Make everything work with the new notation

FiniteSets/Sub.v

@@ -25,10 +25,10 @@ Section sub_classes.
     apply _.
   Defined.  
 
-  Definition hasUnion := forall X Y, C X -> C Y -> C (max_fun X Y).
-  Definition hasIntersection := forall X Y, C X -> C Y -> C (min_fun X Y).
-  Definition hasEmpty := C empty_sub.
-  Definition hasSingleton := forall a, C (singleton a).
+  Definition closedUnion := forall X Y, C X -> C Y -> C (max_fun X Y).
+  Definition closedIntersection := forall X Y, C X -> C Y -> C (min_fun X Y).
+  Definition closedEmpty := C empty_sub.
+  Definition closedSingleton := forall a, C (singleton a).
   Definition hasDecidableEmpty := forall X, C X -> hor (X = empty_sub) (hexists (fun a => X a)).
 End sub_classes.
 
@@ -37,12 +37,12 @@ Section isIn.
   Variable C : (A -> hProp) -> hProp.
 
   Context `{Univalence}.
-  Context {HS : hasSingleton C} {HIn : forall X, C X -> forall a, Decidable (X a)}.
+  Context {HS : closedSingleton C} {HIn : forall X, C X -> forall a, Decidable (X a)}.
 
   Theorem decidable_A_isIn : forall a b : A, Decidable (Trunc (-1) (b = a)).
   Proof.
     intros.
-    unfold Decidable, hasSingleton in *.
+    unfold Decidable, closedSingleton in *.
     pose (HIn (singleton a) (HS a) b).
     destruct s.
     - unfold singleton in t.
@@ -71,13 +71,13 @@ Section intersect.
   Defined.
 
   Context
-    {HI :hasIntersection C} {HE : hasEmpty C}
-    {HS : hasSingleton C} {HDE : hasDecidableEmpty C}.
+    {HI : closedIntersection C} {HE : closedEmpty C}
+    {HS : closedSingleton C} {HDE : hasDecidableEmpty C}.
 
   Theorem decidable_A_intersect : forall a b : A, Decidable (Trunc (-1) (b = a)).
   Proof.
     intros.
-    unfold Decidable, hasEmpty, hasIntersection, hasSingleton, hasDecidableEmpty in *.
+    unfold Decidable, closedEmpty, closedIntersection, closedSingleton, hasDecidableEmpty in *.
     pose (HI (singleton a) (singleton b) (HS a) (HS b)) as IntAB.
     pose (HDE (min_fun (singleton a) (singleton b)) IntAB) as IntE.
     refine (@Trunc_rec _ _ _  _ _ IntE) ; intros [p | p] ; unfold min_fun, singleton in p.

FiniteSets/implementations/lists.v

@@ -5,30 +5,29 @@ Require Import FSets implementations.interface.
 Section Operations.
   Context `{Univalence}.
 
-  Global Instance list_empty : hasEmpty list := fun A => nil.
+  Global Instance list_empty A : hasEmpty (list A) := nil.
 
-  Global Instance list_single : hasSingleton list := fun A a => cons a nil.
+  Global Instance list_single A: hasSingleton (list A) A := fun a => cons a nil.
   
-  Global Instance list_union : hasUnion list.
+  Global Instance list_union A : hasUnion (list A).
   Proof.
-    intros A l1 l2.
+    intros l1 l2.
     induction l1.
     * apply l2.
     * apply (cons a IHl1).
   Defined.
 
-  Global Instance list_membership : hasMembership list.
+  Global Instance list_membership A : hasMembership (list A) A.
   Proof.
-    intros A.
     intros a l.
     induction l as [ | b l IHl].
     - apply False_hp.
     - apply (hor (a = b) IHl).
   Defined.
 
-  Global Instance list_comprehension : hasComprehension list.
+  Global Instance list_comprehension A: hasComprehension (list A) A.
   Proof.
-    intros A ϕ l.
+    intros ϕ l.
     induction l as [ | b l IHl].
     - apply nil.
     - apply (if ϕ b then cons b IHl else IHl).
@@ -36,8 +35,8 @@ Section Operations.
 
   Fixpoint list_to_set A (l : list A) :  FSet A :=
     match l with
-    | nil => E
-    | cons a l => U (L a) (list_to_set A l)
+    | nil => ∅
+    | cons a l => {|a|} ∪ (list_to_set A l)
     end.
 
 End Operations.
@@ -60,10 +59,10 @@ Section ListToSet.
       * apply (tr (inr z2)).
   Defined.
   
-  Definition empty_empty : list_to_set A empty = ∅ := idpath.
+  Definition empty_empty : list_to_set A ∅ = ∅ := idpath.
 
   Lemma filter_comprehension (ϕ : A -> Bool) (l : list A)  :
-    list_to_set A (filter ϕ l) = comprehension ϕ (list_to_set A l).
+    list_to_set A (filter ϕ l) =  {| list_to_set A l & ϕ |}.
   Proof.
     induction l ; cbn in *.
     - reflexivity.
@@ -81,7 +80,7 @@ Section ListToSet.
     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 _)^.
+    - apply (nl _)^.
     - apply (nl _)^.
     - rewrite IHl1.
       apply assoc.

FiniteSets/variations/enumerated.v

@@ -223,7 +223,7 @@ Section enumerated_fset.
     end.
   
   Lemma list_to_fset_ext (ls : list A) (a : A):
-    listExt ls a -> isIn a (list_to_fset ls).
+    listExt ls a -> a ∈ (list_to_fset ls).
   Proof.
     induction ls as [|x xs]; simpl.
     - apply idmap.
@@ -269,7 +269,7 @@ Section fset_dec_enumerated.
     - intros a X Hls.
       strip_truncations. apply tr.
       destruct Hls as [ls Hls].
-      exists (cons a ls). intros b. simpl.
+      exists (cons a ls). intros b. cbn.
       f_ap.
     - intros. apply path_ishprop.
     - intros. apply path_ishprop.
@@ -294,16 +294,16 @@ Section subobjects.
   Definition enumeratedS (P : Sub A) : hProp :=
     enumerated (sigT P).
   
-  Lemma enumeratedS_empty : enumeratedS empty_sub.
+  Lemma enumeratedS_empty : closedEmpty enumeratedS.
   Proof.
     unfold enumeratedS.
     apply tr. exists nil. simpl.
     intros [a Ha]. assumption.
   Defined.
 
-  Lemma enumeratedS_singleton (x : A) : enumeratedS (singleton x).
+  Lemma enumeratedS_singleton : closedSingleton enumeratedS.
   Proof.
-    apply tr. simpl.
+    intros x. apply tr. simpl.
     exists (cons (x;tr idpath) nil).
     intros [y Hxy]. simpl.
     strip_truncations. apply tr.
@@ -417,7 +417,7 @@ Section subobjects.
     end.
 
   Lemma list_weaken_to_fset_ext (P : Sub A) (ls : list (sigT P)) (a : A) (Ha : P a):
-    listExt ls (a;Ha) -> isIn a (list_weaken_to_fset P ls).
+    listExt ls (a;Ha) -> a ∈ (list_weaken_to_fset P ls).
   Proof.
     induction ls as [|[x Hx] xs]; simpl.
     - apply idmap.

FiniteSets/variations/k_finite.v

@@ -6,7 +6,7 @@ Section k_finite.
   Context (A : Type).
   Context `{Univalence}.
 
-  Definition map (X : FSet A) : Sub A := fun a => isIn a X.
+  Definition map (X : FSet A) : Sub A := fun a => a ∈ X.
 
   Global Instance map_injective : IsEmbedding map.
   Proof.
@@ -69,37 +69,35 @@ Section structure_k_finite.
   Context (A : Type).
   Context `{Univalence}.
 
-  Lemma map_union : forall X Y : FSet A, map (U X Y) = max_fun (map X) (map Y).
+  Lemma map_union : forall X Y : FSet A, map (X ∪ Y) = max_fun (map X) (map Y).
   Proof.
     intros.
     unfold map, max_fun.
     reflexivity.
   Defined.
 
-  Lemma k_finite_union : hasUnion (Kf_sub A).
+  Lemma k_finite_union : closedUnion (Kf_sub A).
   Proof.
-    unfold hasUnion, Kf_sub, Kf_sub_intern.
+    unfold closedUnion, Kf_sub, Kf_sub_intern.
     intros.
     destruct X0 as [SX XP].
     destruct X1 as [SY YP].
-    exists (U SX SY).
+    exists (SX ∪ SY).
     rewrite map_union.
     rewrite XP, YP.
     reflexivity.
   Defined.
 
-  Lemma k_finite_empty : hasEmpty (Kf_sub A).
+  Lemma k_finite_empty : closedEmpty (Kf_sub A).
   Proof.
-    unfold hasEmpty, Kf_sub, Kf_sub_intern, map, empty_sub.
-    exists E.
+    exists ∅.
     reflexivity.
   Defined.
 
-  Lemma k_finite_singleton : hasSingleton (Kf_sub A).
+  Lemma k_finite_singleton : closedSingleton (Kf_sub A).
   Proof.
-    unfold hasSingleton, Kf_sub, Kf_sub_intern, map, singleton.
     intro.
-    exists (L a).
+    exists {|a|}.
     cbn.
     apply path_forall.
     intro z. 
@@ -108,7 +106,7 @@ Section structure_k_finite.
 
   Lemma k_finite_hasDecidableEmpty : hasDecidableEmpty (Kf_sub A).
   Proof.
-    unfold hasDecidableEmpty, hasEmpty, Kf_sub, Kf_sub_intern, map.
+    unfold hasDecidableEmpty, closedEmpty, Kf_sub, Kf_sub_intern, map.
     intros.
     destruct X0 as [SX EX].
     rewrite EX.