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)
+    (forall (a : A), a ∈ X -> a ∈ Y)
-      <~> (subset X 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 (X1 ⊆ Y).
-        enough (subset X2 Y).
+        enough (X2 ⊆ Y).
         { split. apply X. apply X0. }
         * apply H2.
           intros a Ha.
-          apply f.
+          refine (f _ (tr _)).
-          apply tr.
           right.
           apply Ha.
         * apply H1.
           intros a Ha.
-          apply f.
+          refine (f _ (tr _)).
-          apply 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)
+    refine (@equiv_compose' _ ((forall a, a ∈ Y -> a ∈ X)
-                               *(forall a, isIn a X -> isIn a Y)) _ _ _).
+                               *(forall a, a ∈ X -> a ∈ Y)) _ _ _).
     - apply equiv_iff_hprop_uncurried.
       split.
       * intros [H1 H2 a].

FiniteSets/fsets/length.v

@@ -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

@@ -54,9 +54,9 @@ Definition comprehension :
   Proof.
     intros P.
     hrecursion.
-- apply E.
+    - apply ∅.
     - intro a.
-  refine (if (P a) then L a else E).
+      refine (if (P a) then {|a|} else ∅).
     - apply U.
     - apply assoc.
     - apply comm.
@@ -71,15 +71,9 @@ Defined.
   Definition isEmpty :
     FSet A -> Bool.
   Proof.
-  hrecursion.
+    simple refine (FSet_rec _ _ _ true (fun _ => false) andb _ _ _ _ _)
-  - apply true.
+    ; try eauto with bool_lattice_hints typeclass_instances.
-  - apply (fun _ => false).
+    intros ; symmetry ; eauto with lattice_hints typeclass_instances.
-  - 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.    
   
 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 (fun a => orb (ϕ a) (ψ a)) x
-                                                     (comprehension ψ 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.
 

FiniteSets/implementations/interface.v

@@ -34,11 +34,11 @@ Section interface.
 
   Class sets :=
     {
-      f_empty : forall A, f A empty = E ;
+      f_empty : forall A, f A empty = ∅ ;
-      f_singleton : forall A a, f A (singleton a) = L a;
+      f_singleton : forall A a, f A (singleton a) = {|a|};
-      f_union : forall A X Y, f A (union X Y) = U (f A X) (f A Y);
+      f_union : forall A X Y, f A (union X Y) = (f A X) ∪ (f A Y);
       f_filter : forall A ϕ X, f A (filter ϕ X) = comprehension ϕ (f A X);
-      f_member : forall A a X, member a X = isIn a (f A X)
+      f_member : forall A a X, member a X = a ∈ (f A X)
     }.
 End interface.
 
@@ -47,9 +47,11 @@ Section properties.
   Variable (T : Type -> Type) (f : forall A, T A -> FSet A).
   Context `{sets T f}.
 
-  Definition set_eq : forall A, T A -> T A -> hProp := fun A X Y =>  (BuildhProp (f A X = f A Y)).
+  Definition set_eq : forall A, T A -> T A -> hProp :=
+    fun A X Y => (BuildhProp (f A X = f A Y)).
 
-  Definition set_subset : forall A, T A -> T A -> hProp := fun A X Y => subset (f A X) (f A Y).
+  Definition set_subset : forall A, T A -> T A -> hProp :=
+    fun A X Y => (f A X) ⊆ (f A Y).
 
   Ltac reduce := intros ; repeat (rewrite ?(f_empty _ _) ; rewrite ?(f_singleton _ _) ;
                          rewrite ?(f_union _ _) ; rewrite ?(f_filter _ _) ;

FiniteSets/implementations/lists.v

@@ -47,7 +47,7 @@ Section ListToSet.
   Context `{Univalence}.
 
   Lemma member_isIn (a : A) (l : list A)  :
-    member a l = isIn a (list_to_set A l).
+    member a l = a ∈ (list_to_set A l).
   Proof.
     induction l ; unfold member in * ; simpl in *.
     - reflexivity.
@@ -60,7 +60,7 @@ Section ListToSet.
       * apply (tr (inr z2)).
   Defined.
   
-  Definition empty_empty : list_to_set A empty = E := idpath.
+  Definition empty_empty : list_to_set A empty = ∅ := idpath.
 
   Lemma filter_comprehension (ϕ : A -> Bool) (l : list A)  :
     list_to_set A (filter ϕ l) = comprehension ϕ (list_to_set A l).
@@ -68,17 +68,17 @@ Section ListToSet.
     induction l ; cbn in *.
     - reflexivity.
     - destruct (ϕ a) ; cbn in * ; unfold list_to_set in IHl.
-      * refine (ap (fun y => U {|a|} y) _).
+      * refine (ap (fun y => {|a|} ∪ y) _).
         apply IHl.
       * rewrite nl.
         apply IHl.
   Defined.
   
-  Definition singleton_single (a : A) : list_to_set A (singleton a) = L a :=
+  Definition singleton_single (a : A) : list_to_set A (singleton a) = {|a|} :=
-    nr (L a).
+    nr {|a|}.
 
   Lemma append_union (l1 l2 : list A) :
-    list_to_set A (union l1 l2) = U (list_to_set A l1) (list_to_set A l2).
+    list_to_set A (union l1 l2) = (list_to_set A l1) ∪ (list_to_set A l2).
   Proof.
     induction l1 ; induction l2 ; cbn.
     - apply (union_idem _)^.

FiniteSets/representations/bad.v

@@ -62,11 +62,11 @@ Module Export FSet.
                   comm x y #
                        uP x y px py = uP y x py px).
     Variable  (nlP : forall (x : FSet A) (px: P x), 
-                  nl x # uP E x eP px = px).
+                  nl x # uP ∅ x eP px = px).
     Variable  (nrP : forall (x : FSet A) (px: P x), 
-                  nr x # uP x E px eP = px).
+                  nr x # uP x ∅ px eP = px).
     Variable  (idemP : forall (x : A), 
-                  idem x # uP (L x) (L x) (lP x) (lP x) = lP x).
+                  idem x # uP {|x|} {|x|} (lP x) (lP x) = lP x).
 
     (* Induction principle *)
     Fixpoint FSet_ind

FiniteSets/representations/definition.v

@@ -42,29 +42,32 @@ Module Export FSet.
   Arguments nl {_} _.
   Arguments nr {_} _.
   Arguments idem {_} _.
+  Notation "{| x |}" :=  (L x).
+  Infix "∪" := U (at level 8, right associativity).
+  Notation "∅" := E.
 
   Section FSet_induction.
     Variable A: Type.
     Variable  (P : FSet A -> Type).
-    Variable  (H :  forall a : FSet A, IsHSet (P a)).
+    Variable  (H :  forall X : FSet A, IsHSet (P X)).
-    Variable  (eP : P E).
+    Variable  (eP : P ∅).
-    Variable  (lP : forall a: A, P (L a)).
+    Variable  (lP : forall a: A, P {|a|}).
-    Variable  (uP : forall (x y: FSet A), P x -> P y -> P (U x y)).
+    Variable  (uP : forall (x y: FSet A), P x -> P y -> P (x ∪ y)).
     Variable  (assocP : forall (x y z : FSet A) 
                                (px: P x) (py: P y) (pz: P z),
                   assoc x y z #
-                        (uP      x    (U y z)          px        (uP y z py pz)) 
+                        (uP x (y ∪ z) px (uP y z py pz)) 
                   = 
-                  (uP   (U x y)    z       (uP x y px py)          pz)).
+                  (uP (x ∪ y) z (uP x y px py) pz)).
     Variable  (commP : forall (x y: FSet A) (px: P x) (py: P y),
                   comm x y #
                        uP x y px py = uP y x py px).
     Variable  (nlP : forall (x : FSet A) (px: P x), 
-                  nl x # uP E x eP px = px).
+                  nl x # uP ∅ x eP px = px).
     Variable  (nrP : forall (x : FSet A) (px: P x), 
-                  nr x # uP x E px eP = px).
+                  nr x # uP x ∅ px eP = px).
     Variable  (idemP : forall (x : A), 
-                  idem x # uP (L x) (L x) (lP x) (lP x) = lP x).
+                  idem x # uP {|x|} {|x|} (lP x) (lP x) = lP x).
     
     (* Induction principle *)
     Fixpoint FSet_ind
@@ -183,9 +186,11 @@ Module Export FSet.
     
   End FSet_recursion.
 
-  Instance FSet_recursion A : HitRecursion (FSet A) := {
+  Instance FSet_recursion A : HitRecursion (FSet A) :=
+    {
       indTy := _; recTy := _; 
-                                                        H_inductor := FSet_ind A; H_recursor := FSet_rec A }.
+      H_inductor := FSet_ind A; H_recursor := FSet_rec A
+    }.
 
 End FSet.