Improved notatio

FiniteSets/representations/bad.v

@@ -1,50 +1,45 @@
 (* This is a /bad/ definition of FSets, without the 0-truncation.
    Here we show that the resulting type is not an h-set. *)
 
-Require Import HoTT.
-Require Import HitTactics.
+Require Import HoTT HitTactics.
+Require Import notation.
+
 Module Export FSet.
   
   Section FSet.
+    Private Inductive FSet (A : Type): Type :=
+    | E : FSet A
+    | L : A -> FSet A
+    | U : FSet A -> FSet A -> FSet A.
+
+    Global Instance fset_empty : hasEmpty FSet := E.
+    Global Instance fset_singleton : hasSingleton FSet := L.
+    Global Instance fset_union : hasUnion FSet := U.
+
     Variable A : Type.
 
-    Private Inductive FSet : Type :=
-    | E : FSet
-    | L : A -> FSet
-    | U : FSet -> FSet -> FSet.
-
-    Notation "{| x |}" :=  (L x).
-    Infix "∪" := U (at level 8, right associativity).
-    Notation "∅" := E.
-
-    Axiom assoc : forall (x y z : FSet ),
+    Axiom assoc : forall (x y z : FSet A),
         x ∪ (y ∪ z) = (x ∪ y) ∪ z.
 
-    Axiom comm : forall (x y : FSet),
+    Axiom comm : forall (x y : FSet A),
         x ∪ y = y ∪ x.
 
-    Axiom nl : forall (x : FSet),
+    Axiom nl : forall (x : FSet A),
         ∅ ∪ x = x.
 
-    Axiom nr : forall (x : FSet),
+    Axiom nr : forall (x : FSet A),
         x ∪ ∅ = x.
 
     Axiom idem : forall (x : A),
-        {| x |} ∪ {|x|} = {|x|}.
+        {|x|} ∪ {|x|} = {|x|}.
 
   End FSet.
 
-  Arguments E {_}.
-  Arguments U {_} _ _.
-  Arguments L {_} _.
   Arguments assoc {_} _ _ _.
   Arguments comm {_} _ _.
   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.
@@ -74,12 +69,11 @@ Module Export FSet.
              {struct x}
       : P x
       := (match x return _ -> _ -> _ -> _ -> _ -> P x with
-          | ∅ => fun _ _ _ _ _ => eP
-          | {|a|} => fun _ _ _ _ _ => lP a
-          | y ∪ z => fun _ _ _ _ _ => uP y z (FSet_ind y) (FSet_ind z)
+          | E => fun _ _ _ _ _ => eP
+          | L a => fun _ _ _ _ _ => lP a
+          | U y z => fun _ _ _ _ _ => uP y z (FSet_ind y) (FSet_ind z)
           end) assocP commP nlP nrP idemP.
 
-
     Axiom FSet_ind_beta_assoc : forall (x y z : FSet A),
         apD FSet_ind (assoc x y z) =
         (assocP x y z (FSet_ind x) (FSet_ind y) (FSet_ind z)).
@@ -192,10 +186,6 @@ Module Export FSet.
 
 End FSet.
 
-Notation "{| x |}" :=  (L x).
-Infix "∪" := U (at level 8, right associativity).
-Notation "∅" := E.
-
 Section set_sphere.
   From HoTT.HIT Require Import Circle.
   Context `{Univalence}.
@@ -274,10 +264,10 @@ Section set_sphere.
     - simpl. reflexivity.
   Defined.
 
-  Lemma FSet_S_ap : (nl (@E A)) = (nr E) -> idpath = loop.
+  Lemma FSet_S_ap : (nl (E A)) = (nr ∅) -> idpath = loop.
   Proof.
     intros H1.
-    enough (ap FSet_to_S (nl E) = ap FSet_to_S (nr E)) as H'.
+    enough (ap FSet_to_S (nl ∅) = ap FSet_to_S (nr ∅)) as H'.
     - rewrite FSet_rec_beta_nl in H'. 
       rewrite FSet_rec_beta_nr in H'.
       simpl in H'. unfold S1op_nr in H'.

FiniteSets/representations/cons_repr.v

@@ -1,35 +1,35 @@
 (* Definition of Finite Sets as via cons lists *)
 Require Import HoTT HitTactics.
+Require Export notation.
 
 Module Export FSetC.
   
   Section FSetC.
+    Private Inductive FSetC (A : Type) : Type :=
+    | Nil : FSetC A
+    | Cns : A ->  FSetC A -> FSetC A.
+
+    Global Instance fset_empty : hasEmpty FSetC := Nil.
+
     Variable A : Type.
-
-    Private Inductive FSetC : Type :=
-    | Nil : FSetC
-    | Cns : A ->  FSetC -> FSetC.
-
+    Arguments Cns {_} _ _.
     Infix ";;" := Cns (at level 8, right associativity).
-    Notation "∅" := Nil.
-
-    Axiom dupl : forall (a: A) (x: FSetC),
+    
+    Axiom dupl : forall (a : A) (x : FSetC A),
         a ;; a ;; x = a ;; x. 
 
-    Axiom comm : forall (a b: A) (x: FSetC),
+    Axiom comm : forall (a b : A) (x : FSetC A),
         a ;; b ;; x = b ;; a ;; x.
 
-    Axiom trunc : IsHSet FSetC.
+    Axiom trunc : IsHSet (FSetC A).
 
   End FSetC.
 
-  Arguments Nil {_}.
   Arguments Cns {_} _ _.
   Arguments dupl {_} _ _.
   Arguments comm {_} _ _ _.
 
   Infix ";;" := Cns (at level 8, right associativity).
-  Notation "∅" := Nil.
 
   Section FSetC_induction.
 
@@ -84,7 +84,6 @@ Module Export FSetC.
       - apply commP. 
     Defined.
 
-
     Definition FSetC_rec_beta_dupl : forall (a: A) (x : FSetC A),
         ap FSetC_rec (dupl a x) = duplP a (FSetC_rec x).
     Proof.
@@ -106,11 +105,12 @@ Module Export FSetC.
   End FSetC_recursion.
 
 
-  Instance FSetC_recursion A : HitRecursion (FSetC A) := {
-                                                          indTy := _; recTy := _; 
-                                                          H_inductor := FSetC_ind A; H_recursor := FSetC_rec A }.
+  Instance FSetC_recursion A : HitRecursion (FSetC A) :=
+    {
+      indTy := _; recTy := _; 
+      H_inductor := FSetC_ind A; H_recursor := FSetC_rec A
+    }.
 
 End FSetC.
 
 Infix ";;" := Cns (at level 8, right associativity).
-Notation "∅" := Nil.

FiniteSets/representations/definition.v

@@ -1,50 +1,44 @@
 (* Definitions of the Kuratowski-finite sets via a HIT *)
-Require Import HoTT.
-Require Import HitTactics.
+Require Import HoTT HitTactics.
+Require Export notation.
 
 Module Export FSet.
   Section FSet.
+    Private Inductive FSet (A : Type) : Type :=
+    | E : FSet A
+    | L : A -> FSet A
+    | U : FSet A -> FSet A -> FSet A.
+
+    Global Instance fset_empty : hasEmpty FSet := E.
+    Global Instance fset_singleton : hasSingleton FSet := L.
+    Global Instance fset_union : hasUnion FSet := U.
+
     Variable A : Type.
-
-    Private Inductive FSet : Type :=
-    | E : FSet
-    | L : A -> FSet
-    | U : FSet -> FSet -> FSet.
-
-    Notation "{| x |}" :=  (L x).
-    Infix "∪" := U (at level 8, right associativity).
-    Notation "∅" := E.
-
-    Axiom assoc : forall (x y z : FSet ),
+    
+    Axiom assoc : forall (x y z : FSet A),
         x ∪ (y ∪ z) = (x ∪ y) ∪ z.
 
-    Axiom comm : forall (x y : FSet),
+    Axiom comm : forall (x y : FSet A),
         x ∪ y = y ∪ x.
 
-    Axiom nl : forall (x : FSet),
+    Axiom nl : forall (x : FSet A),
         ∅ ∪ x = x.
 
-    Axiom nr : forall (x : FSet),
+    Axiom nr : forall (x : FSet A),
         x ∪ ∅ = x.
 
     Axiom idem : forall (x : A),
-        {| x |} ∪ {|x|} = {|x|}.
+        {|x|} ∪ {|x|} = {|x|}.
 
-    Axiom trunc : IsHSet FSet.
+    Axiom trunc : IsHSet (FSet A).
 
   End FSet.
-  
-  Arguments E {_}.
-  Arguments U {_} _ _.
-  Arguments L {_} _.
+
   Arguments assoc {_} _ _ _.
   Arguments comm {_} _ _.
   Arguments nl {_} _.
   Arguments nr {_} _.
-  Arguments idem {_} _.
-  Notation "{| x |}" :=  (L x).
-  Infix "∪" := U (at level 8, right associativity).
-  Notation "∅" := E.
+  Arguments idem {_} _.  
 
   Section FSet_induction.
     Variable A: Type.
@@ -194,10 +188,6 @@ Module Export FSet.
 
 End FSet.
 
-Notation "{| x |}" :=  (L x).
-Infix "∪" := U (at level 8, right associativity).
-Notation "∅" := E.
-
 Lemma union_idem {A : Type} : forall x: FSet A, x ∪ x = x.
 Proof.
   hinduction ; try (intros ; apply set_path2).