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import Mathlib.GroupTheory.Subgroup.Basic
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import Mathlib.GroupTheory.GroupAction.Basic
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import Mathlib.Topology.Basic
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import Mathlib.Topology.Algebra.ConstMulAction
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import Rubin.RigidStabilizer
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import Rubin.InteriorClosure
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namespace Rubin
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open Topology
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/--
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A group action is said to be "locally dense" if for any open set `U` and `p ∈ U`,
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the closure of the orbit of `p` under the `RigidStabilizer G U` contains a neighborhood of `p`.
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The definition provided here is an equivalent one, that does not require using filters.
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See [`LocallyDense.from_rigidStabilizer_in_nhds`] and [`LocallyDense.rigidStabilizer_in_nhds`]
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to translate from/to the original definition.
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A weaker relationship, [`LocallyMoving`], is used whenever possible.
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The main difference between the two is that `LocallyMoving` does not allow us to find a group member
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`g ∈ G` such that `g • p ≠ p` — it only allows us to know that `∃ g ∈ RigidStabilizer G U, g ≠ 1`.
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--/
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class LocallyDense (G α : Type _) [Group G] [TopologicalSpace α] [MulAction G α] :=
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isLocallyDense:
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∀ U : Set α,
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IsOpen U →
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∀ p ∈ U,
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p ∈ interior (closure (MulAction.orbit (RigidStabilizer G U) p))
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#align is_locally_dense Rubin.LocallyDense
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theorem LocallyDense.from_rigidStabilizer_in_nhds (G α : Type _) [Group G] [TopologicalSpace α] [MulAction G α] :
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(∀ U : Set α, IsOpen U → ∀ p ∈ U, closure (MulAction.orbit G•[U] p) ∈ 𝓝 p) →
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LocallyDense G α :=
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by
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intro hyp
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constructor
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intro U U_open p p_in_U
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have closure_in_nhds := hyp U U_open p p_in_U
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rw [mem_nhds_iff] at closure_in_nhds
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rw [mem_interior]
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exact closure_in_nhds
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theorem LocallyDense.rigidStabilizer_in_nhds (G α : Type _) [Group G] [TopologicalSpace α]
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[MulAction G α] [LocallyDense G α]
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{U : Set α} (U_open : IsOpen U) {p : α} (p_in_U : p ∈ U)
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:
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closure (MulAction.orbit G•[U] p) ∈ 𝓝 p :=
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by
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rw [mem_nhds_iff]
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rw [<-mem_interior]
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apply LocallyDense.isLocallyDense <;> assumption
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lemma LocallyDense.elem_from_nonEmpty {G α : Type _} [Group G] [TopologicalSpace α] [MulAction G α] [LocallyDense G α]:
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∀ {U : Set α},
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IsOpen U →
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Set.Nonempty U →
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∃ p ∈ U, p ∈ interior (closure (MulAction.orbit G•[U] p)) :=
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by
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intros U U_open H_ne
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exact ⟨H_ne.some, H_ne.some_mem, LocallyDense.isLocallyDense U U_open H_ne.some H_ne.some_mem⟩
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/--
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This is a stronger statement than `LocallyMoving.get_nontrivial_rist_elem`,
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as here we are able to prove that the nontrivial element does move `p`.
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The condition that `Filer.NeBot (𝓝[≠] p)` is automatically satisfied by the `HasNoIsolatedPoints` class.
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--/
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theorem get_moving_elem_in_rigidStabilizer (G : Type _) {α : Type _}
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[Group G] [TopologicalSpace α] [MulAction G α] [LocallyDense G α]
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[T1Space α] {p : α} [Filter.NeBot (𝓝[≠] p)]
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{U : Set α} (U_open : IsOpen U) (p_in_U : p ∈ U) :
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∃ g : G, g ∈ G•[U] ∧ g • p ≠ p :=
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by
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by_contra g_not_exist
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rw [<-Classical.not_forall_not] at g_not_exist
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simp at g_not_exist
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have orbit_singleton : MulAction.orbit (RigidStabilizer G U) p = {p} := by
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ext x
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rw [MulAction.mem_orbit_iff]
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rw [Set.mem_singleton_iff]
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simp
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constructor
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· intro ⟨g, g_in_rist, g_eq_p⟩
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rw [g_not_exist g g_in_rist] at g_eq_p
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exact g_eq_p.symm
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· intro x_eq_p
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use 1
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rw [x_eq_p, one_smul]
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exact ⟨Subgroup.one_mem _, rfl⟩
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have regular_orbit_empty : interior (closure (MulAction.orbit (RigidStabilizer G U) p)) = ∅ := by
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rw [orbit_singleton]
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rw [closure_singleton]
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rw [interior_singleton]
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have p_in_regular_orbit := LocallyDense.isLocallyDense (G := G) U U_open p p_in_U
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rw [regular_orbit_empty] at p_in_regular_orbit
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exact p_in_regular_orbit
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class LocallyMoving (G α : Type _) [Group G] [TopologicalSpace α] [MulAction G α] :=
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locally_moving: ∀ U : Set α, IsOpen U → Set.Nonempty U → RigidStabilizer G U ≠ ⊥
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#align is_locally_moving Rubin.LocallyMoving
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theorem LocallyMoving.get_nontrivial_rist_elem {G α : Type _}
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[Group G]
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[TopologicalSpace α]
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[MulAction G α]
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[h_lm : LocallyMoving G α]
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{U: Set α}
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(U_open : IsOpen U)
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(U_nonempty : U.Nonempty) :
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∃ x : G, x ∈ G•[U] ∧ x ≠ 1 :=
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by
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have rist_ne_bot := h_lm.locally_moving U U_open U_nonempty
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exact (or_iff_right rist_ne_bot).mp (Subgroup.bot_or_exists_ne_one _)
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theorem LocallyMoving.nontrivial_elem (G α : Type _) [Group G] [TopologicalSpace α]
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[MulAction G α] [LocallyMoving G α] [Nonempty α] :
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∃ g: G, g ≠ 1 :=
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by
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let ⟨g, _, g_ne_one⟩ := (get_nontrivial_rist_elem (G := G) (α := α) isOpen_univ Set.univ_nonempty)
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use g
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theorem LocallyMoving.nontrivial (G α : Type _) [Group G] [TopologicalSpace α]
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[MulAction G α] [LocallyMoving G α] [Nonempty α] : Nontrivial G where
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exists_pair_ne := by
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use 1
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simp only [ne_comm]
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exact nontrivial_elem G α
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theorem LocallyMoving.nonempty_iff_nontrivial (G α : Type _) [Group G] [TopologicalSpace α]
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[MulAction G α] [FaithfulSMul G α] [LocallyMoving G α] : Nonempty α ↔ Nontrivial G :=
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by
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constructor
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· intro; exact LocallyMoving.nontrivial G α
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· intro nontrivial
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by_contra α_empty
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rw [not_nonempty_iff] at α_empty
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let ⟨g, h, g_ne_h⟩ := nontrivial.exists_pair_ne
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apply g_ne_h
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apply FaithfulSMul.eq_of_smul_eq_smul (α := α)
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intro a
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exfalso
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exact α_empty.false a
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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 [ContinuousConstSMul G α]
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variable [FaithfulSMul G α]
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instance dense_locally_moving [T2Space α]
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[H_nip : HasNoIsolatedPoints α]
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[H_ld : LocallyDense G α] :
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LocallyMoving G α
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where
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locally_moving := by
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intros U U_open H_nonempty
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by_contra h_rs
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have ⟨elem, ⟨_, some_in_orbit⟩⟩ := H_ld.elem_from_nonEmpty U_open H_nonempty
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rw [h_rs] at some_in_orbit
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simp at some_in_orbit
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lemma disjoint_nbhd [T2Space α] {g : G} {x : α} (x_moved: g • x ≠ x) :
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∃ U: Set α, IsOpen U ∧ x ∈ U ∧ Disjoint U (g •'' U) :=
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by
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have ⟨V, W, V_open, W_open, gx_in_V, x_in_W, disjoint_V_W⟩ := T2Space.t2 (g • x) x x_moved
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let U := (g⁻¹ •'' V) ∩ W
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use U
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constructor
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exact IsOpen.inter (smulImage_isOpen g⁻¹ V_open) W_open
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constructor
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{
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simp
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rw [mem_inv_smulImage]
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trivial
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}
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{
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apply Set.disjoint_of_subset
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· apply Set.inter_subset_right
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· intro y hy; show y ∈ V
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rw [<-smul_inv_smul g y]
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rw [<-mem_inv_smulImage]
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rw [mem_smulImage] at hy
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simp at hy
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simp
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exact hy.left
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· exact disjoint_V_W.symm
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}
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lemma disjoint_nbhd_in [T2Space α] {g : G} {x : α} {V : Set α}
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(V_open : IsOpen V) (x_in_V : x ∈ V) (x_moved : g • x ≠ x) :
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∃ U : Set α, IsOpen U ∧ x ∈ U ∧ U ⊆ V ∧ Disjoint U (g •'' U) :=
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by
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have ⟨W, W_open, x_in_W, disjoint_W_img⟩ := disjoint_nbhd x_moved
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use W ∩ V
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simp
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constructor
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{
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apply IsOpen.inter <;> assumption
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}
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constructor
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{
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constructor <;> assumption
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}
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show Disjoint (W ∩ V) (g •'' W ∩ V)
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apply Set.disjoint_of_subset
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· exact Set.inter_subset_left W V
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· show g •'' W ∩ V ⊆ g •'' W
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rewrite [smulImage_inter]
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exact Set.inter_subset_left _ _
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· exact disjoint_W_img
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/--
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## Proposition 3.1:
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If a group action is faithful, continuous and "locally moving",
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then `U ⊆ V` if and only if `G•[U] ≤ G•[V]` when `U` and `V` are regular.
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--/
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theorem rigidStabilizer_subset_iff (G : Type _) {α : Type _} [Group G] [TopologicalSpace α]
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[MulAction G α] [ContinuousConstSMul G α] [FaithfulSMul G α]
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[h_lm : LocallyMoving G α]
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{U V : Set α} (U_reg : Regular U) (V_reg : Regular V):
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U ⊆ V ↔ G•[U] ≤ G•[V] :=
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by
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constructor
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exact rigidStabilizer_mono
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intro rist_ss
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by_contra U_not_ss_V
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let W := U \ closure V
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have W_nonempty : Set.Nonempty W := by
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by_contra W_empty
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apply U_not_ss_V
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apply subset_from_diff_closure_eq_empty
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· assumption
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· exact U_reg.isOpen
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· rw [Set.not_nonempty_iff_eq_empty] at W_empty
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exact W_empty
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have W_ss_U : W ⊆ U := by
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simp
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exact Set.diff_subset _ _
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have W_open : IsOpen W := by
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unfold_let
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rw [Set.diff_eq_compl_inter]
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apply IsOpen.inter
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simp
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exact U_reg.isOpen
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have ⟨f, f_in_ristW, f_ne_one⟩ := h_lm.get_nontrivial_rist_elem W_open W_nonempty
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have f_in_ristU : f ∈ RigidStabilizer G U := by
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exact rigidStabilizer_mono W_ss_U f_in_ristW
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have f_notin_ristV : f ∉ RigidStabilizer G V := by
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apply rigidStabilizer_compl f_ne_one
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apply rigidStabilizer_mono _ f_in_ristW
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calc
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W = U ∩ (closure V)ᶜ := by unfold_let; rw [Set.diff_eq_compl_inter, Set.inter_comm]
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_ ⊆ (closure V)ᶜ := Set.inter_subset_right _ _
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_ ⊆ Vᶜ := by
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rw [Set.compl_subset_compl]
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exact subset_closure
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exact f_notin_ristV (rist_ss f_in_ristU)
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/--
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A corollary of the previous theorem is that the rigid stabilizers of two regular sets `U` and `V`
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are equal if and only if `U = V`.
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--/
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theorem rigidStabilizer_eq_iff (G : Type _) [Group G] {α : Type _} [TopologicalSpace α]
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[MulAction G α] [ContinuousConstSMul G α] [FaithfulSMul G α] [LocallyMoving G α]
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{U V : Set α} (U_reg : Regular U) (V_reg : Regular V):
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G•[U] = G•[V] ↔ U = V :=
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by
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constructor
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· intro rist_eq
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apply le_antisymm <;> simp only [Set.le_eq_subset]
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all_goals {
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rw [rigidStabilizer_subset_iff G] <;> try assumption
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rewrite [rist_eq]
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rfl
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}
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· intro H_eq
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rw [H_eq]
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theorem rigidStabilizer_empty_iff (G : Type _) [Group G] {α : Type _} [TopologicalSpace α]
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[MulAction G α] [ContinuousConstSMul G α] [FaithfulSMul G α] [LocallyMoving G α]
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{U : Set α} (U_reg : Regular U) :
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G•[U] = ⊥ ↔ U = ∅ :=
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by
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rw [<-rigidStabilizer_empty (α := α) (G := G)]
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exact rigidStabilizer_eq_iff G U_reg (regular_empty α)
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end Rubin
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