💩 Messy draft code to work towards the end of the proof

I need to change a bunch of things kind of everywhere and clean everything up before continuing

README.md

@@ -1,8 +1,9 @@
 # Lean4 port of the proof of Rubin's Theorem
 
 THis repository contains a (WIP) computer proof of Rubin's Theorem,
-which states that two groups (which satisfy a few conditions) that act on a topological space have a homeomorphism between them,
-such that the homeomorphism preserves the group structure.
+which states that if a group acts on two topological spaces while satisfying enough conditions,
+then there exists a homeomorphism between those two topological spaces,
+such that the homeomorphism preserves the group structure of the group action.
 
 It is based on ["A short proof of Rubin's Theorem"](https://arxiv.org/abs/2203.05930) (James Belk, Luke Elliott and Franceso Matucci),
 and a good part of the computer proof was written in Lean 3 by [Laurent Bartholdi](https://www.math.uni-sb.de/ag/bartholdi/).

Rubin.lean

@@ -742,15 +742,6 @@ variable [MulAction G α] [ContinuousMulAction G α] [FaithfulSMul G α] [Locall
 
 example : TopologicalSpace G := TopologicalSpace.generateFrom (RigidStabilizerBasis.asSets G α)
 
--- TODO: remove
--- proposition_3_4_1
-example {α : Type _} [TopologicalSpace α] [T2Space α] [LocallyCompactSpace α] (F : Ultrafilter α) (p : α):
-  F ≤ 𝓝 p ↔ p ∈ ⋂ (S ∈ F), closure S :=
-by
-  rw [<-Ultrafilter.clusterPt_iff]
-  simp
-  exact clusterPt_iff_forall_mem_closure
-
 theorem proposition_3_4_2 {α : Type _} [TopologicalSpace α] [T2Space α] [LocallyCompactSpace α] (F : Ultrafilter α):
   (∃ p : α, ClusterPt p F) ↔ ∃ S ∈ F, IsCompact (closure S) :=
 by
@@ -1024,13 +1015,385 @@ by
 
 end Ultrafilter
 
-variable {G α β : Type _}
+variable {G α : Type _}
 variable [Group G]
 
 variable [TopologicalSpace α] [MulAction G α] [ContinuousMulAction G α]
-         [TopologicalSpace β] [MulAction G β] [ContinuousMulAction G β]
+
+def IsRigidSubgroup (S : Set G) :=
+  S ≠ {1} ∧ ∃ T : Finset G, S = ⨅ (f ∈ T), AlgebraicCentralizer f
+
+def IsRigidSubgroup.toSubgroup {S : Set G} (S_rigid : IsRigidSubgroup S) : Subgroup G where
+  carrier := S
+  mul_mem' := by
+    let ⟨_, T, S_eq⟩ := S_rigid
+    simp only [S_eq, SetLike.mem_coe]
+    apply Subgroup.mul_mem
+  one_mem' := by
+    let ⟨_, T, S_eq⟩ := S_rigid
+    simp only [S_eq, SetLike.mem_coe]
+    apply Subgroup.one_mem
+  inv_mem' := by
+    let ⟨_, T, S_eq⟩ := S_rigid
+    simp only [S_eq, SetLike.mem_coe]
+    apply Subgroup.inv_mem
+
+theorem IsRigidSubgroup.mem_subgroup {S : Set G} (S_rigid : IsRigidSubgroup S) (g : G):
+  g ∈ S ↔ g ∈ S_rigid.toSubgroup := by rfl
+
+theorem IsRigidSubgroup.toSubgroup_neBot {S : Set G} (S_rigid : IsRigidSubgroup S) :
+  S_rigid.toSubgroup ≠ ⊥ :=
+by
+  intro eq_bot
+  rw [Subgroup.eq_bot_iff_forall] at eq_bot
+  simp only [<-mem_subgroup] at eq_bot
+  apply S_rigid.left
+  rw [Set.eq_singleton_iff_unique_mem]
+  constructor
+  · let ⟨S', S'_eq⟩ := S_rigid.right
+    rw [S'_eq, SetLike.mem_coe]
+    exact Subgroup.one_mem _
+  · assumption
+
+lemma Subgroup.coe_eq (S T : Subgroup G) :
+  (S : Set G) = (T : Set G) ↔ S = T :=
+by
+  constructor
+  · intro h
+    ext x
+    repeat rw [<-Subgroup.mem_carrier]
+    have h₁ : ∀ S : Subgroup G, (S : Set G) = S.carrier := by intro h; rfl
+    repeat rw [h₁] at h
+    rw [h]
+  · intro h
+    rw [h]
+
+def IsRigidSubgroup.algebraicCentralizerBasis {S : Set G} (S_rigid : IsRigidSubgroup S) : AlgebraicCentralizerBasis G := ⟨
+  S_rigid.toSubgroup,
+  by
+    rw [AlgebraicCentralizerBasis.mem_iff' _ (IsRigidSubgroup.toSubgroup_neBot S_rigid)]
+    let ⟨S', S'_eq⟩ := S_rigid.right
+    use S'
+    unfold AlgebraicCentralizerInter₀
+    rw [<-Subgroup.coe_eq, <-S'_eq]
+    rfl
+⟩
+
+theorem IsRigidSubgroup.algebraicCentralizerBasis_val {S : Set G} (S_rigid : IsRigidSubgroup S) :
+  S_rigid.algebraicCentralizerBasis.val = S_rigid.toSubgroup := rfl
+
+section toRegularSupportBasis
+
+variable (α : Type _)
+variable [TopologicalSpace α] [MulAction G α] [ContinuousMulAction G α]
+variable [FaithfulSMul G α] [T2Space α] [LocallyMoving G α]
+
+theorem IsRigidSubgroup.has_regularSupportBasis {S : Set G} (S_rigid : IsRigidSubgroup S) :
+  ∃ (U : RegularSupportBasis α), G•[U.val] = S :=
+by
+  let ⟨S_ne_bot, ⟨T, S_eq⟩⟩ := S_rigid
+  rw [S_eq]
+  simp only [Subgroup.coe_eq]
+  rw [S_eq, <-Subgroup.coe_bot, ne_eq, Subgroup.coe_eq, <-ne_eq] at S_ne_bot
+
+  let T' : Finset (HomeoGroup α) := Finset.map (HomeoGroup.fromContinuous_embedding α) T
+
+  have T'_rsupp_nonempty : Set.Nonempty (RegularSupportInter₀ T') := by
+    unfold RegularSupportInter₀
+    simp only [proposition_2_1 (G := G) (α := α)] at S_ne_bot
+    rw [ne_eq, <-rigidStabilizer_iInter_regularSupport', <-ne_eq] at S_ne_bot
+    let ⟨x, x_in_iInter⟩ := rigidStabilizer_neBot S_ne_bot
+    use x
+    simp
+    simp at x_in_iInter
+    exact x_in_iInter
+
+  let T'' := RegularSupportBasis.fromSeed ⟨T', T'_rsupp_nonempty⟩
+  have T''_val : T''.val = RegularSupportInter₀ T' := rfl
+  use T''
+  rw [T''_val]
+  unfold RegularSupportInter₀
+  simp
+  rw [rigidStabilizer_iInter_regularSupport']
+  simp only [<-proposition_2_1]
+
+noncomputable def IsRigidSubgroup.toRegularSupportBasis {S : Set G} (S_rigid : IsRigidSubgroup S) :
+  RegularSupportBasis α
+:= Exists.choose (IsRigidSubgroup.has_regularSupportBasis α S_rigid)
+
+theorem IsRigidSubgroup.toRegularSupportBasis_eq {S : Set G} (S_rigid : IsRigidSubgroup S):
+  G•[(S_rigid.toRegularSupportBasis α).val] = S :=
+by
+  exact Exists.choose_spec (IsRigidSubgroup.has_regularSupportBasis α S_rigid)
+
+theorem IsRigidSubgroup.toRegularSupportBasis_mono {S T : Set G} (S_rigid : IsRigidSubgroup S)
+  (T_rigid : IsRigidSubgroup T) :
+  S ⊆ T ↔ S_rigid.toRegularSupportBasis α ≤ T_rigid.toRegularSupportBasis α :=
+by
+  rw [RegularSupportBasis.le_def]
+  constructor
+  · intro S_ss_T
+    rw [<-IsRigidSubgroup.toRegularSupportBasis_eq (α := α) S_rigid] at S_ss_T
+    rw [<-IsRigidSubgroup.toRegularSupportBasis_eq (α := α) T_rigid] at S_ss_T
+    simp at S_ss_T
+    rw [<-rigidStabilizer_subset_iff G (RegularSupportBasis.regular _) (RegularSupportBasis.regular _)] at S_ss_T
+    exact S_ss_T
+  · intro Sr_ss_Tr
+    rw [rigidStabilizer_subset_iff G (RegularSupportBasis.regular _) (RegularSupportBasis.regular _)] at Sr_ss_Tr
+    -- TODO: clean that up
+    have Sr_ss_Tr' : (G•[(toRegularSupportBasis α S_rigid).val] : Set G)
+      ⊆ G•[(toRegularSupportBasis α T_rigid).val] :=
+    by
+      simp
+      assumption
+    rw [IsRigidSubgroup.toRegularSupportBasis_eq (α := α) S_rigid] at Sr_ss_Tr'
+    rw [IsRigidSubgroup.toRegularSupportBasis_eq (α := α) T_rigid] at Sr_ss_Tr'
+    assumption
+
+theorem IsRigidSubgroup.toRegularSupportBasis_mono' {S T : Set G} (S_rigid : IsRigidSubgroup S)
+  (T_rigid : IsRigidSubgroup T) :
+  S ⊆ T ↔ (S_rigid.toRegularSupportBasis α : Set α) ⊆ (T_rigid.toRegularSupportBasis α : Set α) :=
+by
+  rw [<-RegularSupportBasis.le_def]
+  rw [<-IsRigidSubgroup.toRegularSupportBasis_mono]
+
+end toRegularSupportBasis
+
+theorem IsRigidSubgroup.conj {U : Set G} (U_rigid : IsRigidSubgroup U) (g : G) : IsRigidSubgroup ((fun h => g * h * g⁻¹) '' U) := by
+  have conj_bijective : ∀ g : G, Function.Bijective (fun h => g * h * g⁻¹) := by
+    intro g
+    constructor
+    · intro x y; simp
+    · intro x
+      use g⁻¹ * x * g
+      group
+
+  constructor
+  · intro H
+    apply U_rigid.left
+    have h₁ : (fun h => g * h * g⁻¹) '' {1} = {1} := by simp
+    rw [<-h₁] at H
+    apply (Set.image_eq_image (conj_bijective g).left).mp H
+  · let ⟨S, S_eq⟩ := U_rigid.right
+    have dec_eq : DecidableEq G := Classical.typeDecidableEq G
+    use Finset.image (fun h => g * h * g⁻¹) S
+    rw [S_eq]
+    simp
+    simp only [Set.image_iInter (conj_bijective _), AlgebraicCentralizer.conj]
+
+def AlgebraicSubsets (V : Set G) : Set (Subgroup G) :=
+  {W ∈ AlgebraicCentralizerBasis G | W ≤ V}
+
+def AlgebraicOrbit (F : Ultrafilter G) (U : Set G) : Set (Subgroup G) :=
+  { (W_rigid.conj g).toSubgroup | (g ∈ U) (W ∈ F) (W_rigid : IsRigidSubgroup W) }
+
+structure RubinFilter (G : Type _) [Group G] where
+  filter : Ultrafilter G
+  -- rigid_basis : ∀ S ∈ filter, ∃ T ⊆ S, IsRigidSubgroup T
+  converges : ∀ U ∈ filter,
+    IsRigidSubgroup U →
+    ∃ V : Set G, IsRigidSubgroup V ∧ V ⊆ U ∧ AlgebraicSubsets V ⊆ AlgebraicOrbit filter U
+
+  -- Only really used to prove that ∀ S : Rigid, T : Rigid, S T ∈ F, S ∩ T : Rigid
+  ne_bot : {1} ∉ filter
+
+instance : Coe (RubinFilter G) (Ultrafilter G) where
+  coe := RubinFilter.filter
+
+section Equivalence
+open Topology
+
+variable (α : Type _)
+variable [TopologicalSpace α] [MulAction G α] [ContinuousMulAction G α]
+variable [FaithfulSMul G α] [T2Space α] [LocallyMoving G α]
+
+-- TODO: either see whether we actually need this step, or change these names to something memorable
+-- This is an attempt to convert a RubinFilter G back to an Ultrafilter α
+def RubinFilter.to_action_filter (F : RubinFilter G) : Filter α :=
+  ⨅ (S : { S : Set G // S ∈ F.filter ∧ IsRigidSubgroup S }), (Filter.principal (S.prop.right.toRegularSupportBasis α))
+
+instance RubinFilter.to_action_filter_neBot {F : RubinFilter G} [Nonempty α] : Filter.NeBot (F.to_action_filter α) :=
+  by
+    unfold to_action_filter
+    rw [Filter.iInf_neBot_iff_of_directed]
+    · intro ⟨S, S_in_F, S_rigid⟩
+      simp
+    · intro ⟨S, S_in_F, S_rigid⟩ ⟨T, T_in_F, T_rigid⟩
+      simp
+      use S ∩ T
+      have ST_in_F : (S ∩ T) ∈ F.filter := by
+        rw [<-Ultrafilter.mem_coe]
+        apply Filter.inter_mem <;> assumption
+      have ST_subgroup : IsRigidSubgroup (S ∩ T) := by
+        constructor
+        swap
+        · let ⟨S_seed, S_eq⟩ := S_rigid.right
+          let ⟨T_seed, T_eq⟩ := T_rigid.right
+          have dec_eq : DecidableEq G := Classical.typeDecidableEq G
+          use S_seed ∪ T_seed
+          rw [Finset.iInf_union, S_eq, T_eq]
+          simp
+        · -- TODO: check if we can't prove this without using F.ne_bot;
+          -- we might be able to use convergence
+          intro ST_eq_bot
+          apply F.ne_bot
+          rw [<-ST_eq_bot]
+          exact ST_in_F
+          -- sorry
+      use ⟨ST_in_F, ST_subgroup⟩
+
+      repeat rw [<-IsRigidSubgroup.toRegularSupportBasis_mono' (α := α)]
+      constructor
+      exact Set.inter_subset_left S T
+      exact Set.inter_subset_right S T
+
+-- lemma RubinFilter.mem_to_action_filter' (F : RubinFilter G) (U : Set α) :
+--   U ∈ F.to_action_filter α ↔ ∃ S : AlgebraicCentralizerBasis G, S.val ∈ F.filter,
+
+lemma RubinFilter.mem_to_action_filter (F : RubinFilter G) {U : Set G} (U_rigid : IsRigidSubgroup U) :
+  U ∈ F.filter ↔ (U_rigid.toRegularSupportBasis α : Set α) ∈ F.to_action_filter α :=
+by
+  unfold to_action_filter
+  constructor
+  · intro U_in_filter
+    apply Filter.mem_iInf_of_mem ⟨U, U_in_filter, U_rigid⟩
+    intro x
+    simp
+  · sorry
+
+noncomputable def RubinFilter.to_action_ultrafilter (F : RubinFilter G) [Nonempty α]: Ultrafilter α :=
+  Ultrafilter.of (F.to_action_filter α)
+
+theorem RubinFilter.to_action_ultrafilter_converges (F : RubinFilter G) [Nonempty α] [LocallyDense G α] [HasNoIsolatedPoints α] [LocallyCompactSpace α] {U : Set G}
+  (U_in_F : U ∈ F.filter) (U_rigid : IsRigidSubgroup U):
+  ∃ p ∈ (U_rigid.toRegularSupportBasis α).val, ClusterPt p (F.to_action_ultrafilter α) :=
+by
+  rw [proposition_3_5 (G := G)]
+
+  let ⟨V, V_rigid, V_ss_U, algsubs_ss_algorb⟩ := F.converges U U_in_F U_rigid
+  let V' := V_rigid.toSubgroup
+  -- TODO: subst V' to simplify the proof?
+
+  use V_rigid.toRegularSupportBasis α
+  constructor
+  {
+    rw [<-IsRigidSubgroup.toRegularSupportBasis_mono]
+    exact V_ss_U
+  }
+
+  unfold RSuppSubsets RSuppOrbit
+  simp
+  intro S S_rsupp_basis S_ss_V
+
+  have S_regular : Regular S := by
+    let ⟨S', S'_eq⟩ := (RegularSupportBasis.mem_asSet S).mp S_rsupp_basis
+    rw [<-S'_eq]
+    exact S'.regular
+
+  have S_nonempty : Set.Nonempty S := by
+    let ⟨S', S'_eq⟩ := (RegularSupportBasis.mem_asSet S).mp S_rsupp_basis
+    rw [<-S'_eq]
+    exact S'.nonempty
+
+  have GS_ss_V : G•[S] ≤ V := by
+    rw [<-V_rigid.toRegularSupportBasis_eq (α := α)]
+    simp
+    rw [<-rigidStabilizer_subset_iff G (α := α) S_regular (RegularSupportBasis.regular _)]
+    assumption
+
+  -- TODO: show that G•[S] ∈ AlgebraicSubsets V
+  have GS_in_algsubs_V : G•[S] ∈ AlgebraicSubsets V := by
+    unfold AlgebraicSubsets
+    simp
+    constructor
+    · rw [rigidStabilizerBasis_eq_algebraicCentralizerBasis (α := α)]
+      let ⟨S', S'_eq⟩ := (RegularSupportBasis.mem_asSet S).mp S_rsupp_basis
+      rw [<-S'_eq]
+      rw [RigidStabilizerBasis.mem_iff' _ (LocallyMoving.locally_moving _ S'.regular.isOpen S'.nonempty)]
+
+      sorry
+    · exact GS_ss_V
+
+  let ⟨g, g_in_U, W, W_in_F, W_rigid, Wconj_eq_GS⟩ := algsubs_ss_algorb GS_in_algsubs_V
+
+  use g
+  constructor
+  {
+    rw [<-SetLike.mem_coe]
+    rw [U_rigid.toRegularSupportBasis_eq (α := α)]
+    assumption
+  }
+
+  use W_rigid.toRegularSupportBasis α
+  constructor
+  · apply Ultrafilter.of_le
+    rw [<-RubinFilter.mem_to_action_filter]
+    assumption
+  · rw [<-rigidStabilizer_eq_iff G]
+    swap
+    {
+      rw [<-smulImage_regular (G := G)]
+      apply RegularSupportBasis.regular
+    }
+    swap
+    {
+      let ⟨S', S'_eq⟩ := (RegularSupportBasis.mem_asSet S).mp S_rsupp_basis
+      rw [<-S'_eq]
+      exact S'.regular
+    }
+    ext i
+    rw [rigidStabilizer_smulImage, <-Wconj_eq_GS, <-IsRigidSubgroup.mem_subgroup, <-SetLike.mem_coe, IsRigidSubgroup.toRegularSupportBasis_eq]
+    simp
+    constructor
+    · intro gig_in_W
+      use g⁻¹ * i * g
+      constructor; exact gig_in_W
+      group
+    · intro ⟨j, j_in_W, gjg_eq_i⟩
+      rw [<-gjg_eq_i]
+      group
+      assumption
+
+end Equivalence
+
+-- TODO: prove that for every rubinfilter, there exists an associated ultrafilter on α that converges
+
+instance RubinFilterSetoid (G : Type _) [Group G] : Setoid (RubinFilter G) where
+  r F F' := ∀ (U : Set G), IsRigidSubgroup U →
+    ((∃ V : Set G, V ≤ U ∧ AlgebraicSubsets V ⊆ AlgebraicOrbit F U)
+    ↔ (∃ V' : Set G, V' ≤ U ∧ AlgebraicSubsets V' ⊆ AlgebraicOrbit F' U))
+  iseqv := by
+    constructor
+    · intros
+      simp
+    · intro F F' h
+      intro U U_rigid
+      symm
+      exact h U U_rigid
+    · intro F₁ F₂ F₃
+      intro h₁₂ h₂₃
+      intro U U_rigid
+      specialize h₁₂ U U_rigid
+      specialize h₂₃ U U_rigid
+      exact Iff.trans h₁₂ h₂₃
+
+def RubinSpace (G : Type _) [Group G] := Quotient (RubinFilterSetoid G)
+
+-- Topology can be generated from the disconnectedness of the filters
+
+variable {β : Type _}
+
+variable [TopologicalSpace β] [MulAction G β] [ContinuousMulAction G β]
+
+
+instance : TopologicalSpace (RubinSpace G) := sorry
+
+instance : MulAction G (RubinSpace G) := sorry
+
+theorem rubin' (hα : RubinAction G α) : EquivariantHomeomorph G α (RubinSpace G) := by sorry
 
 theorem rubin (hα : RubinAction G α) (hβ : RubinAction G β) : EquivariantHomeomorph G α β := by
+  -- by composing rubin' hα
   sorry
 
 end Rubin

Rubin/AlgebraicDisjointness.lean

@@ -149,7 +149,7 @@ lemma smul_inj_moves {ι : Type*} [Fintype ι] [T2Space α]
     apply i_ne_j
     apply f_smul_inj
     group_action
-    group_action at h
+    group at h
     exact h
 
 def smul_inj_nbhd {ι : Type*} [Fintype ι] [T2Space α]
@@ -257,7 +257,7 @@ by
       unfold_let
       rw [h.1]
       rw [smul_eq_iff_eq_inv_smul]
-      group_action
+      group
       symm
       exact same_img
   }
@@ -347,6 +347,51 @@ def AlgebraicSubgroup (f : G) : Set G :=
 def AlgebraicCentralizer (f : G) : Subgroup G :=
   Subgroup.centralizer (AlgebraicSubgroup f)
 
+theorem AlgebraicSubgroup.conj (f g : G) :
+  (fun h => g * h * g⁻¹) '' AlgebraicSubgroup f = AlgebraicSubgroup (g * f * g⁻¹) :=
+by
+  unfold AlgebraicSubgroup
+  rw [Set.image_image]
+  have gxg12_eq : ∀ x : G, g * x^12 * g⁻¹ = (g * x * g⁻¹)^12 := by
+    simp
+  simp only [gxg12_eq]
+  ext x
+  sorry
+  -- unfold IsAlgebraicallyDisjoint
+
+
+@[simp]
+theorem AlgebraicCentralizer.conj (f g : G) :
+  (fun h => g * h * g⁻¹) '' AlgebraicCentralizer f = AlgebraicCentralizer (g * f * g⁻¹) :=
+by
+  unfold AlgebraicCentralizer
+
+  ext x
+  simp [Subgroup.mem_centralizer_iff]
+
+  constructor
+  · intro ⟨y, ⟨x_comm, x_eq⟩⟩
+    intro h h_in_alg
+    rw [<-AlgebraicSubgroup.conj] at h_in_alg
+    simp at h_in_alg
+    let ⟨i, i_in_alg, gig_eq_h⟩ := h_in_alg
+    specialize x_comm i i_in_alg
+    rw [<-gig_eq_h, <-x_eq]
+    group
+    rw [mul_assoc _ i, x_comm, <-mul_assoc]
+  · intro x_comm
+    use g⁻¹ * x * g
+    group
+    simp
+    intro h h_in_alg
+    simp [<-AlgebraicSubgroup.conj] at x_comm
+    specialize x_comm h h_in_alg
+    have h₁ : g⁻¹ * x * g * h = g⁻¹ * (g * h * g⁻¹ * x) * g := by
+      rw [x_comm]
+      group
+    rw [h₁]
+    group
+
 /--
 Finite intersections of [`AlgebraicCentralizer`].
 --/

Rubin/HomeoGroup.lean

@@ -232,19 +232,53 @@ by
   repeat rw [Rubin.smulImage_def]
   simp
 
+def HomeoGroup.fromContinuous_embedding (α : Type _) [TopologicalSpace α] [MulAction G α] [Rubin.ContinuousMulAction G α] [FaithfulSMul G α]: G ↪ (HomeoGroup α) where
+  toFun := fun (g : G) => HomeoGroup.fromContinuous α g
+  inj' := by
+    intro g h fromCont_eq
+    simp at fromCont_eq
+    apply FaithfulSMul.eq_of_smul_eq_smul (α := α)
+    intro x
+    rw [<-fromContinuous_smul, fromCont_eq, fromContinuous_smul]
+
+@[simp]
+theorem HomeoGroup.fromContinuous_embedding_toFun [FaithfulSMul G α] (g : G):
+  HomeoGroup.fromContinuous_embedding α g = HomeoGroup.fromContinuous α g := rfl
+
+def HomeoGroup.fromContinuous_monoidHom (α : Type _) [TopologicalSpace α] [MulAction G α] [Rubin.ContinuousMulAction G α] [FaithfulSMul G α]: G →* (HomeoGroup α) where
+  toFun := fun (g : G) => HomeoGroup.fromContinuous α g
+  map_one' := by
+    simp
+    rw [fromContinuous_one]
+  map_mul' := by
+    simp
+    intros
+    rw [fromContinuous_mul]
+
+-- theorem HomeoGroup.fromContinuous_rigidStabilizer (U : Set α) [FaithfulSMul G α]:
+--   Rubin.RigidStabilizer (HomeoGroup α) U = Subgroup.map (HomeoGroup.fromContinuous_monoidHom α) (Rubin.RigidStabilizer G U) :=
+-- by
+--   ext g
+--   rw [<-Subgroup.mem_carrier]
+--   unfold Rubin.RigidStabilizer
+--   simp
+--   sorry
+
 end ContinuousMulActionCoe
 
 namespace Rubin
 
 section Other
 
+-- TODO: move this somewhere else
 /--
 ## Proposition 3.1
 --/
-theorem homeoGroup_rigidStabilizer_subset_iff {α : Type _} [TopologicalSpace α]
-  [h_lm : LocallyMoving (HomeoGroup α) α]
+theorem rigidStabilizer_subset_iff (G : Type _) {α : Type _} [Group G] [TopologicalSpace α]
+  [MulAction G α] [ContinuousMulAction G α] [FaithfulSMul G α]
+  [h_lm : LocallyMoving G α]
   {U V : Set α} (U_reg : Regular U) (V_reg : Regular V):
-  U ⊆ V ↔ RigidStabilizer (HomeoGroup α) U ≤ RigidStabilizer (HomeoGroup α) V :=
+  U ⊆ V ↔ RigidStabilizer G U ≤ RigidStabilizer G V :=
 by
   constructor
   exact rigidStabilizer_mono
@@ -273,10 +307,10 @@ by
 
   have ⟨f, f_in_ristW, f_ne_one⟩ := h_lm.get_nontrivial_rist_elem W_open W_nonempty
 
-  have f_in_ristU : f ∈ RigidStabilizer (HomeoGroup α) U := by
+  have f_in_ristU : f ∈ RigidStabilizer G U := by
     exact rigidStabilizer_mono W_ss_U f_in_ristW
 
-  have f_notin_ristV : f ∉ RigidStabilizer (HomeoGroup α) V := by
+  have f_notin_ristV : f ∉ RigidStabilizer G V := by
     apply rigidStabilizer_compl f_ne_one
     apply rigidStabilizer_mono _ f_in_ristW
     calc
@@ -288,6 +322,29 @@ by
 
   exact f_notin_ristV (rist_ss f_in_ristU)
 
+theorem rigidStabilizer_eq_iff (G : Type _) [Group G] {α : Type _} [TopologicalSpace α]
+  [MulAction G α] [ContinuousMulAction G α] [FaithfulSMul G α] [LocallyMoving G α]
+  {U V : Set α} (U_reg : Regular U) (V_reg : Regular V):
+  RigidStabilizer G U = RigidStabilizer G V ↔ U = V :=
+by
+  constructor
+  · intro rist_eq
+    apply le_antisymm <;> simp only [Set.le_eq_subset]
+    all_goals {
+      rw [rigidStabilizer_subset_iff G] <;> try assumption
+      rewrite [rist_eq]
+      rfl
+    }
+  · intro H_eq
+    rw [H_eq]
+
+theorem homeoGroup_rigidStabilizer_subset_iff {α : Type _} [TopologicalSpace α]
+  [h_lm : LocallyMoving (HomeoGroup α) α]
+  {U V : Set α} (U_reg : Regular U) (V_reg : Regular V):
+  U ⊆ V ↔ RigidStabilizer (HomeoGroup α) U ≤ RigidStabilizer (HomeoGroup α) V :=
+by
+  rw [rigidStabilizer_subset_iff (HomeoGroup α) U_reg V_reg]
+
 theorem homeoGroup_rigidStabilizer_eq_iff {α : Type _} [TopologicalSpace α]
   [LocallyMoving (HomeoGroup α) α]
   {U V : Set α} (U_reg : Regular U) (V_reg : Regular V):

Rubin/InteriorClosure.lean

@@ -240,4 +240,19 @@ theorem interiorClosure_smulImage {G : Type _} [Group G] [MulAction G α] [Conti
   InteriorClosure (g •'' U) = g •'' InteriorClosure U :=
   interiorClosure_smulImage' g U (continuousMulAction_elem_continuous α g)
 
+theorem smulImage_regular {G : Type _} [Group G] [MulAction G α] [ContinuousMulAction G α]
+  (g : G) (U : Set α) :
+  Regular U ↔ Regular (g •'' U) :=
+by
+  suffices ∀ V : Set α, ∀ h : G, Regular V → Regular (h •'' V) by
+    constructor
+    apply this
+    have U_eq : g⁻¹ •'' (g •'' U) = U := by simp
+    nth_rw 2 [<-U_eq]
+    apply this
+  intro U g U_reg
+  unfold Regular
+  rw [interiorClosure_smulImage]
+  rw [U_reg]
+
 end Rubin

Rubin/RegularSupport.lean

@@ -100,6 +100,31 @@ by
   rw [support_conjugate]
   rw [interiorClosure_smulImage]
 
+theorem rigidStabilizer_iInter_regularSupport (F : Set G) :
+  G•[⋂ (g ∈ F), RegularSupport α g] = (⨅ (g ∈ F), G•[RegularSupport α g]) :=
+by
+  let S := { RegularSupport α g | g ∈ F }
+  have h₁ : ⋂ (g ∈ F), RegularSupport α g = ⋂₀ S := by
+    ext x
+    simp
+  have h₂ : ⨅ (g ∈ F), G•[RegularSupport α g] = ⨅ (s ∈ S), G•[s] := by
+    ext x
+    rw [<-sInf_image]
+    simp
+    rw [Subgroup.mem_iInf]
+    simp only [Subgroup.mem_iInf, and_imp, forall_apply_eq_imp_iff₂]
+
+  rw [h₁, h₂]
+  rw [rigidStabilizer_sInter]
+
+theorem rigidStabilizer_iInter_regularSupport' (F : Finset G) :
+  G•[⋂ (g ∈ F), RegularSupport α g] = (⨅ (g ∈ F), G•[RegularSupport α g]) :=
+by
+  have h₁ : G•[⋂ (g ∈ F), RegularSupport α g] = G•[⋂ (g ∈ (F : Set G)), RegularSupport α g] := by simp
+  have h₂ : ⨅ (g ∈ F), G•[RegularSupport α g] = ⨅ (g ∈ (F : Set G)), G•[RegularSupport α g] := by simp
+
+  rw [h₁, h₂, rigidStabilizer_iInter_regularSupport]
+
 end RegularSupport_continuous
 
 end Rubin

Rubin/RigidStabilizer.lean

@@ -1,5 +1,6 @@
 import Mathlib.Data.Finset.Basic
 import Mathlib.GroupTheory.GroupAction.Basic
+import Mathlib.GroupTheory.GroupAction.FixingSubgroup
 
 import Rubin.Support
 import Rubin.MulActionExt
@@ -11,6 +12,7 @@ def rigidStabilizer' (G : Type _) [Group G] [MulAction G α] (U : Set α) : Set
   {g : G | ∀ x : α, g • x = x ∨ x ∈ U}
 #align rigid_stabilizer' Rubin.rigidStabilizer'
 
+-- TODO: rename to something else? Also check the literature on what this is called
 /--
 A "rigid stabilizer" is a subgroup of `G` associated with a set `U` for which `Support α g ⊆ U` is true for all of its elements.
 
@@ -33,6 +35,14 @@ variable {G α: Type _}
 variable [Group G]
 variable [MulAction G α]
 
+theorem rigidStabilizer_eq_fixingSubgroup_compl (U : Set α) :
+  G•[U] = fixingSubgroup G Uᶜ :=
+by
+  ext g
+  rw [mem_fixingSubgroup_iff, <-Subgroup.mem_carrier]
+  unfold RigidStabilizer
+  simp
+
 theorem rigidStabilizer_support {g : G} {U : Set α} :
   g ∈ RigidStabilizer G U ↔ Support α g ⊆ U :=
 ⟨
@@ -104,6 +114,7 @@ by
       symm at gx_notin_support
       rw [fixes_inv] at gx_notin_support
       rw [<-gx_notin_support]
+      symm
       group_action
       rw [not_mem_support.mp x_notin_support]
     }
@@ -118,8 +129,9 @@ by
       have hx_notin_support := disjoint_not_mem U_disj hx_in_U?
       rw [<-support_inv] at hx_notin_support
       rw [not_mem_support] at hx_notin_support
+      symm at hx_notin_support
       group_action at hx_notin_support
-      rw [<-hx_notin_support]
+      rw [hx_notin_support]
       exact x_fixed
     }
     {
@@ -190,4 +202,15 @@ by
   rw [<-elem_moved_in_support' p h_in_rist]
   assumption
 
+-- TODO: remov ethe need for FaithfulSMul?
+theorem rigidStabilizer_neBot [FaithfulSMul G α] {U : Set α}:
+  G•[U] ≠ ⊥ → Set.Nonempty U :=
+by
+  intro ne_bot
+  by_contra empty
+  apply ne_bot
+  rw [Set.not_nonempty_iff_eq_empty] at empty
+  rw [empty]
+  exact rigidStabilizer_empty G α
+
 end Rubin

Rubin/Support.lean

@@ -15,27 +15,25 @@ The support of a group action of `g: G` on `α` (here generalized to `SMul G α`
 is the set of values `x` in `α` for which `g • x ≠ x`.
 
 This can also be thought of as the complement of the fixpoints of `(g •)`,
-which [`support_eq_not_fixed_by`] provides.
+which [`support_eq_compl_fixedBy`] provides.
 --/
+-- TODO: rename to MulAction.support
 def Support {G : Type _} (α : Type _) [SMul G α] (g : G) :=
   {x : α | g • x ≠ x}
 #align support Rubin.Support
 
-theorem SmulSupport_def {G : Type _} (α : Type _) [SMul G α] {g : G} :
-  Support α g = {x : α | g • x ≠ x} := by tauto
-
 variable {G α: Type _}
 variable [Group G]
 variable [MulAction G α]
 variable {f g : G}
 variable {x : α}
 
-theorem support_eq_not_fixed_by : Support α g = (MulAction.fixedBy α g)ᶜ := by tauto
-#align support_eq_not_fixed_by Rubin.support_eq_not_fixed_by
+theorem support_eq_compl_fixedBy : Support α g = (MulAction.fixedBy α g)ᶜ := by tauto
+#align support_eq_not_fixed_by Rubin.support_eq_compl_fixedBy
 
-theorem support_compl_eq_fixed_by : (Support α g)ᶜ = MulAction.fixedBy α g := by
+theorem fixedBy_eq_compl_support : MulAction.fixedBy α g = (Support α g)ᶜ := by
   rw [<-compl_compl (MulAction.fixedBy _ _)]
-  exact congr_arg (·ᶜ) support_eq_not_fixed_by
+  exact congr_arg (·ᶜ) support_eq_compl_fixedBy
 
 theorem mem_support :
     x ∈ Support α g ↔ g • x ≠ x := by tauto
@@ -304,8 +302,9 @@ by
   rw [<-support_inv, not_mem_support] at gp_notin_supp
   rw [mem_support] at p_in_supp
   apply p_in_supp
+  symm at gp_notin_supp
   group_action at gp_notin_supp
-  exact gp_notin_supp.symm
+  exact gp_notin_supp
 
 theorem elem_moved_in_support' {g : G} (p : α) {U : Set α} (support_in_U : Support α g ⊆ U):
   p ∈ U ↔ g • p ∈ U :=

Rubin/Tactic.lean

@@ -54,18 +54,18 @@ macro_rules
     tsub_self,
     <-mul_assoc,
 
-    one_pow,
-    one_zpow,
-    <-mul_zpow_neg_one,
-    zpow_zero,
-    mul_zpow,
-    zpow_sub,
-    <-zpow_ofNat,
-    <-zpow_neg_one,
-    <-zpow_mul,
-    <-zpow_add_one,
-    <-zpow_one_add,
-    <-zpow_add,
+    -- one_pow,
+    -- one_zpow,
+    -- <-mul_zpow_neg_one,
+    -- zpow_zero,
+    -- mul_zpow,
+    -- zpow_sub,
+    -- <-zpow_ofNat,
+    -- <-zpow_neg_one,
+    -- <-zpow_mul,
+    -- <-zpow_add_one,
+    -- <-zpow_one_add,
+    -- <-zpow_add,
 
     Int.ofNat_add,
     Int.ofNat_mul,

Rubin/Topology.lean

@@ -8,6 +8,11 @@ import Rubin.MulActionExt
 
 namespace Rubin
 
+/--
+Specificies that a group action is continuous, that is, for every group element `g`, `x ↦ g • x` is continuous.
+
+Note that this is a weaker statement than `ContinuousSMul`, as the group `G` is not required to be a topology.
+--/
 class ContinuousMulAction (G α : Type _) [Group G] [TopologicalSpace α] [MulAction G α] where
   continuous : ∀ g : G, Continuous (fun x: α => g • x)
 #align continuous_mul_action Rubin.ContinuousMulAction
@@ -35,12 +40,12 @@ theorem ContinuousMulAction.toHomeomorph_toFun {G : Type _} (α : Type _)
   [Group G] [TopologicalSpace α] [MulAction G α] [ContinuousMulAction G α]
   (g : G) : (ContinuousMulAction.toHomeomorph α g).toFun = fun x => g • x := rfl
 
-
 theorem ContinuousMulAction.toHomeomorph_invFun {G : Type _} (α : Type _)
   [Group G] [TopologicalSpace α] [MulAction G α] [ContinuousMulAction G α]
   (g : G) : (ContinuousMulAction.toHomeomorph α g).invFun = fun x => g⁻¹ • x := rfl
 
 -- TODO: give this a notation?
+-- TODO: coe to / extend MulActionHom
 structure EquivariantHomeomorph (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