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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 Rubin.RigidStabilizer
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namespace Rubin
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open Topology
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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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∀ 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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lemma LocallyDense.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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Set.Nonempty U →
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∃ p ∈ U, p ∈ interior (closure (MulAction.orbit (RigidStabilizer G U) p)) :=
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by
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intros U H_ne
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exact ⟨H_ne.some, H_ne.some_mem, LocallyDense.isLocallyDense U H_ne.some H_ne.some_mem⟩
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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 ∈ RigidStabilizer 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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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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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 _ H_nonempty
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by_contra h_rs
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have ⟨elem, ⟨_, some_in_orbit⟩⟩ := H_ld.nonEmpty 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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{
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-- NOTE: if this is common, then we should make a tactic for solving IsOpen goals
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exact IsOpen.inter (img_open_open g⁻¹ V V_open) W_open
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}
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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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end Rubin
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