Rename smulImage_subset to smulImage_mono

Rubin.lean

@@ -134,7 +134,7 @@ lemma proposition_1_1_1 [h_lm : LocallyMoving G α] [T2Space α] (f g : G) (supp
       _ ⊆ W ∪ (k •'' Support α f₁) := Set.union_subset_union_left _ supp_f₁_subset_W
       _ ⊆ W ∪ (k •'' W) := by
         apply Set.union_subset_union_right
-        exact (smulImage_subset k supp_f₁_subset_W)
+        exact (smulImage_mono k supp_f₁_subset_W)
       _ = W ∪ (f₂ •'' W) := by rw [<-smulImage_eq_of_smul_eq h₂]
       _ ⊆ V ∪ (f₂ •'' W) := Set.union_subset_union_left _ W_in_V
       _ ⊆ V ∪ V := by
@@ -217,7 +217,7 @@ by
   have V_disjoint_smulImage: Disjoint V (f •'' V) := by
     apply Set.disjoint_of_subset
     · exact Set.inter_subset_left _ _
-    · apply smulImage_subset
+    · apply smulImage_mono
       exact Set.inter_subset_left _ _
     · exact disjoint_img_V₀
 
@@ -253,7 +253,7 @@ by
         exact rigidStabilizer_support.mp h_in_ristV
       _ ⊆ V ∪ (f₂ •'' V) := by
         apply Set.union_subset_union_right
-        apply smulImage_subset
+        apply smulImage_mono
         exact rigidStabilizer_support.mp h_in_ristV
   have support_h' : Support α h' ⊆ ⋃(i : Fin 2 × Fin 2), (f₁^(i.1.val) * f₂^(i.2.val)) •'' V := by
     rw [rewrite_Union]
@@ -266,7 +266,7 @@ by
         exact support_f₂_h
       _ ⊆ V ∪ (f₂ •'' V) ∪ (f₁ •'' V ∪ (f₂ •'' V)) := by
         apply Set.union_subset_union_right
-        apply smulImage_subset f₁
+        apply smulImage_mono f₁
         exact support_f₂_h
 
   -- Since h' is nontrivial, it has at least one point p in its support
@@ -367,9 +367,9 @@ by
   have disj : ∀ (i j : Fin n), i ≠ j → Disjoint (g ^ (i : ℕ) •'' V) (g ^ (j : ℕ) •'' V) := by
     intro i j i_ne_j
     apply Set.disjoint_of_subset
-    · apply smulImage_subset
+    · apply smulImage_mono
       apply Set.inter_subset_right
-    · apply smulImage_subset
+    · apply smulImage_mono
       apply Set.inter_subset_right
     exact disj' i j i_ne_j
 

Rubin/AlgebraicDisjointness.lean

@@ -291,11 +291,7 @@ by
   let U := ⋂(p : ι₂), smul_inj_nbhd p.prop f_smul_inj
   use U
 
-  -- The notations provided afterwards tend to be quite ugly because we used `Exists.choose`,
+  repeat' apply And.intro
-  -- but the idea is that this all boils down to applying `Exists.choose_spec`, except in the disjointness case,
-  -- where we transform `Disjoint (f i •'' U) (f j •'' U)` into `Disjoint U ((f i)⁻¹ * f j •'' U)`
-  -- and transform both instances of `U` into `N`, the neighborhood of the chosen `(i, j) ∈ ι₂`
-  repeat' constructor
   · apply isOpen_iInter_of_finite
     intro ⟨⟨i, j⟩, i_ne_j⟩
     apply smul_inj_nbhd_open
@@ -303,6 +299,8 @@ by
     intro ⟨⟨i, j⟩, i_ne_j⟩
     apply smul_inj_nbhd_mem
   · intro i j i_ne_j
+
+    -- We transform `Disjoint (f i •'' U) (f j •'' U)` into `Disjoint N ((f i)⁻¹ * f j •'' N)`
     let N := smul_inj_nbhd i_ne_j f_smul_inj
     have U_subset_N : U ⊆ N := Set.iInter_subset
       (fun (⟨⟨i, j⟩, i_ne_j⟩ : ι₂) => (smul_inj_nbhd i_ne_j f_smul_inj))
@@ -311,7 +309,7 @@ by
     rw [disjoint_comm, smulImage_disjoint_mul]
 
     apply Set.disjoint_of_subset U_subset_N
-    · apply smulImage_subset
+    · apply smulImage_mono
       exact U_subset_N
     · exact smul_inj_nbhd_disjoint i_ne_j f_smul_inj
 

Rubin/SmulImage.lean

@@ -116,15 +116,11 @@ theorem smulImage_inv (g: G) (U V : Set α) : g •'' U = V ↔ U = g⁻¹ •''
     apply SmulImage.inv_congr at h
     exact h
 
--- TODO: rename to smulImage_mono
+theorem smulImage_mono (g : G) {U V : Set α} : U ⊆ V → g •'' U ⊆ g •'' V := by
-theorem smulImage_subset (g : G) {U V : Set α} : U ⊆ V → g •'' U ⊆ g •'' V :=
-  by
   intro h1 x
   rw [Rubin.mem_smulImage, Rubin.mem_smulImage]
   exact fun h2 => h1 h2
-#align smul''_subset Rubin.smulImage_subset
+#align smul''_subset Rubin.smulImage_mono
-
-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
@@ -218,12 +214,12 @@ theorem smulImage_subset_inv {G α : Type _} [Group G] [MulAction G α]
 by
   constructor
   · intro h
-    apply smulImage_subset f⁻¹ at h
+    apply smulImage_mono f⁻¹ at h
     rw [smulImage_mul] at h
     rw [mul_left_inv, one_smulImage] at h
     exact h
   · intro h
-    apply smulImage_subset f at h
+    apply smulImage_mono f at h
     rw [smulImage_mul] at h
     rw [mul_right_inv, one_smulImage] at h
     exact h
@@ -265,11 +261,10 @@ by
 theorem smulImage_disjoint_subset {G α : Type _} [Group G] [MulAction G α]
   {f g : G} {U V : Set α} (h_sub: U ⊆ V):
   Disjoint (f •'' V) (g •'' V) → Disjoint (f •'' U) (g •'' U) :=
-by
+Set.disjoint_of_subset (smulImage_mono _ h_sub) (smulImage_mono _ h_sub)
-  apply Set.disjoint_of_subset (smulImage_subset _ h_sub) (smulImage_subset _ h_sub)
 
 -- States that if `g^i •'' V` and `g^j •'' V` are disjoint for any `i ≠ j` and `x ∈ V`
--- then `g^i • x` will always lie outside of `V`.
+-- then `g^i • x` will always lie outside of `V` if `x ∈ V`.
 lemma smulImage_distinct_of_disjoint_pow {G α : Type _} [Group G] [MulAction G α] {g : G} {V : Set α} {n : ℕ}
   (n_pos : 0 < n)
   (h_disj : ∀ (i j : Fin n), i ≠ j → Disjoint (g ^ (i : ℕ) •'' V) (g ^ (j : ℕ) •'' V)) :

Rubin/Support.lean

@@ -76,7 +76,7 @@ theorem fixed_smulImage_in_support (g : G) {U : Set α} :
 theorem smulImage_subset_in_support (g : G) (U V : Set α) :
     U ⊆ V → Support α g ⊆ V → g •'' U ⊆ V := fun U_in_V support_in_V =>
   Rubin.fixed_smulImage_in_support g support_in_V ▸
-    Rubin.smulImage_subset g U_in_V
+    smulImage_mono g U_in_V
 #align moves_subset_within_support Rubin.smulImage_subset_in_support
 
 theorem support_mul (g h : G) (α : Type _) [MulAction G α] :
@@ -155,10 +155,11 @@ theorem disjoint_support_comm (f g : G) {U : Set α} :
     Support α f ⊆ U → Disjoint U (g •'' U) → ∀ x ∈ U, ⁅f, g⁆ • x = f • x :=
   by
   intro support_in_U disjoint_U x x_in_U
-  have support_conj : Support α (g * f⁻¹ * g⁻¹) ⊆ g •'' U :=
+  have support_conj : Support α (g * f⁻¹ * g⁻¹) ⊆ g •'' U := by
-    ((Rubin.support_conjugate α f⁻¹ g).trans
+    rw [support_conjugate]
-          (Rubin.SmulImage.congr g (Rubin.support_inv α f))).symm ▸
+    apply smulImage_mono
-      Rubin.smulImage_subset g support_in_U
+    rw [support_inv]
+    exact support_in_U
   have goal :=
     (congr_arg (f • ·)
         (Rubin.fixed_of_disjoint x_in_U

Rubin/Topological.lean

@@ -13,7 +13,6 @@ namespace Rubin
 
 section Continuity
 
--- 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)