw [Set.sInter_image]
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rfl
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theorem RegularSupportBasis.mem_iff (S : Set α) :
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S ∈ RegularSupportBasis G α ↔ S.Nonempty ∧ ∃ (seed : Finset G), S = RegularSupport.FiniteInter α seed :=
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by
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simp only [RegularSupportBasis, Set.mem_setOf_eq]
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@[simp]
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theorem RegularSupport.FiniteInter_regular (F : Finset G) :
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Regular (RegularSupport.FiniteInter α F) :=
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by
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rw [RegularSupport.FiniteInter_sInter]
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apply regular_sInter
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· have set_decidable : DecidableEq (Set α) := Classical.typeDecidableEq (Set α)
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let fin : Finset (Set α) := F.image ((fun g => RegularSupport α g))
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apply Set.Finite.ofFinset fin
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simp
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· intro S S_in_set
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simp at S_in_set
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let ⟨g, ⟨_, Heq⟩⟩ := S_in_set
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rw [<-Heq]
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exact regularSupport_regular α g
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@[simp]
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theorem RegularSupportBasis.regular {S : Set α} (S_mem_basis : S ∈ RegularSupportBasis G α) : Regular S := by
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let ⟨_, ⟨seed, S_eq_inter⟩⟩ := (RegularSupportBasis.mem_iff S).mp S_mem_basis
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rw [S_eq_inter]
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apply RegularSupport.FiniteInter_regular
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variable [ContinuousConstSMul G α] [DecidableEq G]
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lemma RegularSupport.FiniteInter_conj (seed : Finset G) (f : G):
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RegularSupport.FiniteInter α (Finset.image (fun g => f * g * f⁻¹) seed) = f •'' RegularSupport.FiniteInter α seed :=
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by
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unfold RegularSupport.FiniteInter
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simp
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conv => {
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rhs
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ext; lhs; ext x; ext; lhs
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ext
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rw [regularSupport_smulImage]
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}
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/--
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A group element `f` acts on sets of `RegularSupportBasis G α`,
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by mapping each element `g` of `S.seed` to `f * g * f⁻¹`
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--/
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noncomputable instance RegularSupportBasis.instSmul : SMul G (RegularSupportBasis G α) where
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smul := fun f S =>
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let new_seed := (Finset.image (fun g => f * g * f⁻¹) S.prop.right.choose)
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⟨
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RegularSupport.FiniteInter α new_seed,
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by
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rw [RegularSupportBasis.mem_iff]
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nth_rw 1 [RegularSupport.FiniteInter_conj, smulImage_nonempty]
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rw [<-S.prop.right.choose_spec]
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constructor
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· exact S.prop.left
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· use new_seed
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⟩
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theorem RegularSupportBasis.smul_eq' (f : G) (S : RegularSupportBasis G α) :
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(f • S).val
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= RegularSupport.FiniteInter α (Finset.image (fun g => f * g * f⁻¹) S.prop.right.choose) := rfl
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theorem RegularSupportBasis.smul_eq (f : G) (S : RegularSupportBasis G α) :
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(f • S).val = f •'' S.val :=
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by
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rw [RegularSupportBasis.smul_eq']
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rw [RegularSupport.FiniteInter_conj]
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rw [<-S.prop.right.choose_spec]
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theorem RegularSupportBasis.smulImage_in_basis {U : Set α} (U_in_basis : U ∈ RegularSupportBasis G α)
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(f : G) : f •'' U ∈ RegularSupportBasis G α :=
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by
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have eq := smul_eq f ⟨U, U_in_basis⟩
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simp only at eq
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rw [<-eq]
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exact Subtype.coe_prop _
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def RegularSupportBasis.fromSingleton [T2Space α] [FaithfulSMul G α] (g : G) (g_ne_one : g ≠ 1) : { S : Set α // S ∈ RegularSupportBasis G α } :=
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let seed : Finset G := {g}
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⟨
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RegularSupport.FiniteInter α seed,
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by
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rw [RegularSupportBasis.mem_iff]
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constructor
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swap
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use seed
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rw [Set.nonempty_iff_ne_empty]
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intro rsupp_empty
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apply g_ne_one
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apply FaithfulSMul.eq_of_smul_eq_smul (α := α)
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intro x
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simp
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rw [<-not_mem_support]
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apply Set.not_mem_subset
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· apply support_subset_regularSupport
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· simp [RegularSupport.FiniteInter] at rsupp_empty
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rw [rsupp_empty]
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exact Set.not_mem_empty x
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⟩
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theorem RegularSupportBasis.fromSingleton_val [T2Space α] [FaithfulSMul G α] (g : G) (g_ne_one : g ≠ 1) :
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(fromSingleton g g_ne_one).val = RegularSupport α g := by simp [fromSingleton, RegularSupport.FiniteInter]
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-- Note: we could potentially implement MulActionHom
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noncomputable instance RegularSupportBasis.instMulAction : MulAction G (RegularSupportBasis G α) where
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one_smul := by
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intro S
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apply Subtype.eq
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rw [RegularSupportBasis.smul_eq]
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rw [one_smulImage]
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mul_smul := by
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intro S f g
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apply Subtype.eq
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repeat rw [RegularSupportBasis.smul_eq]
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rw [smulImage_mul]
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theorem RegularSupportBasis.smul_mono {S T : RegularSupportBasis G α} (f : G) (S_le_T : S.val ⊆ T.val) :
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(f • S).val ⊆ (f • T).val :=
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by
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repeat rw [RegularSupportBasis.smul_eq]
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apply smulImage_mono
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assumption
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section Basis
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open Topology
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variable (G α : Type _)
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variable [Group G]
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variable [TopologicalSpace α] [T2Space α] [LocallyCompactSpace α] [HasNoIsolatedPoints α]
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variable [MulAction G α] [LocallyDense G α] [ContinuousConstSMul G α]
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-- TODO: clean this lemma to not mention W anymore?
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lemma proposition_3_2_subset
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{U : Set α} (U_open : IsOpen U) {p : α} (p_in_U : p ∈ U) :
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∃ (W : Set α), W ∈ 𝓝 p ∧ closure W ⊆ U ∧
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∃ (g : G), g ∈ RigidStabilizer G W ∧ p ∈ RegularSupport α g ∧ RegularSupport α g ⊆ closure W :=
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by
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have U_in_nhds : U ∈ 𝓝 p := by
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rw [mem_nhds_iff]
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use U
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let ⟨W', W'_in_nhds, W'_ss_U, W'_compact⟩ := local_compact_nhds U_in_nhds
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-- This feels like black magic, but okay
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let ⟨W, _W_compact, W_closed, W'_ss_int_W, W_ss_U⟩ := exists_compact_closed_between W'_compact U_open W'_ss_U
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have W_cl_eq_W : closure W = W := IsClosed.closure_eq W_closed
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have W_in_nhds : W ∈ 𝓝 p := by
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rw [mem_nhds_iff]
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use interior W
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repeat' apply And.intro
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· exact interior_subset
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· simp
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· exact W'_ss_int_W (mem_of_mem_nhds W'_in_nhds)
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use W
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repeat' apply And.intro
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exact W_in_nhds
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{
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rw [W_cl_eq_W]
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exact W_ss_U
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}
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have p_in_int_W : p ∈ interior W := W'_ss_int_W (mem_of_mem_nhds W'_in_nhds)
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let ⟨g, g_in_rist, g_moves_p⟩ := get_moving_elem_in_rigidStabilizer G isOpen_interior p_in_int_W
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use g
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repeat' apply And.intro
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· apply rigidStabilizer_mono interior_subset
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simp
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exact g_in_rist
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· rw [<-mem_support] at g_moves_p
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apply support_subset_regularSupport
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exact g_moves_p
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· rw [rigidStabilizer_support] at g_in_rist
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apply subset_trans
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exact regularSupport_subset_closure_support
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apply closure_mono
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apply subset_trans
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exact g_in_rist
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exact interior_subset
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/--
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## Proposition 3.2 : RegularSupportBasis is a topological basis of `α`
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-/
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theorem RegularSupportBasis.isBasis :
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TopologicalSpace.IsTopologicalBasis (RegularSupportBasis G α) :=
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by
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apply TopologicalSpace.isTopologicalBasis_of_isOpen_of_nhds
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{
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intro U U_in_poset
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exact (RegularSupportBasis.regular U_in_poset).isOpen
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}
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intro p U p_in_U U_open
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let ⟨W, _, clW_ss_U, ⟨g, _, p_in_rsupp, rsupp_ss_clW⟩⟩ := proposition_3_2_subset G α U_open p_in_U
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use RegularSupport α g
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repeat' apply And.intro
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· exact ⟨p, p_in_rsupp⟩
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· use {g}
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simp [RegularSupport.FiniteInter]
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· assumption
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· apply subset_trans
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exact rsupp_ss_clW
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exact clW_ss_U
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theorem RegularSupportBasis.closed_inter (b1 b2 : Set α)
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(b1_in_basis : b1 ∈ RegularSupportBasis G α)
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(b2_in_basis : b2 ∈ RegularSupportBasis G α)
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(inter_nonempty : Set.Nonempty (b1 ∩ b2)) :
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(b1 ∩ b2) ∈ RegularSupportBasis G α :=
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by
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unfold RegularSupportBasis
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simp
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constructor
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assumption
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let ⟨_, ⟨s1, b1_eq⟩⟩ := b1_in_basis
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let ⟨_, ⟨s2, b2_eq⟩⟩ := b2_in_basis
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have dec_eq : DecidableEq G := Classical.typeDecidableEq G
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use s1 ∪ s2
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rw [RegularSupport.FiniteInter_sInter]
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rw [Finset.coe_union, Set.image_union, Set.sInter_union]
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repeat rw [<-RegularSupport.FiniteInter_sInter]
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rw [b2_eq, b1_eq]
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theorem RegularSupportBasis.empty_not_mem :
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∅ ∉ RegularSupportBasis G α :=
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by
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intro empty_mem
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rw [RegularSupportBasis.mem_iff] at empty_mem
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exact Set.not_nonempty_empty empty_mem.left
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theorem RegularSupportBasis.univ_mem [Nonempty α]:
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Set.univ ∈ RegularSupportBasis G α :=
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by
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rw [RegularSupportBasis.mem_iff]
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constructor
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exact Set.univ_nonempty
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use ∅
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simp [RegularSupport.FiniteInter]
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end Basis
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end Rubin
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