Some cleaning in notation

FiniteSets/fsets/extensionality.v

@@ -8,7 +8,7 @@ Section ext.
   Context `{Univalence}.
 
   Lemma subset_union_equiv
-    : forall X Y : FSet A, subset X Y <~> U X Y = Y.
+    : forall X Y : FSet A, X ⊆ Y <~> X ∪ Y = Y.
   Proof.
     intros X Y.
     eapply equiv_iff_hprop_uncurried.
@@ -20,8 +20,8 @@ Section ext.
   Defined.
 
   Lemma subset_isIn (X Y : FSet A) :
-    (forall (a : A), isIn a X -> isIn a Y)
-      <~> (subset X Y).
+    (forall (a : A), a ∈ X -> a ∈ Y)
+      <~> (X ⊆ Y).
   Proof.
     eapply equiv_iff_hprop_uncurried.
     split.
@@ -32,19 +32,17 @@ Section ext.
         apply f.
         apply tr ; reflexivity.
       + intros X1 X2 H1 H2 f.
-        enough (subset X1 Y).
-        enough (subset X2 Y).
+        enough (X1 ⊆ Y).
+        enough (X2 ⊆ Y).
         { split. apply X. apply X0. }
         * apply H2.
           intros a Ha.
-          apply f.
-          apply tr.
+          refine (f _ (tr _)).
           right.
           apply Ha.
         * apply H1.
           intros a Ha.
-          apply f.
-          apply tr.
+          refine (f _ (tr _)).
           left.
           apply Ha.
     - hinduction X ;
@@ -64,7 +62,7 @@ Section ext.
 
   (** ** Extensionality proof *)
 
-  Lemma eq_subset' (X Y : FSet A) : X = Y <~> (U Y X = X) * (U X Y = Y).
+  Lemma eq_subset' (X Y : FSet A) : X = Y <~> (Y ∪ X = X) * (X ∪ Y = Y).
   Proof.
     unshelve eapply BuildEquiv.
     { intro H'. rewrite H'. split; apply union_idem. }
@@ -76,20 +74,20 @@ Section ext.
   Defined.
 
   Lemma eq_subset (X Y : FSet A) :
-    X = Y <~> (subset Y X * subset X Y).
+    X = Y <~> (Y ⊆ X * X ⊆ Y).
   Proof.
-    transitivity ((U Y X = X) * (U X Y = Y)).
+    transitivity ((Y ∪ X = X) * (X ∪ Y = Y)).
     apply eq_subset'.
     symmetry.
     eapply equiv_functor_prod'; apply subset_union_equiv.
   Defined.
 
   Theorem fset_ext (X Y : FSet A) :
-    X = Y <~> (forall (a : A), isIn a X = isIn a Y).
+    X = Y <~> (forall (a : A), a ∈ X = a ∈ Y).
   Proof.
     refine (@equiv_compose' _ _ _ _ _) ; [ | apply eq_subset ].
-    refine (@equiv_compose' _ ((forall a, isIn a Y -> isIn a X)
-                               *(forall a, isIn a X -> isIn a Y)) _ _ _).
+    refine (@equiv_compose' _ ((forall a, a ∈ Y -> a ∈ X)
+                               *(forall a, a ∈ X -> a ∈ Y)) _ _ _).
     - apply equiv_iff_hprop_uncurried.
       split.
       * intros [H1 H2 a].

FiniteSets/fsets/length.v

@@ -11,7 +11,7 @@ Section Length.
 
   Opaque isIn_b.
 
-  Definition length (x: FSetC A) : nat.
+  Definition length (x : FSetC A) : nat.
   Proof.
     simple refine (FSetC_ind A _ _ _ _ _ _ x ); simpl.
     - exact 0.
@@ -31,7 +31,7 @@ Section Length.
 
   Definition length_FSet (x: FSet A) := length (FSet_to_FSetC x).
 
-  Lemma length_singleton: forall (a: A), length_FSet (L a) = 1.
+  Lemma length_singleton: forall (a: A), length_FSet ({|a|}) = 1.
   Proof. 
     intro a.
     cbn. reflexivity. 

FiniteSets/fsets/operations.v

@@ -3,84 +3,78 @@ Require Import HoTT HitTactics.
 Require Import representations.definition disjunction lattice.
 
 Section operations.
-Context {A : Type}.
-Context `{Univalence}.
+  Context {A : Type}.
+  Context `{Univalence}.
 
-Definition isIn : A -> FSet A -> hProp.
-Proof.
-intros a.
-hrecursion.
-- exists Empty.
-  exact _.
-- intro a'.
-  exists (Trunc (-1) (a = a')).
-  exact _.
-- apply lor.
-- intros ; symmetry ; apply lor_assoc.
-- apply lor_commutative.
-- apply lor_nl.
-- apply lor_nr.
-- intros ; apply lor_idem.
-Defined.
+  Definition isIn : A -> FSet A -> hProp.
+  Proof.
+    intros a.
+    hrecursion.
+    - exists Empty.
+      exact _.
+    - intro a'.
+      exists (Trunc (-1) (a = a')).
+      exact _.
+    - apply lor.
+    - intros ; symmetry ; apply lor_assoc.
+    - apply lor_commutative.
+    - apply lor_nl.
+    - apply lor_nr.
+    - intros ; apply lor_idem.
+  Defined.
 
-Definition subset : FSet A -> FSet A -> hProp.
-Proof.
-intros X Y.
-hrecursion X.
-- exists Unit.
-  exact _.
-- intros a.
-  apply (isIn a Y).
-- intros X1 X2.
-  exists (prod X1 X2).
-  exact _.
-- intros.
-  apply path_trunctype ; apply equiv_prod_assoc.
-- intros.
-  apply path_trunctype ; apply equiv_prod_symm.
-- intros.
-  apply path_trunctype ; apply prod_unit_l.
-- intros.
-  apply path_trunctype ; apply prod_unit_r.
-- intros a'.
-  apply path_iff_hprop ; cbn.
-  * intros [p1 p2]. apply p1.
-  * intros p.
-    split ; apply p.
-Defined.
+  Definition subset : FSet A -> FSet A -> hProp.
+  Proof.
+    intros X Y.
+    hrecursion X.
+    - exists Unit.
+      exact _.
+    - intros a.
+      apply (isIn a Y).
+    - intros X1 X2.
+      exists (prod X1 X2).
+      exact _.
+    - intros.
+      apply path_trunctype ; apply equiv_prod_assoc.
+    - intros.
+      apply path_trunctype ; apply equiv_prod_symm.
+    - intros.
+      apply path_trunctype ; apply prod_unit_l.
+    - intros.
+      apply path_trunctype ; apply prod_unit_r.
+    - intros a'.
+      apply path_iff_hprop ; cbn.
+      * intros [p1 p2]. apply p1.
+      * intros p.
+        split ; apply p.
+  Defined.
 
-Definition comprehension : 
-  (A -> Bool) -> FSet A -> FSet A.
-Proof.
-intros P.
-hrecursion.
-- apply E.
-- intro a.
-  refine (if (P a) then L a else E).
-- apply U.
-- apply assoc.
-- apply comm.
-- apply nl.
-- apply nr.
-- intros; simpl. 
-  destruct (P x).
-  + apply idem.
-  + apply nl.
-Defined.
+  Definition comprehension : 
+    (A -> Bool) -> FSet A -> FSet A.
+  Proof.
+    intros P.
+    hrecursion.
+    - apply ∅.
+    - intro a.
+      refine (if (P a) then {|a|} else ∅).
+    - apply U.
+    - apply assoc.
+    - apply comm.
+    - apply nl.
+    - apply nr.
+    - intros; simpl. 
+      destruct (P x).
+      + apply idem.
+      + apply nl.
+  Defined.
 
-Definition isEmpty :
-  FSet A -> Bool.
-Proof.
-  hrecursion.
-  - apply true.
-  - apply (fun _ => false).
-  - apply andb.
-  - intros. symmetry. eauto with lattice_hints typeclass_instances.
-  - eauto with bool_lattice_hints typeclass_instances.
-  - eauto with bool_lattice_hints typeclass_instances.
-  - eauto with bool_lattice_hints typeclass_instances.
-  - eauto with bool_lattice_hints typeclass_instances.
-Defined.    
+  Definition isEmpty :
+    FSet A -> Bool.
+  Proof.
+    simple refine (FSet_rec _ _ _ true (fun _ => false) andb _ _ _ _ _)
+    ; try eauto with bool_lattice_hints typeclass_instances.
+    intros ; symmetry ; eauto with lattice_hints typeclass_instances.
+  Defined.    
   
 End operations.
 

FiniteSets/fsets/properties.v

@@ -7,7 +7,7 @@ Section operations_isIn.
   Context {A : Type}.
   Context `{Univalence}.
 
-  Lemma union_idem : forall x: FSet A, U x x = x.
+  Lemma union_idem : forall x: FSet A, x ∪ x = x.
   Proof.
     hinduction ; try (intros ; apply set_path2).
     - apply nl.
@@ -24,7 +24,7 @@ Section operations_isIn.
 
   (** ** Properties about subset relation. *)
   Lemma subset_union (X Y : FSet A) : 
-    subset X Y -> U X Y = Y.
+    X ⊆ Y -> X ∪ Y = Y.
   Proof.
     hinduction X ; try (intros; apply path_forall; intro; apply set_path2).
     - intros. apply nl.
@@ -54,7 +54,7 @@ Section operations_isIn.
   Defined.
   
   Lemma subset_union_l (X : FSet A) :
-    forall Y, subset X (U X Y).
+    forall Y, X ⊆ X ∪ Y.
   Proof.
     hinduction X ; try (repeat (intro; intros; apply path_forall);
                         intro ; apply equiv_hprop_allpath ; apply _).
@@ -69,8 +69,8 @@ Section operations_isIn.
 
   (* simplify it using extensionality *)
   Lemma comprehension_or : forall ϕ ψ (x: FSet A),
-      comprehension (fun a => orb (ϕ a) (ψ a)) x = U (comprehension ϕ x) 
-                                                     (comprehension ψ x).
+      comprehension (fun a => orb (ϕ a) (ψ a)) x
+      = (comprehension ϕ x) ∪ (comprehension ψ x).
   Proof.
     intros ϕ ψ.
     hinduction ; try (intros; apply set_path2). 
@@ -101,15 +101,15 @@ Section properties.
   Context `{Univalence}.
 
   (** isIn properties *)
-  Definition empty_isIn (a: A) : isIn a E -> Empty := idmap.
+  Definition empty_isIn (a: A) : a ∈ E -> Empty := idmap.
   
-  Definition singleton_isIn (a b: A) : isIn a (L b) -> Trunc (-1) (a = b) := idmap.
+  Definition singleton_isIn (a b: A) : a ∈ {|b|} -> Trunc (-1) (a = b) := idmap.
 
   Definition union_isIn (X Y : FSet A) (a : A)
-    : isIn a (U X Y) = isIn a X ∨ isIn a Y := idpath.
+    : a ∈ X ∪ Y = a ∈ X ∨ a ∈ Y := idpath.
 
   Lemma comprehension_isIn (ϕ : A -> Bool) (a : A) : forall X : FSet A,
-      isIn a (comprehension ϕ X) = if ϕ a then isIn a X else False_hp.
+      a ∈ (comprehension ϕ X) = if ϕ a then a ∈ X else False_hp.
   Proof.
     hinduction ; try (intros ; apply set_path2) ; cbn.
     - destruct (ϕ a) ; reflexivity.
@@ -139,7 +139,7 @@ Section properties.
 
   (* The proof can be simplified using extensionality *)
   (** comprehension properties *)
-  Lemma comprehension_false Y : comprehension (fun (_ : A) => false) Y = E.
+  Lemma comprehension_false Y : comprehension (fun (_ : A) => false) Y = ∅.
   Proof.
     hrecursion Y; try (intros; apply set_path2).
     - reflexivity.
@@ -151,7 +151,7 @@ Section properties.
 
   (* Can be simplified using extensionality *)
   Lemma comprehension_subset : forall ϕ (X : FSet A),
-      U (comprehension ϕ X) X = X.
+      (comprehension ϕ X) ∪ X = X.
   Proof.
     intros ϕ.
     hrecursion; try (intros ; apply set_path2) ; cbn.
@@ -171,7 +171,7 @@ Section properties.
       reflexivity.
   Defined.
   
-  Lemma merely_choice : forall X : FSet A, hor (X = E) (hexists (fun a => isIn a X)).
+  Lemma merely_choice : forall X : FSet A, hor (X = ∅) (hexists (fun a => a ∈ X)).
   Proof.
     hinduction; try (intros; apply equiv_hprop_allpath ; apply _).
     - apply (tr (inl idpath)).

FiniteSets/fsets/properties_decidable.v

@@ -29,7 +29,7 @@ Section operations_isIn.
   Defined.
 
   Lemma L_isIn (a b : A) :
-    isIn a {|b|} -> a = b.
+    a ∈ {|b|} -> a = b.
   Proof.
     intros. strip_truncations. assumption.
   Defined.
@@ -63,7 +63,7 @@ Section operations_isIn.
   
   (* Union and membership *)
   Lemma union_isIn_b (X Y : FSet A) (a : A) :
-    isIn_b a (U X Y) = orb (isIn_b a X) (isIn_b a Y).
+    isIn_b a (X ∪ Y) = orb (isIn_b a X) (isIn_b a Y).
   Proof.
     unfold isIn_b ; unfold dec.
     simpl.
@@ -109,7 +109,7 @@ Section SetLattice.
 
   Instance fset_max : maximum (FSet A) := U.
   Instance fset_min : minimum (FSet A) := intersection.
-  Instance fset_bot : bottom (FSet A) := E.
+  Instance fset_bot : bottom (FSet A) := ∅.
   
   Instance lattice_fset : Lattice (FSet A).
   Proof.
@@ -133,7 +133,7 @@ Section comprehension_properties.
   Defined.
   
   (** comprehension properties *)
-  Lemma comprehension_false Y : comprehension (fun (_ : A) => false) Y = E.
+  Lemma comprehension_false Y : comprehension (fun (_ : A) => false) Y = ∅.
   Proof.
     toBool.
   Defined.
@@ -145,7 +145,7 @@ Section comprehension_properties.
   Defined.
   
   Lemma comprehension_subset : forall ϕ (X : FSet A),
-      U (comprehension ϕ X) X = X.
+      (comprehension ϕ X) ∪ X = X.
   Proof.
     toBool.
   Defined.
@@ -159,7 +159,7 @@ Section dec_eq.
   Instance fsets_dec_eq : DecidablePaths (FSet A).
   Proof.
     intros X Y.
-    apply (decidable_equiv' ((subset Y X) * (subset X Y)) (eq_subset X Y)^-1).
+    apply (decidable_equiv' ((Y ⊆ X) * (X ⊆ Y)) (eq_subset X Y)^-1).
     apply decidable_prod.
   Defined.