diff --git a/Rubin.lean b/Rubin.lean index 39bf500..cfb5532 100644 --- a/Rubin.lean +++ b/Rubin.lean @@ -1275,603 +1275,23 @@ by theorem proposition_3_5' {U : Set α} (U_in_basis : U ∈ RegularSupportBasis G α) (F : UltrafilterInBasis (RegularSupportBasis G α)): - (∃ p ∈ U, ClusterPt p F) - ↔ ∃ V : RegularSupportBasis G α, V.val ⊆ U ∧ RSuppSubsets G V.val ⊆ RSuppOrbit F G•[U] := + (∃ p ∈ U, F ≤ nhds p) + ↔ ∃ V ∈ RegularSupportBasis G α, V ⊆ U ∧ RSuppSubsets G V ⊆ RSuppOrbit F G•[U] := by constructor - · simp only [ - F.clusterPt_iff_le_nhds + · intro ex_p + let ⟨⟨V, V_in_basis⟩, V_ss_U, subsets_ss_orbit⟩ := proposition_3_5_1 U_in_basis (F : Filter α) ex_p + exact ⟨V, V_in_basis, V_ss_U, subsets_ss_orbit⟩ + · intro ⟨V, V_in_basis, V_ss_U, subsets_ss_orbit⟩ + simp only [ + <-F.clusterPt_iff_le_nhds (RegularSupportBasis.isBasis G α) (RegularSupportBasis.closed_inter G α) ] - exact proposition_3_5_1 U_in_basis (F : Filter α) - · exact proposition_3_5_2 (F : Filter α) + exact proposition_3_5_2 (F : Filter α) ⟨⟨V, V_in_basis⟩, V_ss_U, subsets_ss_orbit⟩ end Ultrafilter - -/- -variable {G α : Type _} -variable [Group G] - -variable [TopologicalSpace α] [MulAction G α] [ContinuousConstSMul G α] - -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 - -@[simp] -theorem IsRigidSubgroup.mem_subgroup {S : Set G} (S_rigid : IsRigidSubgroup S) (g : G): - g ∈ S_rigid.toSubgroup ↔ g ∈ S := 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] [ContinuousConstSMul G α] -variable [FaithfulSMul G α] [T2Space α] [LocallyMoving G α] - -theorem IsRigidSubgroup.has_regularSupportBasis {S : Set G} (S_rigid : IsRigidSubgroup S) : - ∃ (U : RegularSupportBasis G 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 - - let T' := RegularSupport.FiniteInter α T - - have T'_nonempty : Set.Nonempty T' := by - simp only [RegularSupport.FiniteInter, proposition_2_1 (G := G) (α := α)] at S_ne_bot - rw [ne_eq, <-rigidStabilizer_iInter_regularSupport', <-ne_eq] at S_ne_bot - exact rigidStabilizer_neBot S_ne_bot - - have T'_in_basis : T' ∈ RegularSupportBasis G α := by - constructor - assumption - use T - - use ⟨T', T'_in_basis⟩ - simp [RegularSupport.FiniteInter] - rw [rigidStabilizer_iInter_regularSupport'] - simp only [<-proposition_2_1] - -noncomputable def IsRigidSubgroup.toRegularSupportBasis {S : Set G} (S_rigid : IsRigidSubgroup S) : - RegularSupportBasis G 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_regular {S : Set G} (S_rigid : IsRigidSubgroup S): - Regular (S_rigid.toRegularSupportBasis α).val := -by - apply RegularSupportBasis.regular (G := G) - exact (toRegularSupportBasis α S_rigid).prop - -theorem IsRigidSubgroup.toRegularSupportBasis_nonempty {S : Set G} (S_rigid : IsRigidSubgroup S): - Set.Nonempty (S_rigid.toRegularSupportBasis α).val := -by - exact (Subtype.prop (S_rigid.toRegularSupportBasis α)).left - -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 := -by - rw [rigidStabilizer_subset_iff G (toRegularSupportBasis_regular _ S_rigid) (toRegularSupportBasis_regular _ T_rigid)] - 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 - exact S_ss_T - · intro 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 [<-IsRigidSubgroup.toRegularSupportBasis_mono] - -@[simp] -theorem IsRigidSubgroup.toRegularSupportBasis_rigidStabilizer {S : Set G} (S_rigid : IsRigidSubgroup S) : - G•[(S_rigid.toRegularSupportBasis α : Set α)] = S := -by - sorry - -- TODO: prove that `G•[S_rigid.toRegularSupportBasis] = S` - -@[simp] -theorem IsRigidSubgroup.toRegularSupportBasis_rigidStabilizer' {S : Set G} (S_rigid : IsRigidSubgroup S) (g : G): - g ∈ G•[(S_rigid.toRegularSupportBasis α : Set α)] ↔ g ∈ S := -by - rw [<-SetLike.mem_coe, IsRigidSubgroup.toRegularSupportBasis_rigidStabilizer] - -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 : Filter 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 - -- Issue: It's *really hard* to generate ultrafilters on G that satisfy the other conditions of this rubinfilter - filter : Ultrafilter G - - -- Note: the following condition cannot be met by ultrafilters in G, - -- and doesn't seem to be necessary - -- rigid_basis : ∀ S ∈ filter, ∃ T ⊆ S, IsRigidSubgroup T - - -- Equivalent formulation of convergence - 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 {G : Type _} [Group G] -variable (α : Type _) -variable [TopologicalSpace α] [T2Space α] [MulAction G α] [ContinuousConstSMul G α] -variable [FaithfulSMul G α] [LocallyDense G α] [LocallyCompactSpace α] [HasNoIsolatedPoints 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 - apply IsRigidSubgroup.toRegularSupportBasis_nonempty - · 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 - --- theorem RubinFilter.to_action_filter_converges' (F : RubinFilter G) : --- ∀ U : Set α, U ∈ RegularSupportBasis G α → U ∈ F.to_action_filter → --- ∃ V ⊆ F.to_action_filter, V ⊆ U ∧ - -theorem RubinFilter.to_action_filter_mem {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 - sorry - -theorem RubinFilter.to_action_filter_mem' {F : RubinFilter G} {U : Set α} (U_in_basis : U ∈ RegularSupportBasis G α) : - U ∈ F.to_action_filter α ↔ (G•[U] : Set G) ∈ F.filter := -by - -- trickier to prove but should be possible - sorry - -theorem RubinFilter.to_action_filter_clusterPt [Nonempty α] (F : RubinFilter G) : - ∃ p : α, ClusterPt p (F.to_action_filter α) := -by - have univ_in_basis : Set.univ ∈ RegularSupportBasis G α := by - rw [RegularSupportBasis.mem_iff] - simp - use {} - simp [RegularSupport.FiniteInter] - - have univ_rigid : IsRigidSubgroup (G := G) Set.univ := by - constructor - simp [Set.eq_singleton_iff_unique_mem] - exact LocallyMoving.nontrivial_elem G α - use {} - simp - - suffices ∃ p ∈ Set.univ, ClusterPt p (F.to_action_filter α) by - let ⟨p, _, clusterPt⟩ := this - use p - - apply proposition_3_5_2 (G := G) (α := α) - simp - let ⟨S, S_rigid, _, S_subsets_ss_orbit⟩ := F.converges _ Filter.univ_mem univ_rigid - - use S_rigid.toRegularSupportBasis α - constructor - simp - - unfold RSuppSubsets RSuppOrbit - simp - intro T T_in_basis T_ss_S - - - let T' := G•[T] - have T_regular : Regular T := sorry -- easy - have T'_rigid : IsRigidSubgroup (T' : Set G) := sorry -- provable - have T'_ss_S : (T' : Set G) ⊆ S := sorry -- using one of our lovely theorems - - have T'_in_subsets : T' ∈ AlgebraicSubsets S := by - unfold AlgebraicSubsets - simp - constructor - sorry -- prove that rigid subgroups are in the algebraic centralizer basis - exact T'_ss_S - - let ⟨g, _, W, W_in_F, W_rigid, W_conj⟩ := S_subsets_ss_orbit T'_in_subsets - - use g - constructor - sorry -- TODO: G•[univ] = top - - let W' := W_rigid.toRegularSupportBasis α - have W'_regular : Regular (W' : Set α) := by - apply RegularSupportBasis.regular (G := G) - simp - use W' - - constructor - rw [<-RubinFilter.to_action_filter_mem] - assumption - - rw [<-rigidStabilizer_eq_iff (α := α) (G := G) ((smulImage_regular _ _).mp W'_regular) T_regular] - - ext i - rw [rigidStabilizer_smulImage] - unfold_let at W_conj - rw [<-W_conj] - simp - constructor - · intro - use g⁻¹ * i * g - constructor - assumption - group - · intro ⟨j, j_in_W, gjg_eq_i⟩ - rw [<-gjg_eq_i] - group - assumption - --- theorem RubinFilter.to_action_filter_le_nhds [Nonempty α] (F : RubinFilter G) : --- ∃ p : α, (F.to_action_filter α) ≤ 𝓝 p := --- by --- let ⟨p, p_clusterPt⟩ := to_action_filter_clusterPt α F --- use p --- intro S S_in_nhds --- rw [(RegularSupportBasis.isBasis G α).mem_nhds_iff] at S_in_nhds --- let ⟨T, T_in_basis, p_in_T, T_ss_S⟩ := S_in_nhds - --- suffices T ∈ F.to_action_filter α by --- apply Filter.sets_of_superset (F.to_action_filter α) this T_ss_S - --- rw [RubinFilter.to_action_filter_mem' _ T_in_basis] - --- intro S p_in_S S_open --- sorry - -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 -- pain - -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)] - swap - { - apply Subtype.prop (IsRigidSubgroup.toRegularSupportBasis α U_rigid) - } - - 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_in_basis S_ss_V - -- let ⟨S', S'_eq⟩ := S_in_basis.right - - have S_regular : Regular S := RegularSupportBasis.regular S_in_basis - - have S_nonempty : Set.Nonempty S := S_in_basis.left - - have GS_ss_V : G•[S] ≤ V := by - rw [<-V_rigid.toRegularSupportBasis_eq (α := α)] - simp only [Set.le_eq_subset, SetLike.coe_subset_coe] - rw [<-rigidStabilizer_subset_iff G (α := α) S_regular (IsRigidSubgroup.toRegularSupportBasis_regular _ V_rigid)] - 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⟩ := S_in_basis.right - rw [RigidStabilizerBasis.mem_iff' _ (LocallyMoving.locally_moving _ S_regular.isOpen S_nonempty)] - use S' - rw [RigidStabilizerInter₀, S'_eq, RegularSupport.FiniteInter, rigidStabilizer_iInter_regularSupport'] - · 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 - { - - 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 IsRigidSubgroup.toRegularSupportBasis_regular - } - swap - exact S_regular - - ext i - rw [rigidStabilizer_smulImage, <-Wconj_eq_GS] - 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 - --- Idea: 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.filter U) - ↔ (∃ V' : Set G, V' ≤ U ∧ AlgebraicSubsets V' ⊆ AlgebraicOrbit F'.filter 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 RubinFilterBasis : Set (Set (RubinFilter G)) := - (fun S : Subgroup G => { F : RubinFilter G | (S : Set G) ∈ F.filter }) '' AlgebraicCentralizerBasis G - -instance : TopologicalSpace (RubinFilter G) := TopologicalSpace.generateFrom RubinFilterBasis - -def RubinSpace (G : Type _) [Group G] := Quotient (RubinFilterSetoid G) - -instance : TopologicalSpace (RubinSpace G) := by - unfold RubinSpace - infer_instance - -instance : MulAction G (RubinSpace G) := sorry - -end Equivalence - -section Convert -open Topology - -variable (G α : Type _) -variable [Group G] -variable [TopologicalSpace α] [Nonempty α] [T2Space α] [HasNoIsolatedPoints α] [LocallyCompactSpace α] -variable [MulAction G α] [ContinuousConstSMul G α] [FaithfulSMul G α] [LocallyMoving G α] [LocallyDense G α] - -instance RubinFilter.fromElement_neBot (x : α) : Filter.NeBot (⨅ (S ∈ 𝓝 x), Filter.principal (G•[S] : Set G)) := by sorry - -noncomputable def RubinFilter.fromElement (x : α) : RubinFilter G where - filter := @Ultrafilter.of _ (⨅ (S ∈ 𝓝 x), Filter.principal (G•[S] : Set G)) (RubinFilter.fromElement_neBot G α x) - converges := by - sorry - ne_bot := by - sorry -- this will be fun to try and prove - --- Alternate idea: don't try to compute the associated ultrafilter, and only define a predicate? -theorem RubinFilter.converging_element (F : RubinFilter G) : - ∃ p : α, ClusterPt p (F.to_action_ultrafilter α) := -by - have univ_in_F : Set.univ ∈ F.filter := Filter.univ_mem - have univ_in_basis : IsRigidSubgroup (G := G) Set.univ := by - constructor - sorry -- TODO: prove that Set.univ ≠ {1}, from locallydense - use {} - simp - - let ⟨p, p_in_basis, clusterPt_p⟩ := RubinFilter.to_action_ultrafilter_converges α F univ_in_F univ_in_basis - - use p - -noncomputable def RubinFilter.toElement (F : RubinFilter G) : α := - (F.converging_element G α).choose - -theorem RubinFilter.toElement_equiv (F F' : RubinFilter G) (equiv : F ≈ F'): - F.toElement G α = F'.toElement G α := -by - - sorry - -theorem rubin' (hα : RubinAction G α) : EquivariantHomeomorph G α (RubinSpace G) where - toFun := fun x => ⟦RubinFilter.fromElement (G := G) α x⟧ - invFun := fun f => f.liftOn (RubinFilter.toElement G α) (RubinFilter.toElement_equiv G α) - continuous_toFun := by - simp - constructor - intro S S_open - rw [<-isOpen_coinduced] - -- Note the sneaky different IsOpen's - -- TODO: apply topologicalbasis on both isopen - sorry - continuous_invFun := by - simp - sorry - left_inv := by - intro x - simp - sorry - right_inv := by - intro F - nth_rw 2 [<-Quotient.out_eq F] - rw [Quotient.eq] - simp - sorry - equivariant := by - simp - sorry - -end Convert --/ - - section RubinFilter variable {G : Type _} [Group G] @@ -1890,10 +1310,15 @@ by simp [AlgebraicOrbit] simp only [and_assoc, eq_comm] +def AlgebraicConvergent {G : Type _} [Group G] + (F : Filter G) + (U : Set G) : Prop := + ∃ V ∈ AlgebraicCentralizerBasis G, V ⊆ U ∧ AlgebraicSubsets V ⊆ AlgebraicOrbit F U + structure RubinFilter (G : Type _) [Group G] := filter : UltrafilterInBasis (AlgebraicCentralizerBasis G) - converges : ∃ V ∈ AlgebraicCentralizerBasis G, AlgebraicSubsets V ⊆ AlgebraicOrbit filter Set.univ + converges : AlgebraicConvergent filter.filter Set.univ lemma RegularSupportBasis.empty_not_mem' : ∅ ∉ (RigidStabilizer.inv_order_iso_on G α).toFun '' AlgebraicCentralizerBasis G := by simp @@ -2040,6 +1465,35 @@ by unfold_let rw [RigidStabilizer_leftInv U_in_basis] +theorem IsRubinFilterOf.converges_iff {A : UltrafilterInBasis (RegularSupportBasis G α)} + {B : UltrafilterInBasis (AlgebraicCentralizerBasis G)} + (filter_of : IsRubinFilterOf A B) + {V : Set α} (V_in_basis : V ∈ RegularSupportBasis G α) + : + (∃ p ∈ V, A ≤ nhds p) ↔ + AlgebraicConvergent B.filter G•[V] := +by + unfold AlgebraicConvergent + constructor + · rw [proposition_3_5' V_in_basis] + intro ⟨W, W_in_basis, W_ss_V, subsets_ss_orbit⟩ + use G•[W] + rw [<-filter_of.subsets_ss_orbit W_in_basis] + refine ⟨?GW_in_basis, ?GW_ss_GV, subsets_ss_orbit⟩ + exact AlgebraicCentralizerBasis.mem_of_regularSupportBasis W_in_basis + simp + rwa [<-rigidStabilizer_subset_iff _ (RegularSupportBasis.regular W_in_basis) (RegularSupportBasis.regular V_in_basis)] + · intro ⟨W, W_in_basis, W_ss_GV, subsets_ss_orbit⟩ + rw [<-AlgebraicCentralizerBasis.eq_rist_image (α := α)] at W_in_basis + let ⟨W', W'_in_basis, W'_eq⟩ := W_in_basis + simp only at W'_eq + rw [proposition_3_5' V_in_basis] + use W' + rw [filter_of.subsets_ss_orbit W'_in_basis, W'_eq] + refine ⟨W'_in_basis, ?W'_ss_V, subsets_ss_orbit⟩ + rw [<-W'_eq] at W_ss_GV + simp at W_ss_GV + rwa [<-rigidStabilizer_subset_iff _ (RegularSupportBasis.regular W'_in_basis) (RegularSupportBasis.regular V_in_basis)] at W_ss_GV def RubinFilter.from (F : UltrafilterInBasis (RegularSupportBasis G α)) (F_converges : ∃ p : α, F ≤ nhds p) : RubinFilter G where filter := (F.map_basis @@ -2051,26 +1505,339 @@ def RubinFilter.from (F : UltrafilterInBasis (RegularSupportBasis G α)) (F_conv converges := by let ⟨p, F_le_nhds⟩ := F_converges - have F_clusterPt : ClusterPt p F := by - rw [UltrafilterInBasis.clusterPt_iff_le_nhds] - · assumption - · exact RegularSupportBasis.isBasis G α - · exact RegularSupportBasis.closed_inter G α + have ⟨W, W_in_basis, _, W_subsets_ss_GV_orbit⟩ := (proposition_3_5' (RegularSupportBasis.univ_mem G α) F).mp ⟨p, (Set.mem_univ p), F_le_nhds⟩ - have ⟨⟨W, W_in_basis⟩, W_ss_V, W_subsets_ss_GV_orbit⟩ := (proposition_3_5' (RegularSupportBasis.univ_mem G α) F).mp ⟨p, (Set.mem_univ p), F_clusterPt⟩ - simp only at W_ss_V - simp only at W_subsets_ss_GV_orbit + refine ⟨ + G•[W], + by apply AlgebraicCentralizerBasis.mem_of_regularSupportBasis W_in_basis, + by simp, + ?subsets_ss_orbit + ⟩ - use G•[W] + rw [<-Subgroup.coe_top, <-rigidStabilizer_univ (α := α) (G := G)] + rwa [<-(RubinFilter.from_isRubinFilterOf' F).subsets_ss_orbit W_in_basis] + + +theorem RubinFilter.from_isRubinFilterOf (F : UltrafilterInBasis (RegularSupportBasis G α)) (F_converges : ∃ p : α, F ≤ nhds p): + IsRubinFilterOf F (RubinFilter.from F F_converges).filter := RubinFilter.from_isRubinFilterOf' F + +theorem RubinFilter.map_from_eq (F : UltrafilterInBasis (RegularSupportBasis G α)) (F_converges : ∃ p : α, F ≤ nhds p): + (RubinFilter.from F F_converges).map α = F := +by + apply UltrafilterInBasis.eq_of_le + apply UltrafilterInBasis.le_of_basis_sets + intro S S_in_B S_in_F + + rw [(RubinFilter.from_isRubinFilterOf F F_converges) S S_in_B] at S_in_F + rw [(RubinFilter.map_isRubinFilterOf (RubinFilter.from F F_converges) (α := α)) S S_in_B] + + exact S_in_F + +section Convergence + +variable (α : Type _) +variable [Nonempty α] [TopologicalSpace α] [T2Space α] [HasNoIsolatedPoints α] [LocallyCompactSpace α] +variable [MulAction G α] [ContinuousConstSMul G α] [FaithfulSMul G α] [LocallyDense G α] + +theorem RubinFilter.map_converges (F : RubinFilter G): + ∃ p : α, (F.map α).filter ≤ nhds p := +by + suffices ∃ p ∈ Set.univ, (F.map α).filter ≤ nhds p by + let ⟨p, _, f_le_nhds⟩ := this + exact ⟨p, f_le_nhds⟩ + + rw [proposition_3_5' (RegularSupportBasis.univ_mem G α)] + let ⟨V, V_in_basis, _, subsets_ss_orbit⟩ := F.converges + simp only [Set.subset_univ, true_and, Subtype.exists, exists_prop] + use rigidStabilizer_inv V + refine ⟨(rigidStabilizer_inv_in_basis V).mp V_in_basis, ?subsets_ss_orbit⟩ + rw [(RubinFilter.map_isRubinFilterOf F (α := α)).subsets_ss_orbit + ((rigidStabilizer_inv_in_basis V).mp V_in_basis) + ] + rw [RigidStabilizer_rightInv V_in_basis] + simp + exact subsets_ss_orbit + +theorem RubinFilter.from_map_eq (F : RubinFilter G): + RubinFilter.from (F.map α) (RubinFilter.map_converges α F) = F := +by + rw [mk.injEq] + apply UltrafilterInBasis.eq_of_le + apply UltrafilterInBasis.le_of_basis_sets + intro S S_in_B S_in_F + + rw [(RubinFilter.from_isRubinFilterOf (F.map α) (RubinFilter.map_converges α F)).mem_inv S_in_B] + rw [<-(RubinFilter.map_isRubinFilterOf F (α := α)).mem_inv S_in_B] + exact S_in_F + +noncomputable def RubinFilter.lim (F : RubinFilter G) + : α := Exists.choose (RubinFilter.map_converges F (α := α)) + +theorem RubinFilter.le_nhds_lim (F : RubinFilter G) : + F.map α ≤ nhds (F.lim α) := (RubinFilter.map_converges F (α := α)).choose_spec + +theorem RubinFilter.le_nhds_eq_lim (F : RubinFilter G) (p : α) : + F.map α ≤ nhds p → p = F.lim α := +by + intro F_le_p + have F_le_lim := F.le_nhds_lim (α := α) + by_contra p_ne_lim + rw [<-ne_eq, <-disjoint_nhds_nhds] at p_ne_lim + apply (map F α).ne_bot.ne + exact Filter.empty_mem_iff_bot.mp (p_ne_lim F_le_p F_le_lim trivial) + +lemma RubinFilter.lim_mem_iff (F : RubinFilter G) {T : Set α} (T_in_basis : T ∈ RegularSupportBasis G α) : + F.lim α ∈ T ↔ ∃ V ∈ RegularSupportBasis G α, V ⊆ T ∧ RSuppSubsets G V ⊆ RSuppOrbit (F.map α) G•[T] := +by + rw [<-proposition_3_5' T_in_basis] + + constructor + · intro lim_in_T + use lim α F + exact ⟨lim_in_T, le_nhds_lim α F⟩ + · intro ⟨p, p_in_T, le_nhds_p⟩ + exact (F.le_nhds_eq_lim α p le_nhds_p) ▸ p_in_T + +lemma RubinFilter.exists_nhds_iff_lim_in_set (F : RubinFilter G) (T : Set α) : + (∃ p ∈ T, F.map α ≤ nhds p) ↔ F.lim α ∈ T := +by + constructor + · intro ⟨p, p_in_T, F_le_nhds⟩ + convert p_in_T + exact (F.le_nhds_eq_lim α p F_le_nhds).symm + · intro lim_in_T + exact ⟨lim α F, lim_in_T, le_nhds_lim α F⟩ + +end Convergence + +section Setoid + +/-- +Two rubin filters are equivalent if they share the same behavior in regards to which set their converging point `p` lies in. +--/ +instance RubinFilterSetoid (G : Type _) [Group G] : Setoid (RubinFilter G) where + r F F' := ∀ (U : Set G), U ∈ AlgebraicCentralizerBasis G → + (AlgebraicConvergent F.filter U ↔ AlgebraicConvergent F'.filter U) + iseqv := by constructor - { - apply AlgebraicCentralizerBasis.mem_of_regularSupportBasis W_in_basis - } + · 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 + +lemma RubinFilter.lim_mem_iff_of_eqv {F₁ F₂ : RubinFilter G} (F_equiv : F₁ ≈ F₂) + {S : Set α} (S_in_basis : S ∈ RegularSupportBasis G α) : + F₁.lim α ∈ S ↔ F₂.lim α ∈ S +:= by + have F₁_rubinFilterOf := (RubinFilter.map_isRubinFilterOf F₁ (α := α)) + have F₂_rubinFilterOf := (RubinFilter.map_isRubinFilterOf F₂ (α := α)) + + rw [F₁.lim_mem_iff α S_in_basis, <-proposition_3_5' S_in_basis] + rw [F₁_rubinFilterOf.converges_iff S_in_basis] + rw [F_equiv _ (AlgebraicCentralizerBasis.mem_of_regularSupportBasis S_in_basis)] + rw [<-F₂_rubinFilterOf.converges_iff S_in_basis] + rw [F₂.lim_mem_iff α S_in_basis, <-proposition_3_5' S_in_basis] + +lemma RubinFilter.mem_nhds_lim_iff_of_eqv {F₁ F₂ : RubinFilter G} (F_equiv : F₁ ≈ F₂) + (S : Set α) : S ∈ nhds (F₁.lim α) ↔ S ∈ nhds (F₂.lim α) := +by + suffices ∀ F₁ F₂ : RubinFilter G, F₁ ≈ F₂ → ∀ S : Set α, S ∈ nhds (F₁.lim α) → S ∈ nhds (F₂.lim α) by + constructor + apply this _ _ F_equiv + apply this _ _ (Setoid.symm F_equiv) + + have basis := RegularSupportBasis.isBasis G α + + intro F₁ F₂ F_equiv S S_in_nhds_F₁ + rw [basis.mem_nhds_iff] at S_in_nhds_F₁ + let ⟨T, T_in_basis, lim₁_in_T, T_ss_S⟩ := S_in_nhds_F₁ + + rw [basis.mem_nhds_iff] + use T + refine ⟨T_in_basis, ?lim₂_in_T, T_ss_S⟩ + + rw [lim_mem_iff_of_eqv F_equiv T_in_basis] at lim₁_in_T + exact lim₁_in_T + +theorem RubinFilter.lim_eq_of_eqv {F₁ F₂ : RubinFilter G} (F_equiv : F₁ ≈ F₂) : + F₁.lim α = F₂.lim α := +by + apply RubinFilter.le_nhds_eq_lim + have nhds_lim_in_basis := nhds_in_basis (lim α F₁) (RegularSupportBasis.isBasis G α) + + apply UltrafilterInBasis.le_of_inf_neBot _ (RegularSupportBasis.closed_inter G α) nhds_lim_in_basis + + constructor + intro eq_bot + + rw [Filter.inf_eq_bot_iff] at eq_bot + let ⟨U, U_in_F₂, V, V_in_nhds, UV_empty⟩ := eq_bot + + rw [mem_nhds_lim_iff_of_eqv F_equiv] at V_in_nhds + apply (F₂.map α).ne_bot.ne + rw [<-inf_eq_left.mpr (F₂.le_nhds_lim α)] + rw [Filter.inf_eq_bot_iff] + exact ⟨U, U_in_F₂, V, V_in_nhds, UV_empty⟩ + +theorem RubinFilter.eqv_of_map_converges (F₁ F₂ : RubinFilter G) (p : α): + F₁.map α ≤ nhds p → F₂.map α ≤ nhds p → F₁ ≈ F₂ := +by + intro F₁_le_nhds F₂_le_nhds + intro S S_in_basis + + have F₁_rubinFilterOf := (RubinFilter.map_isRubinFilterOf F₁ (α := α)) + have F₂_rubinFilterOf := (RubinFilter.map_isRubinFilterOf F₂ (α := α)) + + rw [<-AlgebraicCentralizerBasis.eq_rist_image (α := α)] at S_in_basis + let ⟨S', S'_in_basis, S'_eq⟩ := S_in_basis + simp only at S'_eq + rw [<-S'_eq] + + rw [<-F₁_rubinFilterOf.converges_iff S'_in_basis] + rw [<-F₂_rubinFilterOf.converges_iff S'_in_basis] + + rw [F₁.exists_nhds_iff_lim_in_set α S'] + rw [F₂.exists_nhds_iff_lim_in_set α S'] + rw [<-F₁.le_nhds_eq_lim _ _ F₁_le_nhds] + rw [<-F₂.le_nhds_eq_lim _ _ F₂_le_nhds] + + +lemma RubinFilter.fromPoint_converges' (p : α) : + ∃ q : α, ( + UltrafilterInBasis.of + (RegularSupportBasis.closed_inter G α) + (nhds_in_basis p (RegularSupportBasis.isBasis G α)) + ).filter ≤ nhds q := +by + use p + apply UltrafilterInBasis.of_le + +def RubinFilter.fromPoint (p : α) : RubinFilter G := + RubinFilter.from ( + UltrafilterInBasis.of + (RegularSupportBasis.closed_inter G α) + (nhds_in_basis p (RegularSupportBasis.isBasis G α)) + ) + (RubinFilter.fromPoint_converges' p) + +@[simp] +theorem RubinFilter.fromPoint_lim (p : α) : + (RubinFilter.fromPoint (G := G) p).lim α = p := +by + symm + apply RubinFilter.le_nhds_eq_lim + unfold fromPoint + rw [RubinFilter.map_from_eq] + apply UltrafilterInBasis.of_le + +theorem RubinFilter.lim_fromPoint_eqv (F : RubinFilter G) : + RubinFilter.fromPoint (F.lim α) ≈ F := +by + apply eqv_of_map_converges _ _ (F.lim α) + · unfold fromPoint + rw [RubinFilter.map_from_eq] + apply UltrafilterInBasis.of_le + · exact le_nhds_lim α F + +def RubinFilterBasis (G : Type _) [Group G] : Set (Set (RubinFilter G)) := + (fun S : Set G => { F : RubinFilter G | AlgebraicConvergent F.filter S }) '' AlgebraicCentralizerBasis G + +instance : TopologicalSpace (RubinFilter G) := TopologicalSpace.generateFrom (RubinFilterBasis G) + +theorem RubinFilterBasis.mem_iff (S : Set (RubinFilter G)) : + S ∈ RubinFilterBasis G ↔ ∃ B ∈ AlgebraicCentralizerBasis G, ∀ F : RubinFilter G, F ∈ S ↔ AlgebraicConvergent F.filter B := +by + unfold RubinFilterBasis + simp + conv => { + lhs; congr; intro B; rhs + rw [eq_comm, Set.ext_iff] + } + +def RubinSpace (G : Type _) [Group G] := Quotient (RubinFilterSetoid G) + +def RubinSpace.fromPoint (p : α) : RubinSpace G := + ⟦RubinFilter.fromPoint p⟧ + +instance : Membership (RubinFilter G) (RubinSpace G) where + mem := fun F Q => ⟦F⟧ = Q + +theorem RubinSpace.mem_iff (F : RubinFilter G) (Q : RubinSpace G) : + F ∈ Q ↔ ⟦F⟧ = Q := by rfl + +theorem RubinSpace.fromPoint_converges (p : α) : + ∀ F : RubinFilter G, F ∈ RubinSpace.fromPoint (G := G) p → F.lim α = p := +by + intro F + unfold fromPoint + rw [mem_iff, Quotient.eq] + intro F_eqv_from + convert RubinFilter.lim_eq_of_eqv F_eqv_from + clear F_eqv_from + simp + +noncomputable def RubinSpace.lim (Q : RubinSpace G) : α := + Q.liftOn (RubinFilter.lim α) (fun _a _b eqv => RubinFilter.lim_eq_of_eqv eqv) + +theorem RubinSpace.lim_fromPoint (p : α) : + RubinSpace.lim (RubinSpace.fromPoint (G := G) p) = p := +by + unfold lim + let ⟨Q, Q_eq⟩ := (RubinSpace.fromPoint (G := G) p).exists_rep + rw [<-Q_eq] + simp + apply RubinSpace.fromPoint_converges p Q + rwa [mem_iff] + +theorem RubinSpace.fromPoint_lim (Q : RubinSpace G) : + RubinSpace.fromPoint (Q.lim (α := α)) = Q := +by + let ⟨Q', Q'_eq⟩ := Q.exists_rep + rw [<-Q'_eq, lim, fromPoint] + simp + rw [Quotient.eq] + apply RubinFilter.lim_fromPoint_eqv + +instance : TopologicalSpace (RubinSpace G) := by + unfold RubinSpace + infer_instance + +theorem RubinSpace.lim_continuous : Continuous (RubinSpace.lim (G := G) (α := α)) := by + sorry + +theorem RubinSpace.fromPoint_continuous : Continuous (RubinSpace.fromPoint (G := G) (α := α)) := by + sorry + +/-- +The canonical homeomorphism from a topological space that a rubin action acts on to +the rubin space. +--/ +noncomputable def RubinSpace.homeomorph : Homeomorph (RubinSpace G) α where + toFun := RubinSpace.lim + invFun := RubinSpace.fromPoint + + left_inv := RubinSpace.fromPoint_lim + right_inv := RubinSpace.lim_fromPoint + + continuous_toFun := RubinSpace.lim_continuous + continuous_invFun := RubinSpace.fromPoint_continuous + +instance : MulAction G (RubinSpace G) := sorry + +end Setoid + +-- theorem rubin' (hα : RubinAction G α) : EquivariantHomeomorph G α (RubinSpace G) where +-- toFun := fun x => ⟦⟧ +-- invFun := fun S => sorry - rw [<-Subgroup.coe_top, <-rigidStabilizer_univ (α := α) (G := G)] - -- (RegularSupportBasis.univ_mem G α) - rw [<-(RubinFilter.from_isRubinFilterOf' F).subsets_ss_orbit W_in_basis] - assumption end RubinFilter @@ -2079,9 +1846,6 @@ end RubinFilter variable {β : Type _} variable [TopologicalSpace β] [MulAction G β] [ContinuousConstSMul G β] -#check IsOpen.smul - - theorem rubin (hα : RubinAction G α) (hβ : RubinAction G β) : EquivariantHomeomorph G α β := by -- by composing rubin' hα sorry diff --git a/Rubin/AlgebraicDisjointness.lean b/Rubin/AlgebraicDisjointness.lean index 875d531..ecfa8e0 100644 --- a/Rubin/AlgebraicDisjointness.lean +++ b/Rubin/AlgebraicDisjointness.lean @@ -388,10 +388,7 @@ theorem smul_injective_within_period {g : G} {p : α} {n : ℕ} (period_eq_n : Period.period p g = n) : Function.Injective (fun (i : Fin n) => g ^ (i : ℕ) • p) := by - have zpow_fix : (fun (i : Fin n) => g ^ (i : ℕ) • p) = (fun (i : Fin n) => g ^ (i : ℤ) • p) := by - ext x - simp - rw [zpow_fix] + simp only [<-zpow_coe_nat] apply moves_inj intro k one_le_k k_lt_n @@ -434,20 +431,16 @@ by · intro ⟨y, y_disj, x_eq⟩ use g * y * g⁻¹ rw [<-gxg12_eq] - refine ⟨?disj, x_eq⟩ - exact y_disj.conj g + exact ⟨y_disj.conj g, x_eq⟩ · intro ⟨y, y_disj, x_eq⟩ use g⁻¹ * y * g constructor - · rw [(by group : f = g⁻¹ * (g * f * g⁻¹) * g⁻¹⁻¹)] - nth_rw 6 [<-inv_inv g] - exact y_disj.conj g⁻¹ + · convert y_disj.conj g⁻¹ using 1 + all_goals group · nth_rw 3 [<-inv_inv g] simp only [conj_pow] + rw [x_eq] group - group at x_eq - exact x_eq - @[simp] theorem AlgebraicCentralizer.conj (f g : G) : diff --git a/Rubin/Filter.lean b/Rubin/Filter.lean index 57921a9..8022226 100644 --- a/Rubin/Filter.lean +++ b/Rubin/Filter.lean @@ -292,6 +292,21 @@ by any_goals apply B_ss_C all_goals assumption +theorem Filter.InBasis.inf_eq_bot_iff {F G : Filter α} (F_in_basis : F.InBasis B) (G_in_basis : G.InBasis B) : + F ⊓ G = ⊥ ↔ ∃ U ∈ B, ∃ V ∈ B, U ∈ F ∧ V ∈ G ∧ U ∩ V = ∅ := +by + rw [Filter.inf_eq_bot_iff] + constructor + · intro ⟨U, U_in_F, V, V_in_G, UV_empty⟩ + let ⟨U', U'_in_F, U'_in_B, U'_ss_U⟩ := F_in_basis U U_in_F + let ⟨V', V'_in_G, V'_in_B, V'_ss_V⟩ := G_in_basis V V_in_G + refine ⟨U', U'_in_B, V', V'_in_B, U'_in_F, V'_in_G, ?inter_empty⟩ + apply Set.eq_empty_of_subset_empty + apply subset_trans _ (subset_of_eq UV_empty) + exact Set.inter_subset_inter U'_ss_U V'_ss_V + · intro ⟨U, _, V, _, U_in_F, V_in_G, UV_empty⟩ + exact ⟨U, U_in_F, V, V_in_G, UV_empty⟩ + def Filter.InBasis.basis {α : Type _} (F : Filter α) (B : Set (Set α)): Set (Set α) := { S : Set α | S ∈ F ∧ S ∈ B } @@ -1129,6 +1144,23 @@ by apply U.ultra all_goals assumption +theorem UltrafilterInBasis.eq_of_le (U U' : UltrafilterInBasis B) : + U.filter ≤ U'.filter → U = U' := +by + intro U_le_U' + symm + rw [mk.injEq] + exact unique _ U.in_basis U_le_U' + +theorem UltrafilterInBasis.le_of_basis_sets (U U' : UltrafilterInBasis B) : + (∀ S : Set α, S ∈ B → S ∈ U' → S ∈ U) → U.filter ≤ U'.filter := +by + intro h + intro S S_in_U' + let ⟨T, T_in_B, T_in_U', T_ss_S⟩ := U'.in_basis _ S_in_U' + apply Filter.mem_of_superset _ T_ss_S + apply h <;> assumption + theorem UltrafilterInBasis.le_of_inf_neBot (U : UltrafilterInBasis B) (B_closed : ∀ (b1 b2 : Set α), b1 ∈ B → b2 ∈ B → Set.Nonempty (b1 ∩ b2) → b1 ∩ b2 ∈ B) {F : Filter α} (F_in_basis : F.InBasis B) diff --git a/Rubin/RigidStabilizer.lean b/Rubin/RigidStabilizer.lean index d4490cf..d6c5439 100644 --- a/Rubin/RigidStabilizer.lean +++ b/Rubin/RigidStabilizer.lean @@ -147,6 +147,7 @@ by repeat rw [rigidStabilizer_support] rw [Set.subset_inter_iff] +@[simp] theorem rigidStabilizer_empty (G α: Type _) [Group G] [MulAction G α] [FaithfulSMul G α]: G•[(∅ : Set α)] = ⊥ := by @@ -159,6 +160,7 @@ by rw [f_in_rist x (Set.not_mem_empty x)] simp +@[simp] theorem rigidStabilizer_univ (G α: Type _) [Group G] [MulAction G α]: G•[(Set.univ : Set α)] = ⊤ := by