rubin-lean4/Rubin.lean

/-
Copyright (c) 2023 Laurent Bartholdi. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author : Laurent Bartholdi
-/
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Finset.Card
import Mathlib.Data.Fintype.Perm
import Mathlib.GroupTheory.Subgroup.Basic
import Mathlib.GroupTheory.Commutator
import Mathlib.GroupTheory.GroupAction.Basic
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.Perm.Basic
import Mathlib.Topology.Basic
import Mathlib.Topology.Compactness.Compact
import Mathlib.Topology.Separation
import Mathlib.Topology.Homeomorph

import Rubin.Tactic
import Rubin.MulActionExt
import Rubin.SmulImage
import Rubin.Support

#align_import rubin

namespace Rubin
open Rubin.Tactic

-- TODO: find a home
theorem equiv_congr_ne {ι ι' : Type _} (e : ι ≃ ι') {x y : ι} : x ≠ y → e x ≠ e y :=
  by
  intro x_ne_y
  by_contra h
  apply x_ne_y
  convert congr_arg e.symm h <;> simp only [Equiv.symm_apply_apply]
#align equiv.congr_ne Rubin.equiv_congr_ne

----------------------------------------------------------------
section Rubin

variable {G α β : Type _} [Group G]

----------------------------------------------------------------
section Groups

def is_algebraically_disjoint (f g : G) :=
  ∀ h : G,
    ¬Commute f h →
      ∃ f₁ f₂ : G, Commute f₁ g ∧ Commute f₂ g ∧ Commute ⁅f₁, ⁅f₂, h⁆⁆ g ∧ ⁅f₁, ⁅f₂, h⁆⁆ ≠ 1
#align is_algebraically_disjoint Rubin.is_algebraically_disjoint

end Groups

----------------------------------------------------------------
section Actions

variable [MulAction G α]

@[simp]
theorem orbit_bot (G : Type _) [Group G] [MulAction G α] (p : α) :
    MulAction.orbit (⊥ : Subgroup G) p = {p} :=
  by
  ext1
  rw [MulAction.mem_orbit_iff]
  constructor
  · rintro ⟨⟨_, g_bot⟩, g_to_x⟩
    rw [← g_to_x, Set.mem_singleton_iff, Subgroup.mk_smul]
    exact (Subgroup.mem_bot.mp g_bot).symm ▸ one_smul _ _
  exact fun h => ⟨1, Eq.trans (one_smul _ p) (Set.mem_singleton_iff.mp h).symm⟩
#align orbit_bot Rubin.orbit_bot

-- comment by Cedric: would be nicer to define just a subset, and then show it is a subgroup
def rigidStabilizer' (G : Type _) [Group G] [MulAction G α] (U : Set α) : Set G :=
  {g : G | ∀ x : α, g • x = x ∨ x ∈ U}
#align rigid_stabilizer' Rubin.rigidStabilizer'

/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (x «expr ∉ » U) -/
def rigidStabilizer (G : Type _) [Group G] [MulAction G α] (U : Set α) : Subgroup G
    where
  carrier := {g : G | ∀ (x) (_ : x ∉ U), g • x = x}
  mul_mem' ha hb x x_notin_U := by rw [mul_smul, hb x x_notin_U, ha x x_notin_U]
  inv_mem' hg x x_notin_U := smul_eq_iff_inv_smul_eq.mp (hg x x_notin_U)
  one_mem' x _ := one_smul G x
#align rigid_stabilizer Rubin.rigidStabilizer

theorem rist_supported_in_set {g : G} {U : Set α} :
    g ∈ rigidStabilizer G U → Support α g ⊆ U := fun h x x_in_support =>
  by_contradiction (x_in_support ∘ h x)
#align rist_supported_in_set Rubin.rist_supported_in_set

theorem rist_ss_rist {U V : Set α} (V_ss_U : V ⊆ U) :
    (rigidStabilizer G V : Set G) ⊆ (rigidStabilizer G U : Set G) :=
  by
  intro g g_in_ristV x x_notin_U
  have x_notin_V : x ∉ V := by intro x_in_V; exact x_notin_U (V_ss_U x_in_V)
  exact g_in_ristV x x_notin_V
#align rist_ss_rist Rubin.rist_ss_rist

end Actions

----------------------------------------------------------------
section TopologicalActions

variable [TopologicalSpace α] [TopologicalSpace β]

class Topological.ContinuousMulAction (G α : Type _) [Group G] [TopologicalSpace α] extends
    MulAction G α where
  continuous : ∀ g : G, Continuous (@SMul.smul G α _ g)
#align continuous_mul_action Rubin.Topological.ContinuousMulAction

structure Topological.equivariant_homeomorph (G α β : Type _) [Group G] [TopologicalSpace α]
    [TopologicalSpace β] [MulAction G α] [MulAction G β] extends Homeomorph α β where
  equivariant : is_equivariant G toFun
#align equivariant_homeomorph Rubin.Topological.equivariant_homeomorph

theorem Topological.equivariant_fun [MulAction G α] [MulAction G β]
    (h : Rubin.Topological.equivariant_homeomorph G α β) :
    is_equivariant G h.toFun :=
  h.equivariant
#align equivariant_fun Rubin.Topological.equivariant_fun

theorem Topological.equivariant_inv [MulAction G α] [MulAction G β]
    (h : Rubin.Topological.equivariant_homeomorph G α β) :
    is_equivariant G h.invFun :=
  by
  intro g x
  symm
  let e := congr_arg h.invFun (h.equivariant g (h.invFun x))
  rw [h.left_inv _, h.right_inv _] at e
  exact e
#align equivariant_inv Rubin.Topological.equivariant_inv

variable [Rubin.Topological.ContinuousMulAction G α]

theorem Topological.img_open_open (g : G) (U : Set α) (h : IsOpen U)
    [Rubin.Topological.ContinuousMulAction G α] : IsOpen (g •'' U) :=
  by
  rw [Rubin.smulImage_eq_inv_preimage]
  exact Continuous.isOpen_preimage (Rubin.Topological.ContinuousMulAction.continuous g⁻¹) U h
#align img_open_open Rubin.Topological.img_open_open

theorem Topological.support_open (g : G) [TopologicalSpace α] [T2Space α]
    [Rubin.Topological.ContinuousMulAction G α] : IsOpen (Support α g) :=
  by
  apply isOpen_iff_forall_mem_open.mpr
  intro x xmoved
  rcases T2Space.t2 (g • x) x xmoved with ⟨U, V, open_U, open_V, gx_in_U, x_in_V, disjoint_U_V⟩
  exact
    ⟨V ∩ (g⁻¹ •'' U), fun y yW =>
      -- TODO: don't use @-notation here
      @Disjoint.ne_of_mem α U V disjoint_U_V (g • y)
      (mem_inv_smulImage.mp (Set.mem_of_mem_inter_right yW))
      y
      (Set.mem_of_mem_inter_left yW),
      IsOpen.inter open_V (Rubin.Topological.img_open_open g⁻¹ U open_U),
      ⟨x_in_V, mem_inv_smulImage.mpr gx_in_U⟩⟩
#align support_open Rubin.Topological.support_open

end TopologicalActions

----------------------------------------------------------------
section FaithfulActions

variable [MulAction G α] [FaithfulSMul G α]

theorem faithful_moves_point₁ {g : G} (h2 : ∀ x : α, g • x = x) : g = 1 :=
  haveI h3 : ∀ x : α, g • x = (1 : G) • x := fun x => (h2 x).symm ▸ (one_smul G x).symm
  eq_of_smul_eq_smul h3
#align faithful_moves_point Rubin.faithful_moves_point₁

theorem faithful_moves_point'₁ {g : G} (α : Type _) [MulAction G α] [FaithfulSMul G α] :
    g ≠ 1 → ∃ x : α, g • x ≠ x := fun k =>
  by_contradiction fun h => k <| Rubin.faithful_moves_point₁ <| Classical.not_exists_not.mp h
#align faithful_moves_point' Rubin.faithful_moves_point'₁

theorem faithful_rigid_stabilizer_moves_point {g : G} {U : Set α} :
    g ∈ rigidStabilizer G U → g ≠ 1 → ∃ x ∈ U, g • x ≠ x :=
  by
  intro g_rigid g_ne_one
  rcases Rubin.faithful_moves_point'₁ α g_ne_one with ⟨x, xmoved⟩
  exact ⟨x, rist_supported_in_set g_rigid xmoved, xmoved⟩
#align faithful_rist_moves_point Rubin.faithful_rigid_stabilizer_moves_point

theorem ne_one_support_nonempty {g : G} : g ≠ 1 → (Support α g).Nonempty :=
  by
  intro h1
  cases' Rubin.faithful_moves_point'₁ α h1 with x h
  use x
  exact h
#align ne_one_support_nempty Rubin.ne_one_support_nonempty

theorem disjoint_commute {f g : G} :
    Disjoint (Support α f) (Support α g) → Commute f g :=
  by
  intro hdisjoint
  rw [← commutatorElement_eq_one_iff_commute]
  apply @Rubin.faithful_moves_point₁ _ α
  intro x
  rw [commutatorElement_def, mul_smul, mul_smul, mul_smul]
  cases' @or_not (x ∈ Support α f) with hfmoved hffixed
  · rw [smul_eq_iff_inv_smul_eq.mp (not_mem_support.mp (Set.disjoint_left.mp hdisjoint hfmoved)),
      not_mem_support.mp
        (Set.disjoint_left.mp hdisjoint (inv_smul_mem_support hfmoved)),
      smul_inv_smul]
  cases' @or_not (x ∈ Support α g) with hgmoved hgfixed
  · rw [smul_eq_iff_inv_smul_eq.mp
        (not_mem_support.mp <|
          Set.disjoint_right.mp hdisjoint (inv_smul_mem_support hgmoved)),
      smul_inv_smul, not_mem_support.mp hffixed]
  · rw [
      smul_eq_iff_inv_smul_eq.mp (not_mem_support.mp hgfixed),
      smul_eq_iff_inv_smul_eq.mp (not_mem_support.mp hffixed),
      not_mem_support.mp hgfixed,
      not_mem_support.mp hffixed
    ]
#align disjoint_commute Rubin.disjoint_commute

end FaithfulActions

----------------------------------------------------------------
section RubinActions

variable [TopologicalSpace α] [TopologicalSpace β]

def has_no_isolated_points (α : Type _) [TopologicalSpace α] :=
  ∀ x : α, (nhdsWithin x ({x}ᶜ)) ≠ ⊥
#align has_no_isolated_points Rubin.has_no_isolated_points

def is_locally_dense (G α : Type _) [Group G] [TopologicalSpace α] [MulAction G α] :=
  ∀ U : Set α, ∀ p ∈ U, p ∈ interior (closure (MulAction.orbit (rigidStabilizer G U) p))
#align is_locally_dense Rubin.is_locally_dense

structure RubinAction (G α : Type _) extends Group G, TopologicalSpace α, MulAction G α,
    FaithfulSMul G α where
  locally_compact : LocallyCompactSpace α
  hausdorff : T2Space α
  no_isolated_points : Rubin.has_no_isolated_points α
  locallyDense : Rubin.is_locally_dense G α
#align rubin_action Rubin.RubinAction

end RubinActions

----------------------------------------------------------------
section Rubin.Period.period

variable [MulAction G α]

noncomputable def Period.period (p : α) (g : G) : ℕ :=
  sInf {n : ℕ | n > 0 ∧ g ^ n • p = p}
#align period Rubin.Period.period

theorem Period.period_le_fix {p : α} {g : G} {m : ℕ} (m_pos : m > 0)
    (gmp_eq_p : g ^ m • p = p) : 0 < Rubin.Period.period p g ∧ Rubin.Period.period p g ≤ m :=
  by
  constructor
  · by_contra h'; have period_zero : Rubin.Period.period p g = 0; linarith;
    rcases Nat.sInf_eq_zero.1 period_zero with ⟨cont, h_1⟩ | h; linarith;
    exact Set.eq_empty_iff_forall_not_mem.mp h ↑m ⟨m_pos, gmp_eq_p⟩
  exact Nat.sInf_le ⟨m_pos, gmp_eq_p⟩
#align period_le_fix Rubin.Period.period_le_fix

theorem Period.notfix_le_period {p : α} {g : G} {n : ℕ} (n_pos : n > 0)
    (period_pos : Rubin.Period.period p g > 0) (pmoves : ∀ i : ℕ, 0 < i → i < n → g ^ i • p ≠ p) :
    n ≤ Rubin.Period.period p g := by
  by_contra period_le
  exact
    (pmoves (Rubin.Period.period p g) period_pos (not_le.mp period_le))
      (Nat.sInf_mem (Nat.nonempty_of_pos_sInf period_pos)).2
#align notfix_le_period Rubin.Period.notfix_le_period

theorem Period.notfix_le_period' {p : α} {g : G} {n : ℕ} (n_pos : n > 0)
    (period_pos : Rubin.Period.period p g > 0)
    (pmoves : ∀ i : Fin n, 0 < (i : ℕ) → g ^ (i : ℕ) • p ≠ p) : n ≤ Rubin.Period.period p g :=
  Rubin.Period.notfix_le_period n_pos period_pos fun (i : ℕ) (i_pos : 0 < i) (i_lt_n : i < n) =>
    pmoves (⟨i, i_lt_n⟩ : Fin n) i_pos
#align notfix_le_period' Rubin.Period.notfix_le_period'

theorem Period.period_neutral_eq_one (p : α) : Rubin.Period.period p (1 : G) = 1 :=
  by
  have : 0 < Rubin.Period.period p (1 : G) ∧ Rubin.Period.period p (1 : G) ≤ 1 :=
    Rubin.Period.period_le_fix (by norm_num : 1 > 0)
      (by group_action :
        (1 : G) ^ 1 • p = p)
  linarith
#align period_neutral_eq_one Rubin.Period.period_neutral_eq_one

def Period.periods (U : Set α) (H : Subgroup G) : Set ℕ :=
  {n : ℕ | ∃ (p : α) (g : H), p ∈ U ∧ Rubin.Period.period (p : α) (g : G) = n}
#align periods Rubin.Period.periods

-- TODO: split into multiple lemmas
theorem Period.periods_lemmas {U : Set α} (U_nonempty : Set.Nonempty U) {H : Subgroup G}
    (exp_ne_zero : Monoid.exponent H ≠ 0) :
    (Rubin.Period.periods U H).Nonempty ∧
      BddAbove (Rubin.Period.periods U H) ∧
        ∃ (m : ℕ) (m_pos : m > 0), ∀ (p : α) (g : H), g ^ m • p = p :=
  by
  rcases Monoid.exponentExists_iff_ne_zero.2 exp_ne_zero with ⟨m, m_pos, gm_eq_one⟩
  have gmp_eq_p : ∀ (p : α) (g : H), g ^ m • p = p := by
    intro p g; rw [gm_eq_one g];
    group_action
  have periods_nonempty : (Rubin.Period.periods U H).Nonempty := by
    use 1
    let p := Set.Nonempty.some U_nonempty; use p
    use 1
    constructor
    · exact Set.Nonempty.some_mem U_nonempty
    · exact Rubin.Period.period_neutral_eq_one p

  have periods_bounded : BddAbove (Rubin.Period.periods U H) := by
    use m; intro some_period hperiod;
    rcases hperiod with ⟨p, g, hperiod⟩
    rw [← hperiod.2]
    exact (Rubin.Period.period_le_fix m_pos (gmp_eq_p p g)).2
  exact ⟨periods_nonempty, periods_bounded, m, m_pos, gmp_eq_p⟩
#align period_lemma Rubin.Period.periods_lemmas

theorem Period.period_from_exponent (U : Set α) (U_nonempty : U.Nonempty) {H : Subgroup G}
    (exp_ne_zero : Monoid.exponent H ≠ 0) :
    ∃ (p : α) (g : H) (n : ℕ),
      p ∈ U ∧ n > 0 ∧ Rubin.Period.period (p : α) (g : G) = n ∧ n = sSup (Rubin.Period.periods U H) :=
  by
  rcases Rubin.Period.periods_lemmas U_nonempty exp_ne_zero with
    ⟨periods_nonempty, periods_bounded, m, m_pos, gmp_eq_p⟩
  rcases Nat.sSup_mem periods_nonempty periods_bounded with ⟨p, g, hperiod⟩
  use p
  use g
  use sSup (Rubin.Period.periods U H)
  -- TODO: cleanup?
  exact ⟨
    hperiod.1,
    hperiod.2 ▸ (Rubin.Period.period_le_fix m_pos (gmp_eq_p p g)).1,
    hperiod.2,
    rfl
  ⟩
#align period_from_exponent Rubin.Period.period_from_exponent

theorem Period.zero_lt_period_le_Sup_periods {U : Set α} (U_nonempty : U.Nonempty)
    {H : Subgroup G} (exp_ne_zero : Monoid.exponent H ≠ 0) :
    ∀ (p : U) (g : H),
      0 < Rubin.Period.period (p : α) (g : G) ∧
        Rubin.Period.period (p : α) (g : G) ≤ sSup (Rubin.Period.periods U H) :=
  by
  rcases Rubin.Period.periods_lemmas U_nonempty exp_ne_zero with
    ⟨periods_nonempty, periods_bounded, m, m_pos, gmp_eq_p⟩
  intro p g
  have period_in_periods : Rubin.Period.period (p : α) (g : G) ∈ Rubin.Period.periods U H := by
    use p; use g
    simp
  exact
    ⟨(Rubin.Period.period_le_fix m_pos (gmp_eq_p p g)).1,
      le_csSup periods_bounded period_in_periods⟩
#align zero_lt_period_le_Sup_periods Rubin.Period.zero_lt_period_le_Sup_periods

theorem Period.pow_period_fix (p : α) (g : G) : g ^ Rubin.Period.period p g • p = p :=
  by
  cases eq_zero_or_neZero (Rubin.Period.period p g) with
  | inl h => rw [h]; simp
  | inr h =>
    exact
      (Nat.sInf_mem
          (Nat.nonempty_of_pos_sInf
            (Nat.pos_of_ne_zero (@NeZero.ne _ _ (Rubin.Period.period p g) h)))).2
#align pow_period_fix Rubin.Period.pow_period_fix

end Rubin.Period.period

----------------------------------------------------------------
section AlgebraicDisjointness

variable [TopologicalSpace α] [Rubin.Topological.ContinuousMulAction G α] [FaithfulSMul G α]

def Disjointness.IsLocallyMoving (G α : Type _) [Group G] [TopologicalSpace α]
    [MulAction G α] :=
  ∀ U : Set α, IsOpen U → Set.Nonempty U → rigidStabilizer G U ≠ ⊥
#align is_locally_moving Rubin.Disjointness.IsLocallyMoving

-- lemma dense_locally_moving : t2_space α ∧ has_no_isolated_points α ∧ is_locally_dense G α → is_locally_moving G α := begin
--   rintros ⟨t2α,nipα,ildGα⟩ U ioU neU,
--   by_contra,
--   have some_in_U := ildGα U neU.some neU.some_mem,
--   rw [h,orbit_bot G neU.some,@closure_singleton α _ (@t2_space.t1_space α _ t2α) neU.some,@interior_singleton α _ neU.some (nipα neU.some)] at some_in_U,
--   tauto
-- end
-- lemma disjoint_nbhd {g : G} {x : α} [t2_space α] : g • x ≠ x → ∃U : set α, is_open U ∧ x ∈ U ∧ disjoint U (g •'' U) := begin
--   intro xmoved,
--   rcases t2_space.t2 (g • x) x xmoved with ⟨V,W,open_V,open_W,gx_in_V,x_in_W,disjoint_V_W⟩,
--   let U := (g⁻¹ •'' V) ∩ W,
--   use U,
--   split,
--   exact is_open.inter (img_open_open g⁻¹ V open_V) open_W,
--   split,
--   exact ⟨mem_inv_smul''.mpr gx_in_V,x_in_W⟩,
--   exact Set.disjoint_of_subset
--     (Set.inter_subset_right (g⁻¹ •'' V) W)
--     (λ y hy, smul_inv_smul g y ▸ mem_inv_smul''.mp (Set.mem_of_mem_inter_left (mem_smulImage.mp hy)) : g •'' U ⊆ V)
--     disjoint_V_W.symm
-- end
-- lemma disjoint_nbhd_in {g : G} {x : α} [t2_space α] {V : set α} : is_open V → x ∈ V → g • x ≠ x → ∃U : set α, is_open U ∧ x ∈ U ∧ U ⊆ V ∧ disjoint U (g •'' U) := begin
--   intros open_V x_in_V xmoved,
--   rcases disjoint_nbhd xmoved with ⟨W,open_W,x_in_W,disjoint_W⟩,
--   let U := W ∩ V,
--   use U,
--   split,
--   exact is_open.inter open_W open_V,
--   split,
--   exact ⟨x_in_W,x_in_V⟩,
--   split,
--   exact Set.inter_subset_right W V,
--   exact Set.disjoint_of_subset
--     (Set.inter_subset_left W V)
--     ((@smul''_inter _ _ _ _ g W V).symm ▸ Set.inter_subset_left (g •'' W) (g •'' V) : g •'' U ⊆ g •'' W)
--     disjoint_W
-- end
-- lemma rewrite_Union (f : fin 2 × fin 2 → set α) : (⋃(i : fin 2 × fin 2), f i) = (f (0,0) ∪ f (0,1)) ∪ (f (1,0) ∪ f (1,1)) := begin
--   ext,
--   simp only [Set.mem_Union, Set.mem_union],
--   split,
--   { simp only [forall_exists_index],
--     intro i,
--     fin_cases i; simp {contextual := tt}, },
--   { rintro ((h|h)|(h|h)); exact ⟨_, h⟩, },
-- end
-- lemma proposition_1_1_1 (f g : G) (locally_moving : is_locally_moving G α) [t2_space α] : disjoint (support α f) (support α g) → is_algebraically_disjoint f g := begin
--   intros disjoint_f_g h hfh,
--   let support_f := support α f,
-- -- h is not the identity on support α f
--   cases Set.not_disjoint_iff.mp (mt (@disjoint_commute G α _ _ _ _ _) hfh) with x hx,
--   let x_in_support_f := hx.1,
--   let hx_ne_x := mem_support.mp hx.2,
-- -- so since α is Hausdoff there is V nonempty ⊆ support α f with h •'' V disjoint from V
--   rcases disjoint_nbhd_in (support_open f) x_in_support_f hx_ne_x with ⟨V,open_V,x_in_V,V_in_support,disjoint_img_V⟩,
--   let ristV_ne_bot := locally_moving V open_V (Set.nonempty_of_mem x_in_V),
-- -- let f₂ be a nontrivial element of rigid_stabilizer G V
--   rcases (or_iff_right ristV_ne_bot).mp (Subgroup.bot_or_exists_ne_one _) with ⟨f₂,f₂_in_ristV,f₂_ne_one⟩,
-- -- again since α is Hausdorff there is W nonempty ⊆ V with f₂ •'' W disjoint from W
--   rcases faithful_moves_point' α f₂_ne_one with ⟨y,ymoved⟩,
--   let y_in_V : y ∈ V := (rist_supported_in_set f₂_in_ristV) (mem_support.mpr ymoved),
--   rcases disjoint_nbhd_in open_V y_in_V ymoved with ⟨W,open_W,y_in_W,W_in_V,disjoint_img_W⟩,
-- -- let f₁ be a nontrivial element of rigid_stabilizer G W
--   let ristW_ne_bot := locally_moving W open_W (Set.nonempty_of_mem y_in_W),
--   rcases (or_iff_right ristW_ne_bot).mp (Subgroup.bot_or_exists_ne_one _) with ⟨f₁,f₁_in_ristW,f₁_ne_one⟩,
--   use f₁, use f₂,
-- -- note that f₁,f₂ commute with g since their support is in support α f
--   split,
--   exact disjoint_commute (Set.disjoint_of_subset_left (Set.subset.trans (Set.subset.trans (rist_supported_in_set f₁_in_ristW) W_in_V) V_in_support) disjoint_f_g),
--   split,
--   exact disjoint_commute (Set.disjoint_of_subset_left (Set.subset.trans (rist_supported_in_set f₂_in_ristV) V_in_support) disjoint_f_g),
-- -- we claim that [f₁,[f₂,h]] is a nontrivial element of centralizer G g
--   let k := ⁅f₂,h⁆,
-- -- first, h*f₂⁻¹*h⁻¹ is supported on h V, so k := [f₂,h] agrees with f₂ on V
--   have h2 : ∀z ∈ W, f₂•z = k•z := λ z z_in_W,
--     (disjoint_support_comm f₂ h (rist_supported_in_set f₂_in_ristV) disjoint_img_V z (W_in_V z_in_W)).symm,
-- -- then k*f₁⁻¹*k⁻¹ is supported on k W = f₂ W, so [f₁,k] is supported on W ∪ f₂ W ⊆ V ⊆ support f, so commutes with g.
--   have h3 : support α ⁅f₁,k⁆ ⊆ support α f := begin
--     let := (support_comm α k f₁).trans (Set.union_subset_union (rist_supported_in_set f₁_in_ristW) (smul''_subset k $ rist_supported_in_set f₁_in_ristW)),
--     rw [← commutator_element_inv,support_inv,(smul''_eq_of_smul_eq h2).symm] at this,
--     exact (this.trans $ (Set.union_subset_union W_in_V (moves_subset_within_support f₂ W V W_in_V $ rist_supported_in_set f₂_in_ristV)).trans $ eq.subset V.union_self).trans V_in_support
-- end,
--   split,
--   exact disjoint_commute (Set.disjoint_of_subset_left h3 disjoint_f_g),
-- -- finally, [f₁,k] agrees with f₁ on W, so is not the identity.
--   have h4 : ∀z ∈ W, ⁅f₁,k⁆•z = f₁•z :=
--     disjoint_support_comm f₁ k (rist_supported_in_set f₁_in_ristW) (smul''_eq_of_smul_eq h2 ▸ disjoint_img_W),
--   rcases faithful_rist_moves_point f₁_in_ristW f₁_ne_one with ⟨z,z_in_W,z_moved⟩,
--   by_contra h5,
--   exact ((h4 z z_in_W).symm ▸ z_moved : ⁅f₁, k⁆ • z ≠ z) ((congr_arg (λg : G, g•z) h5).trans (one_smul G z)),
-- end
-- @[simp] lemma smul''_mul {g h : G} {U : set α} : g •'' (h •'' U) = (g*h) •'' U :=
--   (mul_smul'' g h U).symm
-- lemma disjoint_nbhd_fin {ι : Type*} [fintype ι] {f : ι → G} {x : α} [t2_space α] : (λi : ι, f i • x).injective → ∃U : set α, is_open U ∧ x ∈ U ∧ (∀i j : ι, i ≠ j → disjoint (f i •'' U) (f j •'' U)) := begin
--   intro f_injective,
--   let disjoint_hyp := λi j (i_ne_j : i≠j), let x_moved : ((f j)⁻¹ * f i) • x ≠ x := begin
--     by_contra,
--     let := smul_congr (f j) h,
--     rw [mul_smul, ← mul_smul,mul_right_inv,one_smul] at this,
--     from i_ne_j (f_injective this),
--   end in disjoint_nbhd x_moved,
--   let ι2 := { p : ι×ι // p.1 ≠ p.2 },
--   let U := ⋂(p : ι2), (disjoint_hyp p.1.1 p.1.2 p.2).some,
--   use U,
--   split,
--   exact is_open_Inter (λp : ι2, (disjoint_hyp p.1.1 p.1.2 p.2).some_spec.1),
--   split,
--   exact Set.mem_Inter.mpr (λp : ι2, (disjoint_hyp p.1.1 p.1.2 p.2).some_spec.2.1),
--   intros i j i_ne_j,
--   let U_inc := Set.Inter_subset (λ p : ι2, (disjoint_hyp p.1.1 p.1.2 p.2).some) ⟨⟨i,j⟩,i_ne_j⟩,
--   let := (disjoint_smul'' (f j) (Set.disjoint_of_subset U_inc (smul''_subset ((f j)⁻¹ * (f i)) U_inc) (disjoint_hyp i j i_ne_j).some_spec.2.2)).symm,
--   simp only [subtype.val_eq_coe, smul''_mul, mul_inv_cancel_left] at this,
--   from this
-- end
-- lemma moves_inj {g : G} {x : α} {n : ℕ} (period_ge_n : ∀ (k : ℤ), 1 ≤ k → k < n → g ^ k • x ≠ x) : function.injective (λ (i : fin n), g ^ (i : ℤ) • x) := begin
--     intros i j same_img,
--     by_contra i_ne_j,
--     let same_img' := congr_arg ((•) (g ^ (-(j : ℤ)))) same_img,
--     simp only [inv_smul_smul] at same_img',
--     rw [← mul_smul,← mul_smul,← zpow_add,← zpow_add,add_comm] at same_img',
--     simp only [add_left_neg, zpow_zero, one_smul] at same_img',
--     let ij := |(i:ℤ) - (j:ℤ)|,
--     rw ← sub_eq_add_neg at same_img',
--     have xfixed : g^ij • x = x := begin
--       cases abs_cases ((i:ℤ) - (j:ℤ)),
--       { rw ← h.1 at same_img', exact same_img' },
--       { rw [smul_eq_iff_inv_smul_eq,← zpow_neg,← h.1] at same_img', exact same_img' }
--     end,
--     have ij_ge_1 : 1 ≤ ij := int.add_one_le_iff.mpr (abs_pos.mpr $ sub_ne_zero.mpr $ norm_num.nat_cast_ne i j ↑i ↑j rfl rfl (fin.vne_of_ne i_ne_j)),
--     let neg_le := int.sub_lt_sub_of_le_of_lt (nat.cast_nonneg i) (nat.cast_lt.mpr (fin.prop _)),
--     rw zero_sub at neg_le,
--     let le_pos := int.sub_lt_sub_of_lt_of_le (nat.cast_lt.mpr (fin.prop _)) (nat.cast_nonneg j),
--     rw sub_zero at le_pos,
--     have ij_lt_n : ij < n := abs_lt.mpr ⟨ neg_le, le_pos ⟩,
--     exact period_ge_n ij ij_ge_1 ij_lt_n xfixed,
-- end
-- lemma int_to_nat (k : ℤ) (k_pos : k ≥ 1) : k = k.nat_abs := begin
--   cases (int.nat_abs_eq k),
--   { exact h },
--   { have : -(k.nat_abs : ℤ) ≤ 0 := neg_nonpos.mpr (int.nat_abs k).cast_nonneg,
--     rw ← h at this, by_contra, linarith }
-- end
-- lemma moves_inj_N {g : G} {x : α} {n : ℕ} (period_ge_n' : ∀ (k : ℕ), 1 ≤ k → k < n → g ^ k • x ≠ x) : function.injective (λ (i : fin n), g ^ (i : ℕ) • x) := begin
--   have period_ge_n : ∀ (k : ℤ), 1 ≤ k → k < n → g ^ k • x ≠ x,
--   { intros k one_le_k k_lt_n,
--     have one_le_k_nat : 1 ≤ k.nat_abs := ((int.coe_nat_le_coe_nat_iff 1 k.nat_abs).1 ((int_to_nat k one_le_k) ▸ one_le_k)),
--     have k_nat_lt_n : k.nat_abs < n := ((int.coe_nat_lt_coe_nat_iff k.nat_abs n).1 ((int_to_nat k one_le_k) ▸ k_lt_n)),
--     have := period_ge_n' k.nat_abs one_le_k_nat k_nat_lt_n,
--     rw [(zpow_coe_nat g k.nat_abs).symm, (int_to_nat k one_le_k).symm] at this,
--     exact this },
--   have := moves_inj period_ge_n,
--   done
-- end
-- lemma moves_1234_of_moves_12 {g : G} {x : α} (xmoves : g^12 • x ≠ x) : function.injective (λi : fin 5, g^(i:ℤ) • x) := begin
--     apply moves_inj,
--     intros k k_ge_1 k_lt_5,
--     by_contra xfixed,
--     have k_div_12 : k * (12 / k) = 12 := begin
--       interval_cases using k_ge_1 k_lt_5; norm_num
--     end,
--     have veryfixed : g^12 • x = x := begin
--       let := smul_zpow_eq_of_smul_eq (12/k) xfixed,
--       rw [← zpow_mul,k_div_12] at this,
--       norm_cast at this
--     end,
--     exact xmoves veryfixed
-- end
-- lemma proposition_1_1_2 (f g : G) [t2_space α] : is_locally_moving G α → is_algebraically_disjoint f g → disjoint (support α f) (support α (g^12)) := begin
--   intros locally_moving alg_disjoint,
-- -- suppose to the contrary that the set U = supp(f) ∩ supp(g^12) is nonempty
--   by_contra not_disjoint,
--   let U := support α f ∩ support α (g^12),
--   have U_nonempty : U.nonempty := Set.not_disjoint_iff_nonempty_inter.mp not_disjoint,
-- -- since X is Hausdorff, we can find a nonempty open set V ⊆ U such that f(V) is disjoint from V and the sets {g^i(V): i=0..4} are pairwise disjoint
--   let x := U_nonempty.some,
--   have five_points : function.injective (λi : fin 5, g^(i:ℤ) • x) := moves_1234_of_moves_12 (mem_support.mp $ (Set.inter_subset_right _ _) U_nonempty.some_mem),
--   rcases disjoint_nbhd_in (is_open.inter (support_open f) (support_open $ g^12)) U_nonempty.some_mem ((Set.inter_subset_left _ _) U_nonempty.some_mem) with ⟨V₀,open_V₀,x_in_V₀,V₀_in_support,disjoint_img_V₀⟩,
--   rcases disjoint_nbhd_fin five_points with ⟨V₁,open_V₁,x_in_V₁,disjoint_img_V₁⟩,
--   simp only at disjoint_img_V₁,
--   let V := V₀ ∩ V₁,
-- -- let h be a nontrivial element of rigid_stabilizer G V, and note that [f,h]≠1 since f(V) is disjoint from V
--   let ristV_ne_bot := locally_moving V (is_open.inter open_V₀ open_V₁) (Set.nonempty_of_mem ⟨x_in_V₀,x_in_V₁⟩),
--   rcases (or_iff_right ristV_ne_bot).mp (Subgroup.bot_or_exists_ne_one _) with ⟨h,h_in_ristV,h_ne_one⟩,
--   have comm_non_trivial : ¬commute f h := begin
--     by_contra comm_trivial,
--     rcases faithful_rist_moves_point h_in_ristV h_ne_one with ⟨z,z_in_V,z_moved⟩,
--     let act_comm := disjoint_support_comm h f (rist_supported_in_set h_in_ristV) (Set.disjoint_of_subset (Set.inter_subset_left V₀ V₁) (smul''_subset f (Set.inter_subset_left V₀ V₁)) disjoint_img_V₀) z z_in_V,
--     rw [commutator_element_eq_one_iff_commute.mpr comm_trivial.symm,one_smul] at act_comm,
--     exact z_moved act_comm.symm,
--   end,
-- -- since g is algebraically disjoint from f, there exist f₁,f₂ ∈ C_G(g) so that the commutator h' = [f1,[f2,h]] is a nontrivial element of C_G(g)
--   rcases alg_disjoint h comm_non_trivial with ⟨f₁,f₂,f₁_commutes,f₂_commutes,h'_commutes,h'_non_trivial⟩,
--   let h' := ⁅f₁,⁅f₂,h⁆⁆,
-- -- now observe that supp([f₂, h]) ⊆ V ∪ f₂(V), and by the same reasoning supp(h')⊆V∪f₁(V)∪f₂(V)∪f₁f₂(V)
--   have support_f₂h : support α ⁅f₂,h⁆ ⊆ V ∪ (f₂ •'' V) := (support_comm α f₂ h).trans (Set.union_subset_union (rist_supported_in_set h_in_ristV) $ smul''_subset f₂ $ rist_supported_in_set h_in_ristV),
--   have support_h' : support α h' ⊆ ⋃(i : fin 2 × fin 2), (f₁^i.1.val*f₂^i.2.val) •'' V := begin
--     let this := (support_comm α f₁ ⁅f₂,h⁆).trans (Set.union_subset_union support_f₂h (smul''_subset f₁ support_f₂h)),
--     rw [smul''_union,← one_smul'' V,← mul_smul'',← mul_smul'',← mul_smul'',mul_one,mul_one] at this,
--     let rw_u := rewrite_Union (λi : fin 2 × fin 2, (f₁^i.1.val*f₂^i.2.val) •'' V),
--     simp only [fin.val_eq_coe, fin.val_zero', pow_zero, mul_one, fin.val_one, pow_one, one_mul] at rw_u,
--     exact rw_u.symm ▸ this,
--   end,
-- -- since h' is nontrivial, it has at least one point p in its support
--   cases faithful_moves_point' α h'_non_trivial with p p_moves,
-- -- since g commutes with h', all five of the points {gi(p):i=0..4} lie in supp(h')
--   have gi_in_support : ∀i : fin 5, g^i.val • p ∈ support α h' := begin
--     intro i,
--     rw mem_support,
--     by_contra p_fixed,
--     rw [← mul_smul,(h'_commutes.pow_right i.val).eq,mul_smul,smul_left_cancel_iff] at p_fixed,
--     exact p_moves p_fixed,
--   end,
-- -- by the pigeonhole principle, one of the four sets V, f₁(V), f₂(V), f₁f₂(V) must contain two of these points, say g^i(p),g^j(p) ∈ k(V) for some 0 ≤ i < j ≤ 4 and k ∈ {1,f₁,f₂,f₁f₂}
--   let pigeonhole : fintype.card (fin 5) > fintype.card (fin 2 × fin 2) := dec_trivial,
--   let choice := λi : fin 5, (Set.mem_Union.mp $ support_h' $ gi_in_support i).some,
--   rcases finset.exists_ne_map_eq_of_card_lt_of_maps_to pigeonhole (λ(i : fin 5) _, finset.mem_univ (choice i)) with ⟨i,_,j,_,i_ne_j,same_choice⟩,
--   clear h_1_w h_1_h_h_w pigeonhole,
--   let k := f₁^(choice i).1.val*f₂^(choice i).2.val,
--   have same_k : f₁^(choice j).1.val*f₂^(choice j).2.val = k := by { simp only at same_choice,
-- rw ← same_choice },
--   have g_i : g^i.val • p ∈ k •'' V := (Set.mem_Union.mp $ support_h' $ gi_in_support i).some_spec,
--   have g_j : g^j.val • p ∈ k •'' V := same_k ▸ (Set.mem_Union.mp $ support_h' $ gi_in_support j).some_spec,
-- -- but since g^(j−i)(V) is disjoint from V and k commutes with g, we know that g^(j−i)k(V) is disjoint from k(V), a contradiction since g^i(p) and g^j(p) both lie in k(V).
--   have g_disjoint : disjoint ((g^i.val)⁻¹ •'' V) ((g^j.val)⁻¹ •'' V) := begin
--     let := (disjoint_smul'' (g^(-(i.val+j.val : ℤ))) (disjoint_img_V₁ i j i_ne_j)).symm,
--     rw [← mul_smul'',← mul_smul'',← zpow_add,← zpow_add] at this,
--     simp only [fin.val_eq_coe, neg_add_rev, coe_coe, neg_add_cancel_right, zpow_neg, zpow_coe_nat, neg_add_cancel_comm] at this,
--     from Set.disjoint_of_subset (smul''_subset _ (Set.inter_subset_right V₀ V₁)) (smul''_subset _ (Set.inter_subset_right V₀ V₁)) this
--   end,
--   have k_commutes : commute k g := commute.mul_left (f₁_commutes.pow_left (choice i).1.val) (f₂_commutes.pow_left (choice i).2.val),
--   have g_k_disjoint : disjoint ((g^i.val)⁻¹ •'' (k •'' V)) ((g^j.val)⁻¹ •'' (k •'' V)) := begin
--     let this := disjoint_smul'' k g_disjoint,
--     rw [← mul_smul'',← mul_smul'',← inv_pow g i.val,← inv_pow g j.val,
--         ← (k_commutes.symm.inv_left.pow_left i.val).eq,
--         ← (k_commutes.symm.inv_left.pow_left j.val).eq,
--        mul_smul'',inv_pow g i.val,mul_smul'' (g⁻¹^j.val) k V,inv_pow g j.val] at this,
--     from this
--   end,
--   exact Set.disjoint_left.mp g_k_disjoint (mem_inv_smul''.mpr g_i) (mem_inv_smul''.mpr g_j)
-- end
-- lemma remark_1_2 (f g : G) : is_algebraically_disjoint f g → commute f g := begin
--   intro alg_disjoint,
--   by_contra non_commute,
--   rcases alg_disjoint g non_commute with ⟨_,_,_,b,_,d⟩,
--   rw [commutator_element_eq_one_iff_commute.mpr b,commutator_element_one_right] at d,
--   tauto
-- end
-- section remark_1_3
-- def G := equiv.perm (fin 2)
-- def σ := equiv.swap (0 : fin 2) (1 : fin 2)
-- example : is_algebraically_disjoint σ σ := begin
--   intro h,
--   fin_cases h,
--   intro hyp1,
--   exfalso,
--   swap, intro hyp2, exfalso,
-- -- is commute decidable? cc,
-- sorry -- dec_trivial
-- sorry -- second sorry needed
-- end
-- end remark_1_3
end AlgebraicDisjointness

----------------------------------------------------------------
section Rubin.RegularSupport.RegularSupport

variable [TopologicalSpace α] [Rubin.Topological.ContinuousMulAction G α]

def RegularSupport.InteriorClosure (U : Set α) :=
  interior (closure U)
#align interior_closure Rubin.RegularSupport.InteriorClosure

theorem RegularSupport.is_open_interiorClosure (U : Set α) :
    IsOpen (Rubin.RegularSupport.InteriorClosure U) :=
  isOpen_interior
#align is_open_interior_closure Rubin.RegularSupport.is_open_interiorClosure

theorem RegularSupport.interiorClosure_mono {U V : Set α} :
    U ⊆ V → Rubin.RegularSupport.InteriorClosure U ⊆ Rubin.RegularSupport.InteriorClosure V :=
  interior_mono ∘ closure_mono
#align interior_closure_mono Rubin.RegularSupport.interiorClosure_mono

def is_regular_open (U : Set α) :=
  Rubin.RegularSupport.InteriorClosure U = U
#align set.is_regular_open Rubin.is_regular_open

theorem is_regular_def (U : Set α) :
    is_regular_open U ↔ Rubin.RegularSupport.InteriorClosure U = U := by rfl
#align set.is_regular_def Rubin.is_regular_def

theorem RegularSupport.IsOpen.in_closure {U : Set α} : IsOpen U → U ⊆ interior (closure U) :=
  by
  intro U_open x x_in_U
  apply interior_mono subset_closure
  rw [U_open.interior_eq]
  exact x_in_U
#align is_open.in_closure Rubin.RegularSupport.IsOpen.in_closure

theorem RegularSupport.IsOpen.interiorClosure_subset {U : Set α} :
    IsOpen U → U ⊆ Rubin.RegularSupport.InteriorClosure U := fun h =>
  (subset_interior_iff_isOpen.mpr h).trans (interior_mono subset_closure)
#align is_open.interior_closure_subset Rubin.RegularSupport.IsOpen.interiorClosure_subset

theorem RegularSupport.regular_interior_closure (U : Set α) :
    is_regular_open (Rubin.RegularSupport.InteriorClosure U) :=
  by
  rw [is_regular_def]
  apply Set.Subset.antisymm
  exact interior_mono ((closure_mono interior_subset).trans (subset_of_eq closure_closure))
  exact (subset_of_eq interior_interior.symm).trans (interior_mono subset_closure)
#align regular_interior_closure Rubin.RegularSupport.regular_interior_closure

def RegularSupport.RegularSupport (α : Type _) [TopologicalSpace α] [MulAction G α] (g : G) :=
  Rubin.RegularSupport.InteriorClosure (Support α g)
#align regular_support Rubin.RegularSupport.RegularSupport

theorem RegularSupport.regularSupport_regular {g : G} :
    is_regular_open (Rubin.RegularSupport.RegularSupport α g) :=
  Rubin.RegularSupport.regular_interior_closure _
#align regular_regular_support Rubin.RegularSupport.regularSupport_regular

theorem RegularSupport.support_subset_regularSupport [T2Space α] (g : G) :
    Support α g ⊆ Rubin.RegularSupport.RegularSupport α g :=
  Rubin.RegularSupport.IsOpen.interiorClosure_subset (Rubin.Topological.support_open g)
#align support_in_regular_support Rubin.RegularSupport.support_subset_regularSupport

theorem RegularSupport.mem_regularSupport (g : G) (U : Set α) :
    is_regular_open U → g ∈ rigidStabilizer G U → Rubin.RegularSupport.RegularSupport α g ⊆ U :=
  fun U_ro g_moves =>
  (is_regular_def _).mp U_ro ▸
    Rubin.RegularSupport.interiorClosure_mono (rist_supported_in_set g_moves)
#align mem_regular_support Rubin.RegularSupport.mem_regularSupport

-- FIXME: Weird naming?
def RegularSupport.AlgebraicCentralizer (f : G) : Set G :=
  {h | ∃ g, h = g ^ 12 ∧ Rubin.is_algebraically_disjoint f g}
#align algebraic_centralizer Rubin.RegularSupport.AlgebraicCentralizer

end Rubin.RegularSupport.RegularSupport

-- ----------------------------------------------------------------
-- section finite_exponent
-- lemma coe_nat_fin {n i : ℕ} (h : i < n) : ∃ (i' : fin n), i = i' := ⟨ ⟨ i, h ⟩, rfl ⟩
-- variables [topological_space α] [continuous_mul_action G α] [has_faithful_smul G α]
-- lemma distinct_images_from_disjoint {g : G} {V : set α} {n : ℕ}
--   (n_pos : 0 < n)
--   (h_disj : ∀ (i j : fin n) (i_ne_j : i ≠ j), disjoint (g ^ (i : ℕ) •'' V) (g ^ (j : ℕ) •'' V)) :
--   ∀ (q : α) (hq : q ∈ V) (i : fin n), (i : ℕ) > 0 → g ^ (i : ℕ) • (q : α) ∉ V :=
-- begin
--   intros q hq i i_pos hcontra,
--   have i_ne_zero : i ≠ (⟨ 0, n_pos ⟩ : fin n), { intro, done },
--   have hcontra' : g ^ (i : ℕ) • (q : α) ∈ g ^ (i : ℕ) •'' V, exact ⟨ q, hq, rfl ⟩,
--   have giq_notin_V := Set.disjoint_left.mp (h_disj i (⟨ 0, n_pos ⟩ : fin n) i_ne_zero) hcontra',
--   exact ((by done : g ^ 0 •'' V = V) ▸ giq_notin_V) hcontra
-- end
-- lemma moves_inj_period {g : G} {p : α} {n : ℕ} (period_eq_n : period p g = n) : function.injective (λ (i : fin n), g ^ (i : ℕ) • p) := begin
--   have period_ge_n : ∀ (k : ℕ), 1 ≤ k → k < n → g ^ k • p ≠ p,
--   { intros k one_le_k k_lt_n gkp_eq_p,
--     have := period_le_fix (nat.succ_le_iff.mp one_le_k) gkp_eq_p,
--     rw period_eq_n at this,
--     linarith },
--   exact moves_inj_N period_ge_n
-- end
-- lemma lemma_2_2 {α : Type u_2} [topological_space α] [continuous_mul_action G α] [has_faithful_smul G α] [t2_space α]
--   (U : set α) (U_open : is_open U) (locally_moving : is_locally_moving G α) :
--   U.nonempty → monoid.exponent (rigid_stabilizer G U) = 0 :=
-- begin
--   intro U_nonempty,
--   by_contra exp_ne_zero,
--   rcases (period_from_exponent U U_nonempty exp_ne_zero) with ⟨ p, g, n, n_pos, hpgn, n_eq_Sup ⟩,
--   rcases disjoint_nbhd_fin (moves_inj_period hpgn) with ⟨ V', V'_open, p_in_V', disj' ⟩,
--   dsimp at disj',
--   let V := U ∩ V',
--   have V_ss_U : V ⊆ U := Set.inter_subset_left U V',
--   have V'_ss_V : V ⊆ V' := Set.inter_subset_right U V',
--   have V_open : is_open V := is_open.inter U_open V'_open,
--   have p_in_V : (p : α) ∈ V := ⟨ subtype.mem p, p_in_V' ⟩,
--   have disj : ∀ (i j : fin n), ¬ i = j → disjoint (↑g ^ ↑i •'' V) (↑g ^ ↑j •'' V),
--   { intros i j i_ne_j W W_ss_giV W_ss_gjV,
--     exact disj' i j i_ne_j
--     (Set.subset.trans W_ss_giV (smul''_subset (↑g ^ ↑i) V'_ss_V))
--     (Set.subset.trans W_ss_gjV (smul''_subset (↑g ^ ↑j) V'_ss_V)) },
--   have ristV_ne_bot := locally_moving V V_open (Set.nonempty_of_mem p_in_V),
--   rcases (or_iff_right ristV_ne_bot).mp (Subgroup.bot_or_exists_ne_one _) with ⟨h,h_in_ristV,h_ne_one⟩,
--   rcases faithful_rist_moves_point h_in_ristV h_ne_one with ⟨ q, q_in_V, hq_ne_q ⟩,
--   have hg_in_ristU : (h : G) * (g : G) ∈ rigid_stabilizer G U := (rigid_stabilizer G U).mul_mem' (rist_ss_rist V_ss_U h_in_ristV) (subtype.mem g),
--   have giq_notin_V : ∀ (i : fin n), (i : ℕ) > 0 → g ^ (i : ℕ) • (q : α) ∉ V := distinct_images_from_disjoint n_pos disj q q_in_V,
--   have giq_ne_q : ∀ (i : fin n), (i : ℕ) > 0 → g ^ (i : ℕ) • (q : α) ≠ (q : α),
--   { intros i i_pos giq_eq_q, exact (giq_eq_q ▸ (giq_notin_V i i_pos)) q_in_V, },
--   have q_in_U : q ∈ U, { have : q ∈ U ∩ V' := q_in_V, exact this.1 },
--   -- We have (hg)^i q = g^i q for all 0 < i < n
--   have pow_hgq_eq_pow_gq : ∀ (i : fin n), (i : ℕ) < n → (h * g) ^ (i : ℕ) • q = (g : G) ^ (i : ℕ) • q,
--   { intros i, induction (i : ℕ) with i',
--     { intro, repeat {rw pow_zero} },
--     { intro succ_i'_lt_n,
--       rw [smul_succ, ih (nat.lt_of_succ_lt succ_i'_lt_n), smul_smul, mul_assoc, ← smul_smul, ← smul_smul, ← smul_succ],
--       have image_q_notin_V : g ^ i'.succ • q ∉ V,
--       { have i'succ_ne_zero := ne_zero.pos i'.succ,
--         exact giq_notin_V (⟨ i'.succ, succ_i'_lt_n ⟩ : fin n) i'succ_ne_zero },
--       exact by_contradiction (λ c, c (by_contradiction (λ c', image_q_notin_V ((rist_supported_in_set h_in_ristV) c')))) } },
--   -- Combined with g^i q ≠ q, this yields (hg)^i q ≠ q for all 0 < i < n
--   have hgiq_ne_q : ∀ (i : fin n), (i : ℕ) > 0 → (h * g) ^ (i : ℕ) • q ≠ q,
--   { intros i i_pos, rw pow_hgq_eq_pow_gq i (fin.is_lt i), by_contra c, exact (giq_notin_V i i_pos) (c.symm ▸ q_in_V) },
--   -- This even holds for i = n
--   have hgnq_ne_q : (h * g) ^ n • q ≠ q,
--   { -- Rewrite (hg)^n q = hg^n q
--     have npred_lt_n : n.pred < n, exact (nat.succ_pred_eq_of_pos n_pos) ▸ (lt_add_one n.pred),
--     rcases coe_nat_fin npred_lt_n with ⟨ i', i'_eq_pred_n ⟩,
--     have hgi'q_eq_gi'q := pow_hgq_eq_pow_gq i' (i'_eq_pred_n ▸ npred_lt_n),
--     have : n = (i' : ℕ).succ := i'_eq_pred_n ▸ (nat.succ_pred_eq_of_pos n_pos).symm,
--     rw [this, smul_succ, hgi'q_eq_gi'q, ← smul_smul, ← smul_succ, ← this],
--     -- Now it follows from g^n q = q and h q ≠ q
--     have n_le_period_qg := notfix_le_period' n_pos ((zero_lt_period_le_Sup_periods U_nonempty exp_ne_zero (⟨ q, q_in_U ⟩ : U) g)).1 giq_ne_q,
--     have period_qg_le_n := (zero_lt_period_le_Sup_periods U_nonempty exp_ne_zero (⟨ q, q_in_U ⟩ : U) g).2, rw ← n_eq_Sup at period_qg_le_n,
--     exact (ge_antisymm period_qg_le_n n_le_period_qg).symm ▸ ((pow_period_fix q (g : G)).symm ▸ hq_ne_q) },
--   -- Finally, we derive a contradiction
--   have period_pos_le_n := zero_lt_period_le_Sup_periods U_nonempty exp_ne_zero (⟨ q, q_in_U ⟩ : U) (⟨ h * g, hg_in_ristU ⟩ : rigid_stabilizer G U),
--   rw ← n_eq_Sup at period_pos_le_n,
--   cases (lt_or_eq_of_le period_pos_le_n.2),
--   { exact (hgiq_ne_q (⟨ (period (q : α) ((h : G) * (g : G))), h_1 ⟩ : fin n) period_pos_le_n.1) (pow_period_fix (q : α) ((h : G) * (g : G))) },
--   { exact hgnq_ne_q (h_1 ▸ (pow_period_fix (q : α) ((h : G) * (g : G)))) }
-- end
-- lemma proposition_2_1 [t2_space α] (f : G) : is_locally_moving G α → (algebraic_centralizer f).centralizer = rigid_stabilizer G (regular_support α f) := sorry
-- end finite_exponent
-- variables [topological_space α] [topological_space β] [continuous_mul_action G α] [continuous_mul_action G β]
-- noncomputable theorem rubin (hα : rubin_action G α) (hβ : rubin_action G β) : equivariant_homeomorph G α β := sorry
end Rubin

end Rubin