Completely fixed notation

FiniteSets/fsets/extensionality.v

@@ -45,7 +45,7 @@ Section ext.
     - intros a Y.
       apply (tr(inl(tr idpath))).
     - intros X1 X2 HX1 HX2 Y.
-      split. 
+      split ; unfold subset in *.
       * rewrite <- assoc. apply HX1.
       * rewrite (comm X1 X2). rewrite <- assoc. apply HX2.
   Defined.
@@ -64,7 +64,7 @@ Section ext.
 
   Lemma subset_isIn (X Y : FSet A) :
     (forall (a : A), a ∈ X -> a ∈ Y)
-      <~> (X ⊆ Y).
+      <~> X ⊆ Y.
   Proof.
     eapply equiv_iff_hprop_uncurried.
     split.

FiniteSets/fsets/isomorphism.v

@@ -45,7 +45,7 @@ Section Iso.
     intros x. 
     hrecursion x; try (intros; apply path_forall; intro; apply set_path2).
     - intros. symmetry. apply nl.
-    - intros a x HR y. rewrite HR. apply assoc.
+    - intros a x HR y. unfold union, fsetc_union in *. rewrite HR. apply assoc.
   Defined.
 
   Lemma repr_iso_id_l: forall (x: FSet A), FSetC_to_FSet (FSet_to_FSetC x) = x.
@@ -56,7 +56,6 @@ Section Iso.
     - intros x y p q. rewrite append_union, p, q. reflexivity.
   Defined.
 
-
   Lemma repr_iso_id_r: forall (x: FSetC A), FSet_to_FSetC (FSetC_to_FSet x) = x.
   Proof.
     hinduction; try (intros; apply set_path2).

FiniteSets/fsets/length.v

@@ -9,24 +9,22 @@ Section Length.
   Context {A_deceq : DecidablePaths A}.
   Context `{Univalence}.
 
-  Opaque isIn_b.
-
   Definition length (x : FSetC A) : nat.
   Proof.
     simple refine (FSetC_ind A _ _ _ _ _ _ x ); simpl.
     - exact 0.
     - intros a y n. 
       pose (y' := FSetC_to_FSet y).
-      exact (if isIn_b a y' then n else (S n)).
-    - intros. rewrite transport_const. cbn.
-      simplify_isIn_b. simpl. reflexivity.
-    - intros. rewrite transport_const. cbn.
+      exact (if a ∈_d y' then n else (S n)).
+    - intros. rewrite transport_const. simpl.
+      simplify_isIn_b. reflexivity.
+    - intros. rewrite transport_const. simpl.
       simplify_isIn_b.
       destruct (dec (a = b)) as [Hab | Hab].
-      + rewrite Hab. simplify_isIn_b. simpl. reflexivity.
+      + rewrite Hab. simplify_isIn_b. reflexivity.
       + rewrite ?L_isIn_b_false; auto.
         ++ simpl. 
-           destruct (isIn_b a (FSetC_to_FSet x0)), (isIn_b b (FSetC_to_FSet x0))
+           destruct (a ∈_d (FSetC_to_FSet x0)), (b ∈_d (FSetC_to_FSet x0))
            ; reflexivity.
         ++ intro p. contradiction (Hab p^).
   Defined.

FiniteSets/fsets/operations.v

@@ -3,12 +3,11 @@ Require Import HoTT HitTactics.
 Require Import representations.definition disjunction lattice.
 
 Section operations.
-  Context {A : Type}.
   Context `{Univalence}.
 
-  Definition isIn : A -> FSet A -> hProp.
+  Global Instance fset_member : forall A, hasMembership (FSet A) A.
   Proof.
-    intros a.
+    intros A a.
     hrecursion.
     - exists Empty.
       exact _.
@@ -23,10 +22,9 @@ Section operations.
     - intros ; apply lor_idem.
   Defined.
 
-  Definition comprehension : 
-    (A -> Bool) -> FSet A -> FSet A.
+  Global Instance fset_comprehension : forall A, hasComprehension (FSet A) A.
   Proof.
-    intros P.
+    intros A P.
     hrecursion.
     - apply ∅.
     - intro a.
@@ -42,7 +40,7 @@ Section operations.
       + apply nl.
   Defined.
 
-  Definition isEmpty :
+  Definition isEmpty (A : Type) :
     FSet A -> Bool.
   Proof.
     simple refine (FSet_rec _ _ _ true (fun _ => false) andb _ _ _ _ _)
@@ -50,7 +48,7 @@ Section operations.
     intros ; symmetry ; eauto with lattice_hints typeclass_instances.
   Defined.
 
-  Definition single_product {B : Type} (a : A) : FSet B -> FSet (A * B).
+  Definition single_product {A B : Type} (a : A) : FSet B -> FSet (A * B).
   Proof.
     hrecursion.
     - apply ∅.
@@ -64,7 +62,7 @@ Section operations.
     - intros ; apply idem.
   Defined.
         
-  Definition product {B : Type} : FSet A -> FSet B -> FSet (A * B).
+  Definition product {A B : Type} : FSet A -> FSet B -> FSet (A * B).
   Proof.
     intros X Y.
     hrecursion X.
@@ -79,28 +77,9 @@ Section operations.
     - intros ; apply union_idem.
   Defined.
    
-End operations.
-
-Section instances_operations.
-  Global Instance fset_comprehension : hasComprehension FSet.
+  Global Instance fset_subset : forall A, hasSubset (FSet A).
   Proof.
-    intros A ϕ X.
-    apply (comprehension ϕ X).
-  Defined.
-
-  Context `{Univalence}.
-
-  Global Instance fset_member : hasMembership FSet.
-  Proof.
-    intros A a X.
-    apply (isIn a X).
-  Defined.
-
-  Context {A : Type}.
-  
-  Definition subset : FSet A -> FSet A -> hProp.
-  Proof.
-    intros X Y.
+    intros A X Y.
     hrecursion X.
     - exists Unit.
       exact _.
@@ -123,6 +102,4 @@ Section instances_operations.
       * intros p.
         split ; apply p.
   Defined.
-End instances_operations.
-
-Infix  "⊆" := subset (at level 10, right associativity).
+End operations.

FiniteSets/fsets/operations_cons_repr.v

@@ -3,8 +3,7 @@ Require Import HoTT HitTactics.
 Require Import representations.cons_repr.
 
 Section operations.
-
-  Global Instance fsetc_union : hasUnion FSetC.
+  Global Instance fsetc_union : forall A, hasUnion (FSetC A).
   Proof.
     intros A x y.
     hinduction x.
@@ -14,6 +13,6 @@ Section operations.
     - apply comm.
   Defined.
 
-  Global Instance fsetc_singleton : hasSingleton FSetC := fun A a => a;;∅.
+  Global Instance fsetc_singleton : forall A, hasSingleton (FSetC A) A := fun A a => a;;∅.
 
 End operations.

FiniteSets/fsets/operations_decidable.v

@@ -7,7 +7,8 @@ Section decidable_A.
   Context {A_deceq : DecidablePaths A}.
   Context `{Univalence}.
   
-  Global Instance isIn_decidable : forall (a : A) (X : FSet A), Decidable (a ∈ X).
+  Global Instance isIn_decidable : forall (a : A) (X : FSet A),
+      Decidable (a ∈ X).
   Proof.
     intros a.
     hinduction ; try (intros ; apply path_ishprop).
@@ -23,26 +24,31 @@ Section decidable_A.
     - intros. apply _. 
   Defined.
 
-  Definition isIn_b (a : A) (X : FSet A) :=
-    match dec (a ∈ X) with
-    | inl _ => true
-    | inr _ => false
-    end.
+  Global Instance fset_member_bool : hasMembership_decidable (FSet A) A.
+  Proof.
+    intros a X.
+    destruct (dec (a ∈ X)).
+    - apply true.
+    - apply false.
+  Defined.
 
   Global Instance subset_decidable : forall (X Y : FSet A), Decidable (X ⊆ Y).
   Proof.
     hinduction ; try (intros ; apply path_ishprop).
     - intro ; apply _.
-    - intros. apply _. 
     - intros ; apply _. 
+    - intros. unfold subset in *. apply _.
   Defined.
 
-  Definition subset_b (X Y : FSet A) :=
-    match dec (X ⊆ Y) with
-    | inl _ => true
-    | inr _ => false
-    end.
+  Global Instance fset_subset_bool : hasSubset_decidable (FSet A).
+  Proof.
+    intros X Y.
+    destruct (dec (X ⊆ Y)).
+    - apply true.
+    - apply false.
+  Defined.
 
-  Definition intersection (X Y : FSet A) : FSet A := comprehension (fun a => isIn_b a Y) X. 
+  Global Instance fset_intersection : hasIntersection (FSet A)
+    := fun X Y => {|X & (fun a => a ∈_d Y)|}. 
 
 End decidable_A.

FiniteSets/fsets/properties.v

@@ -150,7 +150,7 @@ Section properties.
   Defined.  
 
   (** comprehension properties *)
-  Lemma comprehension_false X : {|X & (fun (_ : A) => false)|} = ∅.
+  Lemma comprehension_false : forall (X : FSet A), {|X & fun _ => false|} = ∅.
   Proof.
     toHProp.
   Defined.
@@ -172,7 +172,7 @@ Section properties.
   Defined.
 
   Lemma comprehension_all : forall (X : FSet A),
-      comprehension (fun a => true) X = X.
+      {|X & fun _ => true|} = X.
   Proof.
     toHProp.
   Defined.

FiniteSets/fsets/properties_decidable.v

@@ -8,13 +8,13 @@ Section operations_isIn.
 
   Context {A : Type} `{DecidablePaths A} `{Univalence}.
 
-  Lemma ext : forall (S T : FSet A), (forall a, isIn_b a S = isIn_b a T) -> S = T.
+  Lemma ext : forall (S T : FSet A), (forall a, a ∈_d S = a ∈_d T) -> S = T.
   Proof.
     intros X Y H2.
     apply fset_ext.
     intro a.
     specialize (H2 a).
-    unfold isIn_b, dec in H2.
+    unfold member_dec, fset_member_bool, dec in H2.
     destruct (isIn_decidable a X) ; destruct (isIn_decidable a Y).
     - apply path_iff_hprop ; intro ; assumption.
     - contradiction (true_ne_false).
@@ -23,7 +23,7 @@ Section operations_isIn.
   Defined.
 
   Lemma empty_isIn (a : A) :
-    isIn_b a ∅ = false.
+    a ∈_d ∅ = false.
   Proof.
     reflexivity.
   Defined.
@@ -35,9 +35,9 @@ Section operations_isIn.
   Defined.
 
   Lemma L_isIn_b_true (a b : A) (p : a = b) :
-    isIn_b a {|b|} = true.
+    a ∈_d {|b|} = true.
   Proof.
-    unfold isIn_b, dec.
+    unfold member_dec, fset_member_bool, dec.
     destruct (isIn_decidable a {|b|}) as [n | n] .
     - reflexivity.
     - simpl in n.
@@ -45,15 +45,15 @@ Section operations_isIn.
   Defined.
 
   Lemma L_isIn_b_aa (a : A) :
-    isIn_b a {|a|} = true.
+    a ∈_d {|a|} = true.
   Proof.
     apply L_isIn_b_true ; reflexivity.
   Defined.
 
   Lemma L_isIn_b_false (a b : A) (p : a <> b) :
-    isIn_b a {|b|} = false.
+    a ∈_d {|b|} = false.
   Proof.
-    unfold isIn_b, dec.
+    unfold member_dec, fset_member_bool, dec in *.
     destruct (isIn_decidable a {|b|}).
     - simpl in t.
       strip_truncations.
@@ -63,24 +63,25 @@ Section operations_isIn.
   
   (* Union and membership *)
   Lemma union_isIn_b (X Y : FSet A) (a : A) :
-    isIn_b a (X ∪ Y) = orb (isIn_b a X) (isIn_b a Y).
+    a ∈_d (X ∪ Y) = orb (a ∈_d X) (a ∈_d Y).
   Proof.
-    unfold isIn_b ; unfold dec.
+    unfold member_dec, fset_member_bool, dec.
     simpl.
     destruct (isIn_decidable a X) ; destruct (isIn_decidable a Y) ; reflexivity.
   Defined.
 
   Lemma comprehension_isIn_b (Y : FSet A) (ϕ : A -> Bool) (a : A) :
-    isIn_b a {|Y & ϕ|} = andb (isIn_b a Y) (ϕ a).
+    a ∈_d {|Y & ϕ|} = andb (a ∈_d Y) (ϕ a).
   Proof.
-    unfold isIn_b, dec ; simpl.
+    unfold member_dec, fset_member_bool, dec ; simpl.
     destruct (isIn_decidable a {|Y & ϕ|}) as [t | t]
     ; destruct (isIn_decidable a Y) as [n | n] ; rewrite comprehension_isIn in t
-    ; destruct (ϕ a) ; try reflexivity ; try contradiction.
+    ; destruct (ϕ a) ; try reflexivity ; try contradiction
+    ; try (contradiction (n t)) ; contradiction (t n).
   Defined.
 
   Lemma intersection_isIn_b (X Y: FSet A) (a : A) :
-    isIn_b a (intersection X Y) = andb (isIn_b a X) (isIn_b a Y).
+    a ∈_d (intersection X Y) = andb (a ∈_d X) (a ∈_d Y).
   Proof.
     apply comprehension_isIn_b.
   Defined.
@@ -114,7 +115,7 @@ Section SetLattice.
   Instance fset_min : minimum (FSet A).
   Proof.
     intros x y.
-    apply (intersection x y).
+    apply (x ∩ y).
   Defined.
   
   Instance fset_bot : bottom (FSet A).

FiniteSets/notation.v

@@ -48,31 +48,44 @@ Arguments neutralityR {_} {_} {_} {_} _.
 Arguments absorb {_} {_} {_} {_} _ _.
 
 Section structure.
-  Variable (T : Type -> Type).
+  Variable (T A : Type).
   
   Class hasMembership : Type :=
-    member : forall A : Type, A -> T A -> hProp.
+    member : A -> T -> hProp.
+
+  Class hasMembership_decidable : Type :=
+    member_dec : A -> T -> Bool.
+
+  Class hasSubset : Type :=
+    subset : T -> T -> hProp.
+
+  Class hasSubset_decidable : Type :=
+    subset_dec : T -> T -> Bool.
 
   Class hasEmpty : Type :=
-    empty : forall A, T A.
+    empty : T.
 
   Class hasSingleton : Type :=
-    singleton : forall A, A -> T A.
+    singleton : A -> T.
   
   Class hasUnion : Type :=
-    union : forall A, T A -> T A -> T A.
+    union : T -> T -> T.
 
   Class hasIntersection : Type :=
-    intersection : forall A, T A -> T A -> T A.
+    intersection : T -> T -> T.
 
   Class hasComprehension : Type :=
-    filter : forall A, (A -> Bool) -> T A -> T A.
+    filter : (A -> Bool) -> T -> T.
 End structure.
 
 Arguments member {_} {_} {_} _ _.
-Arguments empty {_} {_} {_}.
+Arguments subset {_} {_} _ _.
+Arguments member_dec {_} {_} {_} _ _.
+Arguments subset_dec {_} {_} _ _.
+Arguments empty {_} {_}.
 Arguments singleton {_} {_} {_} _.
-Arguments union {_} {_} {_} _ _.
+Arguments union {_} {_} _ _.
+Arguments intersection {_} {_} _ _.
 Arguments filter {_} {_} {_} _ _.
 
 Notation "∅" := empty.
@@ -87,3 +100,6 @@ Notation "( X ∩ )" := (intersection X) (only parsing).
 Notation "( ∩ Y )" := (fun X => X ∩ Y) (only parsing).
 Notation "{| X & ϕ |}" := (filter ϕ X).
 Infix "∈" := member (at level 9, right associativity).
+Infix  "⊆" := subset (at level 10, right associativity).
+Infix "∈_d" := member_dec (at level 9, right associativity).
+Infix  "⊆_d" := subset_dec (at level 10, right associativity).

FiniteSets/representations/bad.v

@@ -12,9 +12,9 @@ Module Export FSet.
     | 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.
+    Global Instance fset_empty : forall A, hasEmpty (FSet A) := E.
+    Global Instance fset_singleton : forall A, hasSingleton (FSet A) A := L.
+    Global Instance fset_union : forall A, hasUnion (FSet A) := U.
 
     Variable A : Type.
 

FiniteSets/representations/cons_repr.v

@@ -9,7 +9,7 @@ Module Export FSetC.
     | Nil : FSetC A
     | Cns : A ->  FSetC A -> FSetC A.
 
-    Global Instance fset_empty : hasEmpty FSetC := Nil.
+    Global Instance fset_empty : forall A,hasEmpty (FSetC A) := Nil.
 
     Variable A : Type.
     Arguments Cns {_} _ _.

FiniteSets/representations/definition.v

@@ -9,9 +9,9 @@ Module Export FSet.
     | 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.
+    Global Instance fset_empty : forall A, hasEmpty (FSet A) := E.
+    Global Instance fset_singleton : forall A, hasSingleton (FSet A) A := L.
+    Global Instance fset_union : forall A, hasUnion (FSet A) := U.
 
     Variable A : Type.