✨ Start working on homeomorphic groups

Rubin/HomeoGroup.lean

@@ -0,0 +1,186 @@
+import Mathlib.Logic.Equiv.Defs
+import Mathlib.Topology.Basic
+import Mathlib.Topology.Homeomorph
+
+-- TODO: extract ContinuousMulAction into its own file, or into MulActionExt?
+import Rubin.Topological
+import Rubin.RegularSupport
+
+structure HomeoGroup (α : Type _) [TopologicalSpace α] extends
+  Homeomorph α α
+
+variable {α : Type _}
+variable [TopologicalSpace α]
+
+def HomeoGroup.coe : HomeoGroup α -> Homeomorph α α := HomeoGroup.toHomeomorph
+
+def HomeoGroup.from : Homeomorph α α -> HomeoGroup α := HomeoGroup.mk
+
+instance homeoGroup_coe : Coe (HomeoGroup α) (Homeomorph α α) where
+  coe := HomeoGroup.coe
+
+instance homeoGroup_coe₂ : Coe (Homeomorph α α) (HomeoGroup α) where
+  coe := HomeoGroup.from
+
+def HomeoGroup.toPerm : HomeoGroup α → Equiv.Perm α := fun g => g.coe.toEquiv
+
+instance homeoGroup_coe_perm : Coe (HomeoGroup α) (Equiv.Perm α) where
+  coe := HomeoGroup.toPerm
+
+@[simp]
+theorem HomeoGroup.toPerm_def (g : HomeoGroup α) : g.coe.toEquiv = (g : Equiv.Perm α) := rfl
+
+@[simp]
+theorem HomeoGroup.mk_coe (g : HomeoGroup α) : HomeoGroup.mk (g.coe) = g := rfl
+
+@[simp]
+theorem HomeoGroup.eq_iff_coe_eq {f g : HomeoGroup α} : f.coe = g.coe ↔ f = g := by
+  constructor
+  {
+    intro f_eq_g
+    rw [<-HomeoGroup.mk_coe f]
+    rw [f_eq_g]
+    simp
+  }
+  {
+    intro f_eq_g
+    unfold HomeoGroup.coe
+    rw [f_eq_g]
+  }
+
+@[simp]
+theorem HomeoGroup.from_toHomeomorph (m : Homeomorph α α) : (HomeoGroup.from m).toHomeomorph = m := rfl
+
+instance homeoGroup_one : One (HomeoGroup α) where
+  one := HomeoGroup.from (Homeomorph.refl α)
+
+theorem HomeoGroup.one_def : (1 : HomeoGroup α) = (Homeomorph.refl α : HomeoGroup α) := rfl
+
+instance homeoGroup_inv : Inv (HomeoGroup α) where
+  inv := fun g => HomeoGroup.from (g.coe.symm)
+
+@[simp]
+theorem HomeoGroup.inv_def (g : HomeoGroup α) : (Homeomorph.symm g.coe : HomeoGroup α) = g⁻¹ := rfl
+
+theorem HomeoGroup.coe_inv {g : HomeoGroup α} : HomeoGroup.coe (g⁻¹) = (HomeoGroup.coe g).symm := rfl
+
+instance homeoGroup_mul : Mul (HomeoGroup α) where
+  mul := fun a b => ⟨b.toHomeomorph.trans a.toHomeomorph⟩
+
+theorem HomeoGroup.coe_mul {f g : HomeoGroup α} : HomeoGroup.coe (f * g) = (HomeoGroup.coe g).trans (HomeoGroup.coe f) := rfl
+
+@[simp]
+theorem HomeoGroup.mul_def (f g : HomeoGroup α) : HomeoGroup.from ((HomeoGroup.coe g).trans (HomeoGroup.coe f)) = f * g := rfl
+
+instance homeoGroup_group : Group (HomeoGroup α) where
+  mul_assoc := by
+    intro a b c
+    rw [<-HomeoGroup.eq_iff_coe_eq]
+    repeat rw [HomeoGroup_coe_mul]
+    rfl
+  mul_one := by
+    intro a
+    rw [<-HomeoGroup.eq_iff_coe_eq]
+    rw [HomeoGroup.coe_mul]
+    rfl
+  one_mul := by
+    intro a
+    rw [<-HomeoGroup.eq_iff_coe_eq]
+    rw [HomeoGroup.coe_mul]
+    rfl
+  mul_left_inv := by
+    intro a
+    rw [<-HomeoGroup.eq_iff_coe_eq]
+    rw [HomeoGroup.coe_mul]
+    rw [HomeoGroup.coe_inv]
+    simp
+    rfl
+
+/--
+The HomeoGroup trivially has a continuous and faithful `MulAction` on the underlying topology `α`.
+--/
+instance homeoGroup_smul₁ : SMul (HomeoGroup α) α where
+  smul := fun g x => g.toFun x
+
+@[simp]
+theorem HomeoGroup.smul₁_def (f : HomeoGroup α) (x : α) : f.toFun x = f • x := rfl
+
+@[simp]
+theorem HomeoGroup.smul₁_def' (f : HomeoGroup α) (x : α) : f.toHomeomorph x = f • x := rfl
+
+@[simp]
+theorem HomeoGroup.coe_toFun_eq_smul₁ (f : HomeoGroup α) (x : α) : FunLike.coe (HomeoGroup.coe f) x = f • x := rfl
+
+instance homeoGroup_mulAction₁ : MulAction (HomeoGroup α) α where
+  one_smul := by
+    intro x
+    rfl
+  mul_smul := by
+    intro f g x
+    rfl
+
+instance homeoGroup_mulAction₁_continuous : Rubin.ContinuousMulAction (HomeoGroup α) α where
+  continuous := by
+    intro h
+    constructor
+    intro S S_open
+    conv => {
+      congr; ext
+      congr; ext
+      rw [<-HomeoGroup.smul₁_def']
+    }
+    simp only [Homeomorph.isOpen_preimage]
+    exact S_open
+
+instance homeoGroup_mulAction₁_faithful : FaithfulSMul (HomeoGroup α) α where
+  eq_of_smul_eq_smul := by
+    intro f g hyp
+    rw [<-HomeoGroup.eq_iff_coe_eq]
+    ext x
+    simp
+    exact hyp x
+
+namespace Rubin
+
+variable {α : Type _}
+variable [TopologicalSpace α]
+
+-- Note that the condition that the resulting set is non-empty is introduced later in `RegularInter`
+-- TODO: rename!!!
+def RegularInterElem (S : Finset (HomeoGroup α)): Set α :=
+  ⋂₀ ((fun (g : HomeoGroup α) => RegularSupport α g) '' S)
+
+def RegularInter (α : Type _) [TopologicalSpace α]: Type* :=
+  { S : Set α //
+    Set.Nonempty S
+    ∧ ∃ (seed : Finset (HomeoGroup α)), S = RegularInterElem seed
+  }
+
+@[simp]
+theorem regularInter_open (S : RegularInter α) : Set.Nonempty S.val := S.prop.left
+
+@[simp]
+theorem regularInter_regular (S : RegularInter α) : Regular S.val := by
+  have ⟨seed, S_from_seed⟩ := S.prop.right
+  rw [S_from_seed]
+  unfold RegularInterElem
+
+  apply regular_sInter
+  · have set_decidable : DecidableEq (Set α) := Classical.typeDecidableEq (Set α)
+    let fin : Finset (Set α) := seed.image ((fun g => RegularSupport α g))
+
+    apply Set.Finite.ofFinset fin
+    simp
+  · intro S S_in_set
+    simp at S_in_set
+    let ⟨g, ⟨_, Heq⟩⟩ := S_in_set
+    rw [<-Heq]
+    exact regularSupport_regular α g
+
+-- TODO:
+-- def RegularInter.smul : HomeoGroup α → RegularInter α -> RegularInter α
+
+-- instance homeoGroup_smul₂ : SMul (HomeoGroup α) (RegularInter α) where
+--   smul := fun g x =>
+
+end Rubin

Rubin/InteriorClosure.lean

@@ -1,6 +1,8 @@
 import Mathlib.Topology.Basic
 import Mathlib.Topology.Separation
 
+import Rubin.SmulImage
+
 namespace Rubin
 
 variable {α : Type _}
@@ -189,4 +191,100 @@ by
   · exact (interiorClosure_subset U'_open) u_in_U'
   · exact (interiorClosure_subset V'_open) v_in_V'
 
+theorem regular_inter {U V : Set α} : Regular U → Regular V → Regular (U ∩ V) :=
+by
+  intro U_reg V_reg
+  simp
+  have UV_open : IsOpen (U ∩ V) := IsOpen.inter U_reg.isOpen V_reg.isOpen
+
+  apply Set.eq_of_subset_of_subset
+  · simp
+    constructor
+    · nth_rw 2 [<-U_reg]
+      apply interiorClosure_mono
+      simp
+    · nth_rw 2 [<-V_reg]
+      apply interiorClosure_mono
+      simp
+  · apply interiorClosure_subset
+    exact UV_open
+
+theorem regular_sInter {S : Set (Set α)} (S_finite : Set.Finite S) (all_reg : ∀ U ∈ S, Regular U):
+  Regular (⋂₀ S) :=
+Set.Finite.induction_on' S_finite (by simp) (by
+  intro U S' U_in_S _ _ IH
+  rw [Set.sInter_insert]
+  apply regular_inter
+  · exact all_reg _ U_in_S
+  · exact IH
+)
+
+theorem smulImage_interior {G : Type _} [Group G] [MulAction G α]
+  (g : G) (U : Set α)
+  (g_continuous : ∀ S : Set α, IsOpen S → IsOpen (g •'' S) ∧ IsOpen (g⁻¹ •'' S)):
+  interior (g •'' U) = g •'' interior U :=
+by
+  unfold interior
+  rw [smulImage_sUnion]
+  simp
+  ext x
+  simp
+  constructor
+  · intro ⟨T, ⟨T_open, T_sub⟩, x_in_T⟩
+    use g⁻¹ •'' T
+    repeat' apply And.intro
+    · exact (g_continuous T T_open).right
+    · rw [smulImage_subset_inv]
+      rw [inv_inv]
+      exact T_sub
+    · rw [smulImage_mul, mul_right_inv, one_smulImage]
+      exact x_in_T
+  · intro ⟨T, ⟨T_open, T_sub⟩, x_in_T⟩
+    use g •'' T
+    repeat' apply And.intro
+    · exact (g_continuous T T_open).left
+    · apply smulImage_mono
+      exact T_sub
+    · exact x_in_T
+
+theorem smulImage_closure {G : Type _} [Group G] [MulAction G α]
+  (g : G) (U : Set α)
+  (g_continuous : ∀ S : Set α, IsOpen S → IsOpen (g •'' S) ∧ IsOpen (g⁻¹ •'' S)):
+  closure (g •'' U) = g •'' closure U :=
+by
+  have g_continuous' : ∀ S : Set α, IsClosed S → IsClosed (g •'' S) ∧ IsClosed (g⁻¹ •'' S) := by
+    intro S S_closed
+    rw [<-isOpen_compl_iff] at S_closed
+    repeat rw [<-isOpen_compl_iff]
+    repeat rw [smulImage_compl]
+    exact g_continuous _ S_closed
+  unfold closure
+  rw [smulImage_sInter]
+  simp
+  ext x
+  simp
+  constructor
+  · intro IH T' T T_closed U_ss_T T'_eq
+    rw [<-T'_eq]
+    clear T' T'_eq
+    apply IH
+    · exact (g_continuous' _ T_closed).left
+    · apply smulImage_mono
+      exact U_ss_T
+  · intro IH T T_closed gU_ss_T
+    apply IH
+    · exact (g_continuous' _ T_closed).right
+    · rw [<-smulImage_subset_inv]
+      exact gU_ss_T
+    · simp
+
+theorem interiorClosure_smulImage {G : Type _} [Group G] [MulAction G α]
+  (g : G) (U : Set α)
+  (g_continuous : ∀ S : Set α, IsOpen S → IsOpen (g •'' S) ∧ IsOpen (g⁻¹ •'' S)) :
+  InteriorClosure (g •'' U) = g •'' InteriorClosure U :=
+by
+  simp
+  rw [<-smulImage_interior _ _ g_continuous]
+  rw [<-smulImage_closure _ _ g_continuous]
+
 end Rubin

Rubin/MulActionExt.lean

@@ -71,4 +71,13 @@ by
     rw [<-mul_smul, mul_right_inv, one_smul]
   }
 
+lemma exists_smul_ne {G : Type _} (α : Type _) [Group G] [MulAction G α] [h_f : FaithfulSMul G α]
+  {f g : G} (f_ne_g : f ≠ g) : ∃ (x : α), f • x ≠ g • x :=
+by
+  by_contra h
+  rw [Classical.not_exists_not] at h
+  apply f_ne_g
+  apply h_f.eq_of_smul_eq_smul
+  exact h
+
 end Rubin

Rubin/RegularSupport.lean

@@ -63,6 +63,13 @@ by
   apply interiorClosure_mono
   exact h
 
+theorem regularSupport_subset_closure_support {g : G} :
+  RegularSupport α g ⊆ closure (Support α g) :=
+by
+  unfold RegularSupport
+  simp
+  exact interior_subset
+
 theorem regularSupport_subset_of_rigidStabilizer {g : G} {U : Set α} (U_reg : Regular U) :
   g ∈ RigidStabilizer G U → RegularSupport α g ⊆ U :=
 by
@@ -86,4 +93,12 @@ by
 
 end RegularSupport_continuous
 
+theorem regularSupport_smulImage {G α : Type _} [Group G] [TopologicalSpace α] [ContinuousMulAction G α]
+  {f g : G} :
+  f •'' (RegularSupport α g) = RegularSupport α (f * g * f⁻¹) :=
+by
+  unfold RegularSupport
+  rw [support_conjugate]
+  rw [interiorClosure_smulImage _ _ (continuousMulAction_elem_continuous α f)]
+
 end Rubin

Rubin/SmulImage.lean

@@ -105,9 +105,18 @@ by
   repeat rw [<-smulImage_mul]
   exact SmulImage.congr g⁻¹ h
 
-theorem smulImage_inv {g: G} {U V : Set α} (img_eq : g •'' U = g •'' V) : U = V :=
+theorem smulImage_inv (g: G) (U V : Set α) : g •'' U = V ↔ U = g⁻¹ •'' V := by
-  SmulImage.inv_congr g img_eq
+  nth_rw 2 [<-one_smulImage (G := G) U]
+  rw [<-mul_left_inv g, <-smulImage_mul]
+  constructor
+  · intro h
+    rw [SmulImage.congr]
+    exact h
+  · intro h
+    apply SmulImage.inv_congr at h
+    exact h
 
+-- TODO: rename to smulImage_mono
 theorem smulImage_subset (g : G) {U V : Set α} : U ⊆ V → g •'' U ⊆ g •'' V :=
   by
   intro h1 x
@@ -115,6 +124,8 @@ theorem smulImage_subset (g : G) {U V : Set α} : U ⊆ V → g •'' U ⊆ g
   exact fun h2 => h1 h2
 #align smul''_subset Rubin.smulImage_subset
 
+def smulImage_mono (g : G) {U V : Set α} : U ⊆ V → g •'' U ⊆ g •'' V := smulImage_subset g
+
 theorem smulImage_union (g : G) {U V : Set α} : g •'' U ∪ V = (g •'' U) ∪ (g •'' V) :=
   by
   ext
@@ -129,6 +140,56 @@ theorem smulImage_inter (g : G) {U V : Set α} : g •'' U ∩ V = (g •'' U)
     Rubin.mem_smulImage, Set.mem_inter_iff]
 #align smul''_inter Rubin.smulImage_inter
 
+theorem smulImage_sUnion (g : G) {S : Set (Set α)} : g •'' (⋃₀ S) = ⋃₀ {g •'' T | T ∈ S} :=
+by
+  ext x
+  constructor
+  · intro h
+    rw [mem_smulImage, Set.mem_sUnion] at h
+    rw [Set.mem_sUnion]
+    let ⟨T, ⟨T_in_S, ginv_x_in_T⟩⟩ := h
+    simp
+    use T
+    constructor; trivial
+    rw [mem_smulImage]
+    exact ginv_x_in_T
+  · intro h
+    rw [Set.mem_sUnion] at h
+    rw [mem_smulImage, Set.mem_sUnion]
+    simp at h
+    let ⟨T, ⟨T_in_S, x_in_gT⟩⟩ := h
+    use T
+    constructor; trivial
+    rw [<-mem_smulImage]
+    exact x_in_gT
+
+theorem smulImage_sInter (g : G) {S : Set (Set α)} : g •'' (⋂₀ S) = ⋂₀ {g •'' T | T ∈ S} :=
+by
+  ext x
+  constructor
+  · intro h
+    rw [mem_smulImage, Set.mem_sInter] at h
+    rw [Set.mem_sInter]
+    simp
+    intro T T_in_S
+    rw [mem_smulImage]
+    exact h T T_in_S
+  · intro h
+    rw [Set.mem_sInter] at h
+    rw [mem_smulImage, Set.mem_sInter]
+    intro T T_in_S
+    rw [<-mem_smulImage]
+    simp at h
+    exact h T T_in_S
+
+@[simp]
+theorem smulImage_compl (g : G) (U : Set α) : (g •'' U)ᶜ = g •'' Uᶜ :=
+by
+  ext x
+  rw [Set.mem_compl_iff]
+  repeat rw [mem_smulImage]
+  rw [Set.mem_compl_iff]
+
 theorem smulImage_eq_inv_preimage {g : G} {U : Set α} : g •'' U = (g⁻¹ • ·) ⁻¹' U :=
   by
   ext

Rubin/Support.lean

@@ -229,4 +229,32 @@ by
   apply Set.eq_empty_iff_forall_not_mem.mp support_empty
   exact h
 
+theorem support_eq: Support α f = Support α g ↔ ∀ (x : α), (f • x = x ∧ g • x = x) ∨ (f • x ≠ x ∧ g • x ≠ x) := by
+  constructor
+  · intro h
+    intro x
+    by_cases x_in? : x ∈ Support α f
+    · right
+      have gx_ne_x := by rw [h] at x_in?; exact x_in?
+      exact ⟨x_in?, gx_ne_x⟩
+    · left
+      have fx_eq_x : f • x = x := by rw [<-not_mem_support]; exact x_in?
+      have gx_eq_x : g • x = x := by rw [<-not_mem_support, <-h]; exact x_in?
+      exact ⟨fx_eq_x, gx_eq_x⟩
+  · intro h
+    ext x
+    constructor
+    · intro fx_ne_x
+      rw [mem_support] at fx_ne_x
+      rw [mem_support]
+      cases h x with
+      | inl h₁ => exfalso; exact fx_ne_x h₁.left
+      | inr h₁ => exact h₁.right
+    · intro gx_ne_x
+      rw [mem_support] at gx_ne_x
+      rw [mem_support]
+      cases h x with
+      | inl h₁ => exfalso; exact gx_ne_x h₁.right
+      | inr h₁ => exact h₁.left
+
 end Rubin

Rubin/Topological.lean

@@ -13,12 +13,23 @@ namespace Rubin
 
 section Continuity
 
--- TODO: don't have this extend MulAction
+-- TODO: don't have this extend MulAction and move this somewhere else
 class ContinuousMulAction (G α : Type _) [Group G] [TopologicalSpace α] extends
     MulAction G α where
   continuous : ∀ g : G, Continuous (fun x: α => g • x)
 #align continuous_mul_action Rubin.ContinuousMulAction
 
+theorem continuousMulAction_elem_continuous {G : Type _} (α : Type _)
+  [Group G] [TopologicalSpace α] [hc : ContinuousMulAction G α] (g : G):
+  ∀ (S : Set α), IsOpen S → IsOpen (g •'' S) ∧ IsOpen ((g⁻¹) •'' S) :=
+by
+  intro S S_open
+  repeat rw [smulImage_eq_inv_preimage]
+  rw [inv_inv]
+  constructor
+  · exact (hc.continuous g⁻¹).isOpen_preimage _ S_open
+  · exact (hc.continuous g).isOpen_preimage _ S_open
+
 -- TODO: give this a notation?
 structure EquivariantHomeomorph (G α β : Type _) [Group G] [TopologicalSpace α]
     [TopologicalSpace β] [MulAction G α] [MulAction G β] extends Homeomorph α β where