Improved notatio

FiniteSets/fsets/properties.v

@@ -9,17 +9,23 @@ Section characterize_isIn.
   Context `{Univalence}.
 
   (** isIn properties *)
-  Definition empty_isIn (a: A) : a ∈ E -> Empty := idmap.
+  Definition empty_isIn (a: A) : a ∈ ∅ -> Empty := idmap.
   
   Definition singleton_isIn (a b: A) : a ∈ {|b|} -> Trunc (-1) (a = b) := idmap.
 
   Definition union_isIn (X Y : FSet A) (a : A)
     : a ∈ X ∪ Y = a ∈ X ∨ a ∈ Y := idpath.
 
-  Lemma comprehension_isIn (ϕ : A -> Bool) (a : A) : forall X : FSet A,
-      a ∈ (comprehension ϕ X) = if ϕ a then a ∈ X else False_hp.
+  Lemma comprehension_union (ϕ : A -> Bool) : forall X Y : FSet A,
+      {|X ∪ Y & ϕ|} = {|X & ϕ|} ∪ {|Y & ϕ|}.
   Proof.
-    hinduction ; try (intros ; apply set_path2) ; cbn.
+    reflexivity.
+  Defined.  
+
+  Lemma comprehension_isIn (ϕ : A -> Bool) (a : A) : forall X : FSet A,
+      a ∈ {|X & ϕ|} = if ϕ a then a ∈ X else False_hp.
+  Proof.
+    hinduction ; try (intros ; apply set_path2).
     - destruct (ϕ a) ; reflexivity.
     - intros b.
       assert (forall c d, ϕ a = c -> ϕ b = d ->
@@ -36,6 +42,8 @@ Section characterize_isIn.
       }
       apply (X (ϕ a) (ϕ b) idpath idpath).
     - intros X Y H1 H2.
+      rewrite comprehension_union.
+      rewrite union_isIn.
       rewrite H1, H2.
       destruct (ϕ a).
       * reflexivity.
@@ -44,11 +52,14 @@ Section characterize_isIn.
            destruct Z ; assumption.
         ** intros ; apply tr ; right ; assumption.
   Defined.
+End characterize_isIn.
 
-  Context {B : Type}.
-
+Section product_isIn.
+  Context {A B : Type}.
+  Context `{Univalence}.
+  
   Lemma isIn_singleproduct (a : A) (b : B) (c : A) : forall (Y : FSet B),
-      isIn (a, b) (single_product c Y) = land (BuildhProp (Trunc (-1) (a = c))) (isIn b Y).
+      (a, b) ∈ (single_product c Y) = land (BuildhProp (Trunc (-1) (a = c))) (b ∈ Y).
   Proof.
     hinduction ; try (intros ; apply path_ishprop).
     - apply path_hprop ; symmetry ; apply prod_empty_r.
@@ -62,16 +73,16 @@ Section characterize_isIn.
         rewrite Z1, Z2.
         apply (tr idpath).
     - intros X1 X2 HX1 HX2.
-      apply path_iff_hprop.
-      *  intros X.
-         strip_truncations.
-         destruct X as [H1 | H1] ; rewrite ?HX1, ?HX2 in H1 ; destruct H1 as [H1 H2].
-         ** split.
-            *** apply H1.
-            *** apply (tr(inl H2)).
-         ** split.
-            *** apply H1.
-            *** apply (tr(inr H2)).
+      apply path_iff_hprop ; rewrite ?union_isIn.
+      * intros X.
+        strip_truncations.
+        destruct X as [H1 | H1] ; rewrite ?HX1, ?HX2 in H1 ; destruct H1 as [H1 H2].
+        ** split.
+           *** apply H1.
+           *** apply (tr(inl H2)).
+        ** split.
+           *** apply H1.
+           *** apply (tr(inr H2)).
       * intros [H1 H2].
         strip_truncations.
         apply tr.
@@ -84,15 +95,16 @@ Section characterize_isIn.
   Defined.
   
   Definition isIn_product (a : A) (b : B) (X : FSet A) (Y : FSet B) :
-    isIn (a,b) (product X Y) = land (isIn a X) (isIn b Y).
+    (a,b) ∈ (product X Y) = land (a ∈ X) (b ∈ Y).
   Proof.
     hinduction X ; try (intros ; apply path_ishprop).
     - apply path_hprop ; symmetry ; apply prod_empty_l.
     - intros.
       apply isIn_singleproduct.
     - intros X1 X2 HX1 HX2.
+      rewrite (union_isIn (product X1 Y)).
       rewrite HX1, HX2.
-      apply path_iff_hprop.
+      apply path_iff_hprop ; rewrite ?union_isIn.
       * intros X.
         strip_truncations.
         destruct X as [[H3 H4] | [H3 H4]] ; split ; try (apply H4).
@@ -104,7 +116,7 @@ Section characterize_isIn.
         ** left ; split ; assumption.
         ** right ; split ; assumption.
   Defined.
-End characterize_isIn.
+End product_isIn.
 
 Ltac simplify_isIn :=
   repeat (rewrite union_isIn
@@ -120,8 +132,17 @@ Section properties.
   Context {A : Type}.
   Context `{Univalence}.
 
-  Instance: bottom(FSet A) := ∅.
-  Instance: maximum(FSet A) := U.
+  Instance: bottom(FSet A).
+  Proof.
+    unfold bottom.
+    apply ∅.
+  Defined.
+
+  Instance: maximum(FSet A).
+  Proof.
+    intros x y.
+    apply (x ∪ y).
+  Defined.
 
   Global Instance joinsemilattice_fset : JoinSemiLattice (FSet A).
   Proof.
@@ -129,26 +150,32 @@ Section properties.
   Defined.  
 
   (** comprehension properties *)
-  Lemma comprehension_false Y : comprehension (fun (_ : A) => false) Y = ∅.
+  Lemma comprehension_false X : {|X & (fun (_ : A) => false)|} = ∅.
   Proof.
     toHProp.
   Defined.
 
   Lemma comprehension_subset : forall ϕ (X : FSet A),
-      (comprehension ϕ X) ∪ X = X.
+      {|X & ϕ|} ∪ X = X.
   Proof.
     toHProp.
     destruct (ϕ a) ; eauto with lattice_hints typeclass_instances.
   Defined.
 
-  Lemma comprehension_or : forall ϕ ψ (x: FSet A),
-      comprehension (fun a => orb (ϕ a) (ψ a)) x
-      = (comprehension ϕ x) ∪ (comprehension ψ x).
+  Lemma comprehension_or : forall ϕ ψ (X : FSet A),
+      {|X & (fun a => orb (ϕ a) (ψ a))|}
+      = {|X & ϕ|} ∪ {|X & ψ|}.
   Proof.
     toHProp.
     symmetry ; destruct (ϕ a) ; destruct (ψ a)
     ; eauto with lattice_hints typeclass_instances.
   Defined.
+
+  Lemma comprehension_all : forall (X : FSet A),
+      comprehension (fun a => true) X = X.
+  Proof.
+    toHProp.
+  Defined.
   
   Lemma merely_choice : forall X : FSet A, hor (X = ∅) (hexists (fun a => a ∈ X)).
   Proof.
@@ -161,7 +188,7 @@ Section properties.
       destruct TX as [XE | HX] ; destruct TY as [YE | HY] ; try(strip_truncations ; apply tr).
       * refine (tr (inl _)).
         rewrite XE, YE.
-        apply (union_idem E).
+        apply (union_idem ∅).
       * destruct HY as [a Ya].
         refine (inr (tr _)).
         exists a.