Port the codebase to HottClasses

Initial work + use the latest version of HoTT

FiniteSets/implementations/lists.v

@@ -62,7 +62,7 @@ Section ListToSet.
   Definition empty_empty : list_to_set A ∅ = ∅ := idpath.
 
   Lemma filter_comprehension (ϕ : A -> Bool) (l : list A)  :
-    list_to_set A (filter ϕ l) =  {| list_to_set A l & ϕ |}.
+    list_to_set A (filter ϕ l) =  {| list_to_set A l | ϕ |}.
   Proof.
     induction l ; cbn in *.
     - reflexivity.
@@ -170,11 +170,11 @@ Section refinement_examples.
   Defined.
 
   Lemma exist_elim (X : list A) (ϕ : A -> hProp)
-    : list_exist ϕ X -> hexists (fun a => a ∈ X * ϕ a).
+    : list_exist ϕ X -> hexists (fun a => a ∈ X * ϕ a)%type.
   Proof.
     rewrite list_exist_set.
     assert (hexists (fun a : A => a ∈ (list_to_set A X) * ϕ a)
-            -> hexists (fun a : A => a ∈ X * ϕ a))
+            -> hexists (fun a : A => a ∈ X * ϕ a))%type
       as H2.
     {
       intros H1.

FiniteSets/interfaces/lattice_examples.v

@@ -1,5 +1,5 @@
 (** Some examples of lattices. *)
-Require Import HoTT lattice_interface.
+Require Export HoTT lattice_interface.
 
 (** [Bool] is a lattice. *)
 Section BoolLattice.
@@ -10,14 +10,18 @@ Section BoolLattice.
     ; auto
     ; try contradiction.
 
-  Instance maximum_bool : maximum Bool := orb.
+  Instance maximum_bool : Join Bool := orb.
-  Instance minimum_bool : minimum Bool := andb.
+  Instance minimum_bool : Meet Bool := andb.
-  Instance bottom_bool : bottom Bool := false.
+  Instance bottom_bool : Bottom Bool := false.
 
+  Global Instance boundedjoinsemilattice_bool : BoundedJoinSemiLattice Bool.
+  Proof. repeat (split ; (apply _ || solve_bool)). Defined.
+  Global Instance meetsemilattice_bool : MeetSemiLattice Bool.
+  Proof. repeat (split ; (apply _ || solve_bool)). Defined.
+  Global Instance boundedmeetsemilattice_bool : @BoundedSemiLattice Bool (⊓) true.
+  Proof. repeat (split ; (apply _ || solve_bool)). Defined.
   Global Instance lattice_bool : Lattice Bool.
-  Proof.
+  Proof. split ; (apply _ || solve_bool). Defined.
-    split ; solve_bool.
-  Defined.
 
   Definition and_true : forall b, andb b true = b.
   Proof.
@@ -50,12 +54,12 @@ End BoolLattice.
 
 Create HintDb bool_lattice_hints.
 Hint Resolve associativity : bool_lattice_hints.
-Hint Resolve (associativity _ _ _)^ : bool_lattice_hints.
+(* Hint Resolve (associativity _ _ _)^ : bool_lattice_hints. *)
 Hint Resolve commutativity : bool_lattice_hints.
-Hint Resolve absorb : bool_lattice_hints.
+Hint Resolve absorption : bool_lattice_hints.
 Hint Resolve idempotency : bool_lattice_hints.
-Hint Resolve neutralityL : bool_lattice_hints.
+Hint Resolve left_identity : bool_lattice_hints.
-Hint Resolve neutralityR : bool_lattice_hints.
+Hint Resolve right_identity : bool_lattice_hints.
 
 Hint Resolve
      associativity
@@ -66,25 +70,51 @@ Hint Resolve
 (** If [B] is a lattice, then [A -> B] is a lattice. *)
 Section fun_lattice.
   Context {A B : Type}.
-  Context `{Lattice B}.
+  Context `{BJoin : Join B}.
+  Context `{BMeet : Meet B}.
+  Context `{@Lattice B BJoin BMeet}.
   Context `{Funext}.
+  Context `{BBottom : Bottom B}.
 
-  Global Instance max_fun : maximum (A -> B) :=
+  Global Instance bot_fun : Bottom (A -> B)
-    fun (f g : A -> B) (a : A) => max_L0 (f a) (g a).
+    := fun _ => ⊥.
 
-  Global Instance min_fun : minimum (A -> B) :=
+  Global Instance join_fun : Join (A -> B) :=
-    fun (f g : A -> B) (a : A) => min_L0 (f a) (g a).
+    fun (f g : A -> B) (a : A) => (f a) ⊔ (g a).
 
-  Global Instance bot_fun : bottom (A -> B)
+  Global Instance meet_fun : Meet (A -> B) :=
-    := fun _ => empty_L.
+    fun (f g : A -> B) (a : A) => (f a) ⊓ (g a).
 
   Ltac solve_fun :=
     compute ; intros ; apply path_forall ; intro ;
     eauto with lattice_hints typeclass_instances.
 
+  Create HintDb test1.
+  Lemma associativity_lat `{Lattice A} (x y z : A) :
+    x ⊓ (y ⊓ z) = x ⊓ y ⊓ z.
+  Proof. apply associativity. Defined.
+  Hint Resolve associativity : test1.
+  Hint Resolve associativity_lat : test1.
+
   Global Instance lattice_fun : Lattice (A -> B).
   Proof.
-    split ; solve_fun.
+    repeat (split; try (apply _)).
+    eauto with test1.
+    (* TODO *)
+    all: solve_fun.
+    apply associativity.
+    apply commutativity.
+    apply idempotency. apply _.
+    apply associativity.
+    apply commutativity.
+    apply idempotency. apply _.    
+  Defined.
+
+  Global Instance boundedjoinsemilattice_fun
+   `{@BoundedJoinSemiLattice B BJoin BBottom} :
+    BoundedJoinSemiLattice (A -> B).
+  Proof.
+    repeat split; try apply _; try solve_fun.
   Defined.
 End fun_lattice.
 
@@ -92,24 +122,26 @@ End fun_lattice.
 Section sub_lattice.
   Context {A : Type} {P : A -> hProp}.
   Context `{Lattice A}.
-  Context {Hmax : forall x y, P x -> P y -> P (max_L0 x y)}.
+  Context `{Bottom A}.
-  Context {Hmin : forall x y, P x -> P y -> P (min_L0 x y)}.
+  Context {Hmax : forall x y, P x -> P y -> P (x ⊔ y)}.
-  Context {Hbot : P empty_L}.
+  Context {Hmin : forall x y, P x -> P y -> P (x ⊓ y)}.
+  Context {Hbot : P ⊥}.
 
   Definition AP : Type := sig P.
 
-  Instance botAP : bottom AP := (empty_L ; Hbot).
+  Instance botAP : Bottom AP.
+  Proof. refine (⊥ ↾ _). apply Hbot. Defined.
 
-  Instance maxAP : maximum AP :=
+  Instance maxAP : Join AP :=
     fun x y =>
       match x, y with
-      | (a ; pa), (b ; pb) => (max_L0 a b ; Hmax a b pa pb)
+      | (a ; pa), (b ; pb) => (a ⊔ b ; Hmax a b pa pb)
       end.
 
-  Instance minAP : minimum AP :=
+  Instance minAP : Meet AP :=
     fun x y =>
       match x, y with
-      | (a ; pa), (b ; pb) => (min_L0 a b ; Hmin a b pa pb)
+      | (a ; pa), (b ; pb) => (a ⊓ b ; Hmin a b pa pb)
       end.
 
   Instance hprop_sub : forall c, IsHProp (P c).
@@ -127,19 +159,25 @@ Section sub_lattice.
 
   Global Instance lattice_sub : Lattice AP.
   Proof.
-    split ; solve_sub.
+    repeat (split ; try (apply _ || solve_sub)).
+    apply associativity.
+    apply commutativity.
+    apply idempotency. apply _.
+    apply associativity.
+    apply commutativity.
+    apply idempotency. apply _.
   Defined.
 End sub_lattice.
 
-Instance lor : maximum hProp := fun X Y => BuildhProp (Trunc (-1) (sum X Y)).
+Instance lor : Join hProp := fun X Y => BuildhProp (Trunc (-1) (sum X Y)).
 
 Delimit Scope logic_scope with L.
 Notation "A ∨ B" := (lor A B) (at level 20, right associativity) : logic_scope.
 Arguments lor _%L _%L.
 Open Scope logic_scope.
 
-Instance land : minimum hProp := fun X Y => BuildhProp (prod X Y).
+Instance land : Meet hProp := fun X Y => BuildhProp (prod X Y).
-Instance lfalse : bottom hProp := False_hp.
+Instance lfalse : Bottom hProp := False_hp.
 
 Notation "A ∧ B" := (land A B) (at level 20, right associativity) : logic_scope.
 Arguments land _%L _%L.
@@ -181,17 +219,18 @@ Section hPropLattice.
     intros X Y Z.
     symmetry.
     apply path_hprop.
+    symmetry.
     apply equiv_prod_assoc.
   Defined.
 
-  Instance lor_idempotent : Idempotent lor.
+  Instance lor_idempotent : BinaryIdempotent lor.
   Proof.
     intros X.
     apply path_iff_hprop ; lor_intros
     ; try(refine (tr(inl _))) ; auto.
   Defined.
 
-  Instance land_idempotent : Idempotent land.
+  Instance land_idempotent : BinaryIdempotent land.
   Proof.
     intros X.
     apply path_iff_hprop ; cbn.
@@ -199,14 +238,14 @@ Section hPropLattice.
     - intros a ; apply (pair a a).
   Defined.
 
-  Instance lor_neutrall : NeutralL lor lfalse.
+  Instance lor_neutrall : LeftIdentity lor lfalse.
   Proof.
     intros X.
     apply path_iff_hprop ; lor_intros ; try contradiction
     ; try (refine (tr(inr _))) ; auto.
   Defined.
 
-  Instance lor_neutralr : NeutralR lor lfalse.
+  Instance lor_neutralr : RightIdentity lor lfalse.
   Proof.
     intros X.
     apply path_iff_hprop ; lor_intros ; try contradiction
@@ -235,16 +274,17 @@ Section hPropLattice.
       * apply (tr (inl X)).
   Defined.
 
-  Global Instance lattice_hprop : Lattice hProp :=
+  Global Instance lattice_hprop : Lattice hProp.
-    { commutative_min := _ ;
+  Proof. repeat (split ; try apply _). Defined.
-      commutative_max := _ ;
+
-      associative_min := _ ;
+  Global Instance bounded_jsl_hprop : BoundedJoinSemiLattice hProp.
-      associative_max := _ ;
+  Proof. repeat (split ; try apply _). Qed.
-      idempotent_min := _ ;
+
-      idempotent_max := _ ;
+  Global Instance top_hprop : Top hProp := Unit_hp.
-      neutralL_max := _ ;
+  Global Instance bounded_msl_hprop : @BoundedSemiLattice hProp (⊓) ⊤.
-      neutralR_max := _ ;
+  Proof.
-      absorption_min_max := _ ;
+    repeat (split; try apply _); cbv.
-      absorption_max_min := _
+    - intros X. apply path_trunctype ; apply prod_unit_l.
-  }.
+    - intros X. apply path_trunctype ; apply prod_unit_r.
+  Defined.
 End hPropLattice.

FiniteSets/interfaces/lattice_interface.v

@@ -1,119 +1,60 @@
 (** Interface for lattices and join semilattices. *)
 Require Import HoTT.
+From HoTT.Classes.interfaces Require Export abstract_algebra canonical_names.
+From HoTT.Classes Require Export theory.lattices.
 
-(** Some preliminary notions to define lattices. *)
+(* (** Join semilattices as a typeclass. They only have a join operator. *) *)
-Section binary_operation.
+(* Section JoinSemiLattice. *)
-  Definition operation (A : Type) := A -> A -> A.
+(*   Variable A : Type. *)
+(*   Context {max_L : Join A} {empty_L : Bottom A}. *)
 
-  Variable (A : Type)
+(*   Class JoinSemiLattice := *)
-           (f : operation A).
+(*     { *)
+(*       commutative_max_js :> Commutative max_L ; *)
+(*       associative_max_js :> Associative max_L ; *)
+(*       idempotent_max_js :> BinaryIdempotent max_L ; *)
+(*       neutralL_max_js :> LeftIdentity max_L empty_L ; *)
+(*       neutralR_max_js :> RightIdentity max_L empty_L ; *)
+(*     }. *)
+(* End JoinSemiLattice. *)
 
-  Class Commutative :=
+(* Arguments JoinSemiLattice _ {_} {_}. *)
-    commutativity : forall x y, f x y = f y x.
 
-  Class Associative :=
+(* Create HintDb joinsemilattice_hints. *)
-    associativity : forall x y z, f (f x y) z = f x (f y z).
+(* Hint Resolve associativity : joinsemilattice_hints. *)
+(* Hint Resolve (associativity _ _ _)^ : joinsemilattice_hints. *)
+(* Hint Resolve commutativity : joinsemilattice_hints. *)
+(* Hint Resolve idempotency : joinsemilattice_hints. *)
+(* Hint Resolve neutralityL : joinsemilattice_hints. *)
+(* Hint Resolve neutralityR : joinsemilattice_hints. *)
 
-  Class Idempotent :=
+(* (** Lattices as a typeclass which have both a join and a meet. *) *)
-    idempotency : forall x, f x x = x.
+(* Section Lattice. *)
+(*   Variable A : Type. *)
+(*   Context {max_L : maximum A} {min_L : minimum A} {empty_L : bottom A}. *)
 
-  Variable g : operation A.
+(*   Class Lattice := *)
+(*     { *)
+(*       commutative_min :> Commutative min_L ; *)
+(*       commutative_max :> Commutative max_L ; *)
+(*       associative_min :> Associative min_L ; *)
+(*       associative_max :> Associative max_L ; *)
+(*       idempotent_min :> Idempotent min_L ; *)
+(*       idempotent_max :> Idempotent max_L ; *)
+(*       neutralL_max :> NeutralL max_L empty_L ; *)
+(*       neutralR_max :> NeutralR max_L empty_L ; *)
+(*       absorption_min_max :> Absorption min_L max_L ; *)
+(*       absorption_max_min :> Absorption max_L min_L *)
+(*     }. *)
+(* End Lattice. *)
 
-  Class Absorption :=
+(* Arguments Lattice _ {_} {_} {_}. *)
-    absorb : forall x y, f x (g x y) = x.
-
-  Variable (n : A).
-
-  Class NeutralL :=
-    neutralityL : forall x, f n x = x.
-
-  Class NeutralR :=
-    neutralityR : forall x, f x n = x.
-End binary_operation.
-
-Arguments Commutative {_} _.
-Arguments Associative {_} _.
-Arguments Idempotent {_} _.
-Arguments NeutralL {_} _ _.
-Arguments NeutralR {_} _ _.
-Arguments Absorption {_} _ _.
-Arguments commutativity {_} {_} {_} _ _.
-Arguments associativity {_} {_} {_} _ _ _.
-Arguments idempotency {_} {_} {_} _.
-Arguments neutralityL {_} {_} {_} {_} _.
-Arguments neutralityR {_} {_} {_} {_} _.
-Arguments absorb {_} {_} {_} {_} _ _.
-
-(** The operations in a lattice. *)
-Section lattice_operations.
-  Variable (A : Type).
-
-  Class maximum :=
-    max_L : operation A.
-
-  Class minimum :=
-    min_L : operation A.
-
-  Class bottom :=
-    empty : A.
-End lattice_operations.
-
-Arguments max_L {_} {_} _.
-Arguments min_L {_} {_} _.
-Arguments empty {_}.
-
-(** Join semilattices as a typeclass. They only have a join operator. *)
-Section JoinSemiLattice.
-  Variable A : Type.
-  Context {max_L : maximum A} {empty_L : bottom A}.
-
-  Class JoinSemiLattice :=
-    {
-      commutative_max_js :> Commutative max_L ;
-      associative_max_js :> Associative max_L ;
-      idempotent_max_js :> Idempotent max_L ;
-      neutralL_max_js :> NeutralL max_L empty_L ;
-      neutralR_max_js :> NeutralR max_L empty_L ;
-    }.
-End JoinSemiLattice.
-
-Arguments JoinSemiLattice _ {_} {_}.
-
-Create HintDb joinsemilattice_hints.
-Hint Resolve associativity : joinsemilattice_hints.
-Hint Resolve (associativity _ _ _)^ : joinsemilattice_hints.
-Hint Resolve commutativity : joinsemilattice_hints.
-Hint Resolve idempotency : joinsemilattice_hints.
-Hint Resolve neutralityL : joinsemilattice_hints.
-Hint Resolve neutralityR : joinsemilattice_hints.
-
-(** Lattices as a typeclass which have both a join and a meet. *)
-Section Lattice.
-  Variable A : Type.
-  Context {max_L : maximum A} {min_L : minimum A} {empty_L : bottom A}.
-
-  Class Lattice :=
-    {
-      commutative_min :> Commutative min_L ;
-      commutative_max :> Commutative max_L ;
-      associative_min :> Associative min_L ;
-      associative_max :> Associative max_L ;
-      idempotent_min :> Idempotent min_L ;
-      idempotent_max :> Idempotent max_L ;
-      neutralL_max :> NeutralL max_L empty_L ;
-      neutralR_max :> NeutralR max_L empty_L ;
-      absorption_min_max :> Absorption min_L max_L ;
-      absorption_max_min :> Absorption max_L min_L
-    }.
-End Lattice.
-
-Arguments Lattice _ {_} {_} {_}.
 
 Create HintDb lattice_hints.
 Hint Resolve associativity : lattice_hints.
-Hint Resolve (associativity _ _ _)^ : lattice_hints.
+(* Hint Resolve (associativity _ _ _)^ : lattice_hints. *)
 Hint Resolve commutativity : lattice_hints.
-Hint Resolve absorb : lattice_hints.
+Hint Resolve absorption : lattice_hints.
 Hint Resolve idempotency : lattice_hints.
-Hint Resolve neutralityL : lattice_hints.
+Hint Resolve left_identity : lattice_hints.
-Hint Resolve neutralityR : lattice_hints.
+Hint Resolve right_identity : lattice_hints.

FiniteSets/interfaces/monad_interface.v

@@ -1,5 +1,7 @@
 (** The structure of a monad. *)
+(* TODO REMOVE THIS *)
 Require Import HoTT.
+Require Export HoTT.Classes.interfaces.monad.
 
 Section functor.
   Variable (F : Type -> Type).
@@ -12,34 +14,8 @@ Section functor.
     }.
 End functor.
 
-Section monad_operations.
+Section monad_join.
-  Variable (F : Type -> Type).
+  Context `{Monad M}.
 
-  Class hasPure :=
+  Definition mjoin {A} (m : M (M A)) : M A := bind m id.
-    {
+End monad_join.
-      pure : forall (A : Type), A -> F A
-    }.
-
-  Class hasBind :=
-    {
-      bind : forall (A : Type), F(F A) -> F A
-    }.
-End monad_operations.
-
-Arguments pure {_} {_} _ _.
-Arguments bind {_} {_} _ _.
-
-Section monad.
-  Variable (F : Type -> Type).
-  Context `{Functor F} `{hasPure F} `{hasBind F}.
-
-  Class Monad :=
-    {
-      bind_assoc : forall {A : Type},
-        bind A o bind (F A) = bind A o fmap F (bind A) ;
-      bind_neutral_l : forall {A : Type},
-          bind A o pure (F A) = idmap ;
-      bind_neutral_r : forall {A : Type},
-          bind A o fmap F (pure A) = idmap
-    }.
-End monad.

FiniteSets/interfaces/set_interface.v

@@ -16,7 +16,7 @@ Section interface.
       f_empty : forall A, f A ∅ = ∅ ;
       f_singleton : forall A a, f A (singleton a) = {|a|};
       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) = {| f A X & φ |};
+      f_filter : forall A φ X, f A (filter φ X) = {| f A X | φ |};
       f_member : forall A a X, member a X = a ∈ (f A X)
     }.
 
@@ -175,12 +175,12 @@ Section quotient_properties.
   Proof.
     intros ϕ X.
     apply (quotient_iso _)^-1.
-    simple refine ({|_ & ϕ|}).
+    simple refine ({|_ | ϕ|}).
     apply (quotient_iso (f A) X).
   Defined.
 
   Definition well_defined_filter (A : Type) (ϕ : A -> Bool) (X : T A) :
-    {|class_of _ X & ϕ|} = class_of _ {|X & ϕ|}.
+    {|class_of _ X | ϕ|} = class_of _ {|X | ϕ|}.
   Proof.
     rewrite <- (eissect (quotient_iso _)).
     simpl.
@@ -224,30 +224,35 @@ Section quotient_properties.
       apply (ap _ HXY).
   Defined.
 
-  Instance View_max A : maximum (View A).
+  Instance join_view A : Join (View A).
   Proof.
     apply view_union.
   Defined.
 
-  Hint Unfold Commutative Associative Idempotent NeutralL NeutralR View_max view_union.
+  Local Hint Unfold Commutative Associative HeteroAssociative Idempotent BinaryIdempotent LeftIdentity RightIdentity join_view view_union sg_op join_is_sg_op meet_is_sg_op.
 
-  Instance bottom_view A : bottom (View A).
+  Instance bottom_view A : Bottom (View A).
   Proof.
     apply View_empty.
   Defined.
 
-  Ltac sq1 := autounfold ; intros ; try f_ap
+  Ltac sq1 := autounfold; intros;
-                         ; rewrite ?(eisretr (quotient_iso _))
+              unfold view_union; try f_ap;
-                         ; eauto with lattice_hints typeclass_instances.
+              rewrite ?(eisretr (quotient_iso _)).
     
-  Ltac sq2 := autounfold ; intros
+  Ltac sq2 := autounfold; intros;
-              ; rewrite <- (eissect (quotient_iso _)), ?(eisretr (quotient_iso _))
+              unfold view_union;
-              ; f_ap ; simpl
+              rewrite <- (eissect (quotient_iso _)), ?(eisretr (quotient_iso _));
-              ; reduce T ; eauto with lattice_hints typeclass_instances.
+              f_ap; simpl.
 
-  Global Instance view_lattice A : JoinSemiLattice (View A).
+  Global Instance view_lattice A : BoundedJoinSemiLattice (View A).
   Proof.
-    split ; try sq1 ; try sq2.
+    repeat split; try apply _.
+    - sq1. apply associativity.
+    - sq2. apply left_identity.
+    - sq2. apply right_identity.
+    - sq1. apply commutativity.
+    - sq2. apply binary_idempotent.
   Defined.
 
 End quotient_properties.
@@ -316,7 +321,8 @@ Section properties.
     
   Ltac via_quotient := intros ; apply reflect_f_eq ; simpl
                        ; rewrite <- ?(well_defined_union T _), <- ?(well_defined_empty T _)
-                       ; eauto with lattice_hints typeclass_instances.  
+                       ; try (apply commutativity || apply associativity || apply binary_idempotent || apply left_identity || apply right_identity).
+                       (* TODO ; eauto with lattice_hints typeclass_instances. *)
 
   Lemma union_comm : forall A (X Y : T A),
       set_eq f (X ∪ Y) (Y ∪ X).
@@ -328,6 +334,7 @@ Section properties.
     set_eq f ((X ∪ Y) ∪ Z) (X ∪ (Y ∪ Z)).
   Proof.
     via_quotient.
+    symmetry. apply associativity.
   Defined.
 
   Lemma union_idem : forall A (X : T A),

FiniteSets/interfaces/set_names.v

@@ -53,7 +53,7 @@ Infix "∩" := intersection (at level 8, right associativity).
 Notation "( ∩ )" := intersection (only parsing).
 Notation "( X ∩ )" := (intersection X) (only parsing).
 Notation "( ∩ Y )" := (fun X => X ∩ Y) (only parsing).
-Notation "{| X & ϕ |}" := (filter ϕ X).
+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).

FiniteSets/kuratowski/kuratowski_sets.v

@@ -139,11 +139,17 @@ Section relations.
     - apply False_hp.
     - apply (fun a' => merely(a = a')).
     - apply lor.
+      (* TODO *)
     - eauto with lattice_hints typeclass_instances.
+      apply associativity.
     - eauto with lattice_hints typeclass_instances.
+      apply commutativity.
     - eauto with lattice_hints typeclass_instances.
+      apply left_identity.
     - eauto with lattice_hints typeclass_instances.
+      apply right_identity.
     - eauto with lattice_hints typeclass_instances.
+      intros. simpl. apply binary_idempotent.
   Defined.
 
   (** Subset relation of finite sets. *)
@@ -153,15 +159,15 @@ Section relations.
     hrecursion X.
     - apply Unit_hp.
     - apply (fun a => a ∈ Y).
-    - intros X1 X2.
+    - apply land.
-      exists (prod X1 X2).
+      (* TODO *)
-      exact _.
     - eauto with lattice_hints typeclass_instances. 
+      apply associativity.
     - eauto with lattice_hints typeclass_instances.
-    - intros.
+      apply commutativity.
-      apply path_trunctype ; apply prod_unit_l.
+    - apply left_identity.
-    - intros.
+    - apply right_identity.
-      apply path_trunctype ; apply prod_unit_r.
     - eauto with lattice_hints typeclass_instances.
+      intros. apply binary_idempotent.
   Defined.
 End relations.

FiniteSets/kuratowski/length.v

@@ -6,7 +6,7 @@ Section length.
   
   Definition length : FSet A -> nat.
     simple refine (FSet_cons_rec _ _ _ _ _ _).
-    - apply 0.
+    - apply 0%nat.
     - intros a X n.
       apply (if a ∈_d X then n else (S n)).
     - intros X a n.
@@ -295,7 +295,7 @@ End length_sum.
 Section length_zero_one.
   Context {A : Type} `{Univalence} `{MerelyDecidablePaths A}.
 
-  Lemma Z_not_S n : S n = 0 -> Empty.
+  Lemma Z_not_S n : S n = 0%nat -> Empty.
   Proof.
     refine (@equiv_path_nat (n.+1) 0)^-1.
   Defined.
@@ -311,7 +311,7 @@ Section length_zero_one.
     apply ((equiv_path_nat)^-1 X).
   Defined.
   
-  Theorem length_zero : forall (X : FSet A) (HX : length X = 0), X = ∅.
+  Theorem length_zero : forall (X : FSet A) (HX : length X = 0%nat), X = ∅.
   Proof.
     simple refine (FSet_cons_ind (fun Z => _) _ _ _ _ _)
     ; try (intros ; apply path_ishprop) ; simpl.
@@ -326,7 +326,7 @@ Section length_zero_one.
       * contradiction (Z_not_S _ HaX).
   Defined.  
 
-  Theorem length_one : forall (X : FSet A) (HX : length X = 1), hexists (fun a => X = {|a|}).
+  Theorem length_one : forall (X : FSet A) (HX : length X = 1%nat), hexists (fun a => X = {|a|}).
   Proof.
     simple refine (FSet_cons_ind (fun Z => _) _ _ _ _ _)
     ; try (intros ; apply path_ishprop) ; simpl.

FiniteSets/kuratowski/operations.v

@@ -19,25 +19,20 @@ Section operations.
     - apply (idem o f).
   Defined.
 
-  Global Instance fset_pure : hasPure FSet.
+  Global Instance return_fset : Return FSet := fun _ a => {|a|}.
-  Proof.
-    split.
-    apply (fun _ a => {|a|}).
-  Defined.  
 
-  Global Instance fset_bind : hasBind FSet.
+  Global Instance bind_fset : Bind FSet.
   Proof.
-    split.
+    intros A B m f.
-    intros A.
+    hrecursion m.
-    hrecursion.
     - exact ∅.
-    - exact idmap.
+    - exact f.
     - apply (∪).
     - apply assoc.
     - apply comm.
     - apply nl.
     - apply nr.
-    - apply union_idem.
+    - intros; apply union_idem.
   Defined.
 
   (** Set-theoretical operations. *)
@@ -110,8 +105,8 @@ Section operations.
     - intros a f.
       apply {|f (a; tr idpath)|}.
     - intros X1 X2 HX1 HX2 f.
-      pose (fX1 := fun Z : {a : A & a ∈ X1} => f (pr1 Z;tr (inl (pr2 Z)))).
+      pose (fX1 := fun Z : {a : A | a ∈ X1} => f (pr1 Z;tr (inl (pr2 Z)))).
-      pose (fX2 := fun Z : {a : A & a ∈ X2} => f (pr1 Z;tr (inr (pr2 Z)))).
+      pose (fX2 := fun Z : {a : A | a ∈ X2} => f (pr1 Z;tr (inr (pr2 Z)))).
       specialize (HX1 fX1).
       specialize (HX2 fX2).
       apply (HX1 ∪ HX2).
@@ -201,9 +196,9 @@ Section operations_decidable.
   Defined.
 
   Global Instance fset_intersection : hasIntersection (FSet A)
-    := fun X Y => {|X & (fun a => a ∈_d Y)|}.
+    := fun X Y => {|X | fun a => a ∈_d Y |}.
 
-  Definition difference := fun X Y => {|X & (fun a => negb a ∈_d Y)|}.
+  Definition difference := fun X Y => {|X | fun a => negb a ∈_d Y|}.
 End operations_decidable.
 
 Section FSet_cons_rec.

FiniteSets/kuratowski/properties.v

@@ -6,7 +6,7 @@ Require Import lattice_interface lattice_examples monad_interface.
 Section monad_fset.
   Context `{Funext}.
 
-  Global Instance fset_functorish : Functorish FSet.
+  Global Instance functorish_fset : Functorish FSet.
   Proof.
     simple refine (Build_Functorish _ _ _).
     - intros ? ? f.
@@ -19,7 +19,7 @@ Section monad_fset.
       ; try (intros ; apply path_ishprop).
   Defined.
 
-  Global Instance fset_functor : Functor FSet.
+  Global Instance functor_fset : Functor FSet.
   Proof.
     split.
     intros.
@@ -30,24 +30,18 @@ Section monad_fset.
     ; try (intros ; apply path_ishprop).
   Defined.
 
-  Global Instance fset_monad : Monad FSet.
+  Global Instance monad_fset : Monad FSet.
   Proof.
     split.
-    - intros.
+    - intros. reflexivity.
-      apply path_forall.
+    - intros A X.
-      intro X.
+      hrecursion X;
-      hrecursion X ; try (intros; f_ap) ;
+        try (intros; compute[bind ret bind_fset return_fset]; simpl; f_ap);
-        try (intros; apply path_ishprop).
+        try (intros; apply path_ishprop); try reflexivity.
-    - intros.
+    - intros A B C X f g.
-      apply path_forall.
+      hrecursion X;
-      intro X.
+        try (intros; compute[bind ret bind_fset return_fset]; simpl; f_ap);
-      hrecursion X ; try (intros; f_ap) ;
+        try (intros; apply path_ishprop); try reflexivity.
-        try (intros; apply path_ishprop).
-    - intros.
-      apply path_forall.
-      intro X.
-      hrecursion X ; try (intros; f_ap) ;
-        try (intros; apply path_ishprop).
   Defined.
 
   Lemma fmap_isIn `{Univalence} {A B : Type} (f : A -> B) (a : A) (X : FSet A) :
@@ -64,7 +58,7 @@ Section monad_fset.
       + right. by apply HX1.
   Defined.
 
-  Instance fmap_Surjection `{Univalence} {A B : Type} (f : A -> B)
+  Instance surjection_fmap `{Univalence} {A B : Type} (f : A -> B)
     {Hsurj : IsSurjection f} : IsSurjection (fmap FSet f).
   Proof.
     apply BuildIsSurjection.
@@ -87,8 +81,8 @@ Section monad_fset.
       f_ap.
   Defined.
 
-  Lemma bind_isIn `{Univalence} {A : Type} (X : FSet (FSet A)) (a : A) :
+  Lemma mjoin_isIn `{Univalence} {A : Type} (X : FSet (FSet A)) (a : A) :
-    (exists x, x ∈ X * a ∈ x) -> a ∈ (bind _ X).
+    (exists x, prod (x ∈ X) (a ∈ x)) -> a ∈ (mjoin X).
   Proof.
     hinduction X;
       try (intros; apply path_forall; intro; apply path_ishprop).
@@ -253,13 +247,13 @@ Section properties_membership.
     : a ∈ X ∪ Y = a ∈ X ∨ a ∈ Y := idpath.
 
   Lemma comprehension_union (ϕ : A -> Bool) : forall X Y : FSet A,
-      {|X ∪ Y & ϕ|} = {|X & ϕ|} ∪ {|Y & ϕ|}.
+      {| (X ∪ Y) | ϕ|} = {|X | ϕ|} ∪ {|Y | ϕ|}.
   Proof.
     reflexivity.
   Defined.
 
   Lemma comprehension_isIn (ϕ : A -> Bool) (a : A) : forall X : FSet A,
-      a ∈ {|X & ϕ|} = if ϕ a then a ∈ X else False_hp.
+      a ∈ {|X | ϕ|} = if ϕ a then a ∈ X else False_hp.
   Proof.
     hinduction ; try (intros ; apply set_path2).
     - destruct (ϕ a) ; reflexivity.
@@ -386,8 +380,8 @@ Section properties_membership.
         ** apply p.
         ** apply path_ishprop.
     - intros X1 X2 HX1 HX2 f b.
-      pose (fX1 := fun Z : {a : A & a ∈ X1} => f (pr1 Z;tr (inl (pr2 Z)))).
+      pose (fX1 := fun Z : {a : A | a ∈ X1} => f (pr1 Z;tr (inl (pr2 Z)))).
-      pose (fX2 := fun Z : {a : A & a ∈ X2} => f (pr1 Z;tr (inr (pr2 Z)))).
+      pose (fX2 := fun Z : {a : A | a ∈ X2} => f (pr1 Z;tr (inr (pr2 Z)))).
       specialize (HX1 fX1 b).
       specialize (HX2 fX2 b).
       apply path_iff_hprop.
@@ -461,8 +455,8 @@ Section properties_membership.
         apply tr.
         unfold IsEmbedding, hfiber in *.
         specialize (f_inj (f a)).
-        pose ((a;idpath (f a)) : {x : A & f x = f a}) as p1.
+        pose ((a;idpath (f a)) : {x : A | f x = f a}) as p1.
-        pose ((b;Hfa^) : {x : A & f x = f a}) as p2.
+        pose ((b;Hfa^) : {x : A | f x = f a}) as p2.
         assert (p1 = p2) as Hp1p2.
         { apply f_inj. }
         apply (ap (fun x => x.1) Hp1p2).
@@ -485,31 +479,18 @@ Ltac toHProp :=
 Section fset_join_semilattice.
   Context {A : Type}.
 
-  Instance: bottom(FSet A).
+  Global Instance bottom_fset : Bottom (FSet A) := ∅.
-  Proof.
-    unfold bottom.
-    apply ∅.
-  Defined.
 
-  Instance: maximum(FSet A).
+  Global Instance join_fset : Join (FSet A) := fun x y => x ∪ y.
-  Proof.
-    intros x y.
-    apply (x ∪ y).
-  Defined.
 
-  Global Instance joinsemilattice_fset : JoinSemiLattice (FSet A).
+  Global Instance boundedjoinsemilattice_fset : BoundedJoinSemiLattice (FSet A).
   Proof.
-    split.
+    repeat split; try apply _; cbv.
-    - intros ? ?.
+    - apply assoc.
-      apply comm.
+    - apply nl.
-    - intros ? ? ?.
+    - apply nr.
-      apply (assoc _ _ _)^.
+    - apply comm.
-    - intros ?.
+    - apply union_idem.
-      apply union_idem.
-    - intros x.
-      apply nl.
-    - intros ?.
-      apply nr.
   Defined.  
 End fset_join_semilattice.
 
@@ -569,10 +550,10 @@ Section properties_membership_decidable.
   Defined.
 
   Lemma comprehension_isIn_d (Y : FSet A) (ϕ : A -> Bool) (a : A) :
-    a ∈_d {|Y & ϕ|} = andb (a ∈_d Y) (ϕ a).
+    a ∈_d {|Y | ϕ|} = andb (a ∈_d Y) (ϕ a).
   Proof.
     unfold member_dec, fset_member_bool, dec ; simpl.
-    destruct (isIn_decidable a {|Y & ϕ|}) as [t | t]
+    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
     ; try (contradiction (n t)) ; contradiction (t n).
@@ -692,27 +673,14 @@ Section set_lattice.
   Context `{MerelyDecidablePaths A}.
   Context `{Univalence}.
 
-  Instance fset_max : maximum (FSet A).
+  Global Instance meet_fset : Meet (FSet A) := intersection.
-  Proof.
-    intros x y.
-    apply (x ∪ y).
-  Defined.
 
-  Instance fset_min : minimum (FSet A).
+  Global Instance lattice_fset : Lattice (FSet A).
   Proof.
-    intros x y.
+    repeat split; try apply _;
-    apply (x ∩ y).
+      compute[sg_op meet_is_sg_op meet_fset];
-  Defined.
+      toBool.
-
+    apply associativity.
-  Instance fset_bot : bottom (FSet A).
-  Proof.
-    unfold bottom.
-    apply ∅.
-  Defined.
-
-  Instance lattice_fset : Lattice (FSet A).
-  Proof.
-    split ; toBool.
   Defined.
 End set_lattice.
 
@@ -720,7 +688,7 @@ End set_lattice.
 Section dec_eq.
   Context {A : Type} `{DecidablePaths A} `{Univalence}.
 
-  Instance fsets_dec_eq : DecidablePaths (FSet A).
+  Global Instance dec_eq_fset : DecidablePaths (FSet A).
   Proof.
     intros X Y.
     apply (decidable_equiv' ((Y ⊆ X) * (X ⊆ Y)) (eq_subset X Y)^-1).
@@ -732,29 +700,32 @@ End dec_eq.
 Section comprehension_properties.
   Context {A : Type} `{Univalence}.
 
-  Lemma comprehension_false : forall (X : FSet A), {|X & fun _ => false|} = ∅.
+  Lemma comprehension_false : forall (X : FSet A), {|X | fun _ => false|} = ∅.
   Proof.
     toHProp.
   Defined.
 
   Lemma comprehension_subset : forall ϕ (X : FSet A),
-      {|X & ϕ|} ∪ X = X.
+      {|X | ϕ|} ∪ X = X.
   Proof.
     toHProp.
     destruct (ϕ a) ; eauto with lattice_hints typeclass_instances.
+    - apply binary_idempotent.
+    - apply left_identity.
   Defined.
 
   Lemma comprehension_or : forall ϕ ψ (X : FSet A),
-      {|X & (fun a => orb (ϕ a) (ψ a))|}
+      {|X | (fun a => orb (ϕ a) (ψ a))|}
-      = {|X & ϕ|} ∪ {|X & ψ|}.
+      = {|X | ϕ|} ∪ {|X | ψ|}.
   Proof.
     toHProp.
     symmetry ; destruct (ϕ a) ; destruct (ψ a)
-    ; eauto with lattice_hints typeclass_instances.
+    (* ; eauto with lattice_hints typeclass_instances; *)
+    ; simpl; (apply binary_idempotent || apply left_identity || apply right_identity).
   Defined.
 
   Lemma comprehension_all : forall (X : FSet A),
-      {|X & fun _ => true|} = X.
+      {|X | fun _ => true|} = X.
   Proof.
     toHProp.
   Defined.

FiniteSets/misc/dec_fset.v

@@ -13,12 +13,16 @@ Section quantifiers.
     - intros X Y.
       apply (BuildhProp (X * Y)).
     - eauto with lattice_hints typeclass_instances.
+    (* TODO eauto hints *)
+      apply associativity.
     - eauto with lattice_hints typeclass_instances.
+      apply commutativity.
     - intros.
       apply path_trunctype ; apply prod_unit_l.
     - intros.
       apply path_trunctype ; apply prod_unit_r.
     - eauto with lattice_hints typeclass_instances.
+      intros. apply binary_idempotent.
   Defined.
 
   Lemma all_intro X : forall (HX : forall x, x ∈ X -> P x), all X.
@@ -62,10 +66,16 @@ Section quantifiers.
     - apply P.
     - apply lor.
     - eauto with lattice_hints typeclass_instances.
+    (* TODO eauto with .. *)
+      apply associativity.
     - eauto with lattice_hints typeclass_instances.
+      apply commutativity.
     - eauto with lattice_hints typeclass_instances.
+      apply left_identity.
     - eauto with lattice_hints typeclass_instances. 
+      apply right_identity.
     - eauto with lattice_hints typeclass_instances.
+      intros. apply binary_idempotent.
   Defined.
 
   Lemma exist_intro X a : a ∈ X -> P a -> exist X.
@@ -84,7 +94,7 @@ Section quantifiers.
       * apply (inr (HX2 t Pa)).
   Defined.
 
-  Lemma exist_elim X : exist X -> hexists (fun a => a ∈ X * P a).
+  Lemma exist_elim X : exist X -> hexists (fun a => a ∈ X * P a)%type.
   Proof.
     hinduction X ; try (intros ; apply path_ishprop).
     - contradiction.
@@ -132,7 +142,7 @@ Section simple_example.
   Context `{Univalence}.
 
   Definition P : nat -> hProp := fun n => BuildhProp(n = n).
-  Definition X : FSet nat := {|0|} ∪ {|1|}.
+  Definition X : FSet nat := {|0%nat|} ∪ {|1%nat|}.
 
   Definition simple_example : all P X.
   Proof.

FiniteSets/misc/dec_kuratowski.v

@@ -66,12 +66,12 @@ Section length.
   Context {A : Type} `{Univalence}.
 
   Variable (length : FSet A -> nat)
-           (length_singleton : forall (a : A), length {|a|} = 1)
+           (length_singleton : forall (a : A), length {|a|} = 1%nat)
-           (length_one : forall (X : FSet A), length X = 1 -> hexists (fun a => X = {|a|})).
+           (length_one : forall (X : FSet A), length X = 1%nat -> hexists (fun a => X = {|a|})).
 
   Theorem dec_length (a b : A) : Decidable(merely(a = b)).
   Proof.
-    destruct (dec (length ({|a|} ∪ {|b|}) = 1)).
+    destruct (dec (length ({|a|} ∪ {|b|}) = 1%nat)).
     - refine (inl _).
       pose (length_one _ p) as Hp.
       simple refine (Trunc_rec _ Hp).

FiniteSets/misc/ordered.v

@@ -1,84 +1,54 @@
 (** If [A] has a total order, then we can pick the minimum of finite sets. *)
 Require Import HoTT HitTactics.
+Require Import HoTT.Classes.interfaces.orders.
 Require Import kuratowski.kuratowski_sets kuratowski.operations kuratowski.properties.
 
-Definition relation A := A -> A -> Type.
-
-Section TotalOrder.
-  Class IsTop (A : Type) (R : relation A) (a : A) :=
-    top_max : forall x, R x a.
-  
-  Class LessThan (A : Type) :=
-    leq : relation A.
-  
-  Class Antisymmetric {A} (R : relation A) :=
-    antisymmetry : forall x y, R x y -> R y x -> x = y.
-  
-  Class Total {A} (R : relation A) :=
-    total : forall x y,  x = y \/ R x y \/ R y x. 
-  
-  Class TotalOrder (A : Type) {R : LessThan A} :=
-    { TotalOrder_Reflexive :> Reflexive R | 2 ;
-      TotalOrder_Antisymmetric :> Antisymmetric R | 2; 
-      TotalOrder_Transitive :> Transitive R | 2;
-      TotalOrder_Total :> Total R | 2; }.
-End TotalOrder.
-
 Section minimum.
   Context {A : Type}.
-  Context `{TotalOrder A}.
+  Context (Ale : Le A).
+  Context `{@TotalOrder A Ale}.
+
+  Class IsTop `{Top A} := is_top : forall (x : A), x ≤ ⊤.
 
   Definition min (x y : A) : A.
   Proof.
-    destruct (@total _ R _ x y).
+    destruct (total _ x y).
     - apply x.
-    - destruct s as [s | s].
+    - apply y.
-      * apply x.
-      * apply y.
   Defined.
 
-  Lemma min_spec1 x y : R (min x y) x.
+  Lemma min_spec1 x y : (min x y) ≤ x.
   Proof.
     unfold min.
-    destruct (total x y) ; simpl.
+    destruct (total _ x y) ; simpl.
     - reflexivity.
-    - destruct s as [ | t].
+    - assumption.
-      * reflexivity.
-      * apply t.
   Defined.
 
-  Lemma min_spec2 x y z : R z x -> R z y -> R z (min x y).
+  Lemma min_spec2 x y z : z ≤ x -> z ≤ y -> z ≤ (min x y).
   Proof.
     intros.
     unfold min.
-    destruct (total x y) as [ | s].
+    destruct (total _ x y); simpl; assumption.
-    * assumption.
-    * try (destruct s) ; assumption.
   Defined.
   
   Lemma min_comm x y : min x y = min y x.
   Proof.
     unfold min.
-    destruct (total x y) ; destruct (total y x) ; simpl.
+    destruct (total _ x y) ; destruct (total _ y x) ; simpl; try reflexivity.
-    - assumption.
+    + apply antisymmetry with (≤); [apply _|assumption..].
-    - destruct s as [s | s] ; auto.
+    + apply antisymmetry with (≤); [apply _|assumption..].
-    - destruct s as [s | s] ; symmetry ; auto.
-    - destruct s as [s | s] ; destruct s0 as [s0 | s0] ; try reflexivity.
-      * apply (@antisymmetry _ R _ _) ; assumption.
-      * apply (@antisymmetry _ R _ _) ; assumption.
   Defined.
 
   Lemma min_idem x : min x x = x.
   Proof.
     unfold min.
-    destruct (total x x) ; simpl.
+    destruct (total _ x x) ; simpl; reflexivity.
-    - reflexivity.
-    - destruct s ; reflexivity.
   Defined.
 
   Lemma min_assoc x y z : min (min x y) z = min x (min y z).
   Proof.
-    apply (@antisymmetry _ R _ _).
+    apply antisymmetry with (≤). apply _.
     - apply min_spec2.
       * etransitivity ; apply min_spec1.
       * apply min_spec2.
@@ -96,43 +66,36 @@ Section minimum.
         ; rewrite min_comm ; apply min_spec1.
   Defined.
   
-  Variable (top : A).
+  Lemma min_nr `{IsTop} x : min x ⊤ = x.
-  Context `{IsTop A R top}.
-
-  Lemma min_nr x : min x top = x.
   Proof.
     intros.
     unfold min.
-    destruct (total x top).
+    destruct (total _ x ⊤).
     - reflexivity.
-    - destruct s.
+    - apply antisymmetry with (≤). apply _.
-      * reflexivity.
+      + assumption.
-      * apply (@antisymmetry _ R _ _).
+      + apply is_top.
-        ** assumption.
-        ** refine (top_max _). apply _.
   Defined.
 
-  Lemma min_nl x : min top x = x.
+  Lemma min_nl `{IsTop} x : min ⊤ x = x.
   Proof.
     rewrite min_comm.
     apply min_nr.
   Defined.
 
-  Lemma min_top_l x y : min x y = top -> x = top.
+  Lemma min_top_l `{IsTop} x y : min x y = ⊤ -> x = ⊤.
   Proof.
     unfold min.
-    destruct (total x y).
+    destruct (total _ x y) as [?|s].
     - apply idmap.
-    - destruct s as [s | s].
+    - intros X.
-      * apply idmap.
-      * intros X.
       rewrite X in s.
-        apply (@antisymmetry _ R _ _).
+      apply antisymmetry with (≤). apply _.
-        ** apply top_max.
+      * apply is_top.
-        ** assumption.
+      * assumption.
   Defined.
   
-  Lemma min_top_r x y : min x y = top -> y = top.
+  Lemma min_top_r `{IsTop} x y : min x y = ⊤ -> y = ⊤.
   Proof.
     rewrite min_comm.
     apply min_top_l.
@@ -144,72 +107,60 @@ Section add_top.
   Variable (A : Type).
   Context `{TotalOrder A}.
 
-  Definition Top := A + Unit.
+  Definition Topped := (A + Unit)%type.
-  Definition top : Top := inr tt.
+  Global Instance top_topped : Top Topped := inr tt.
 
-  Global Instance RTop : LessThan Top.
+  Global Instance RTop : Le Topped.
   Proof.
-    unfold relation.
+    intros [a1 | ?] [a2 | ?].
-    induction 1 as [a1 | ] ; induction 1 as [a2 | ].
+    - apply (a1 ≤ a2).
-    - apply (R a1 a2).
     - apply Unit_hp.
     - apply False_hp.
     - apply Unit_hp.
   Defined.
 
-  Global Instance rtop_hprop :
+  Instance rtop_hprop :
-    is_mere_relation A R -> is_mere_relation Top RTop.
+    is_mere_relation A R -> is_mere_relation Topped RTop.
   Proof.
-    intros P a b.
+    intros ? P a b.
-    destruct a ; destruct b ; apply _.
+    destruct a ; destruct b ; simpl; apply _.
   Defined.
   
-  Global Instance RTopOrder : TotalOrder Top.
+  Global Instance RTopOrder : TotalOrder RTop.
   Proof.
-    split.
+    repeat split; try apply _.
-    - intros x ; induction x ; unfold RTop ; simpl.
+    - intros [?|?] ; cbv.
       * reflexivity.
       * apply tt.
-    - intros x y ; induction x as [a1 | ] ; induction y as [a2 | ] ; unfold RTop ; simpl
+    - intros [a1 | []] [a2 | []] [a3 | []]; cbv
-      ; try contradiction.
+      ; try contradiction; try (by intros; apply tt).
-      * intros ; f_ap.
+      apply transitivity.
-        apply (@antisymmetry _ R _ _) ; assumption.
+    - intros [a1 |[]] [a2 |[]]; cbv; try contradiction.
-      * intros ; induction b ; induction b0.
+      + intros. f_ap. apply antisymmetry with Ale; [apply _|assumption..].
-        reflexivity.
+      + intros; reflexivity.
-    - intros x y z ; induction x as [a1 | b1] ; induction y as [a2 | b2]
+    - intros [a1|[]] [a2|[]]; cbv.
-      ; induction z as [a3 | b3] ; unfold RTop ; simpl
+      * apply total. apply _.
-      ; try contradiction ; intros ; try (apply tt).
+      * apply (inl tt).
-      transitivity a2 ; assumption.
+      * apply (inr tt).
-    - intros x y.
+      * apply (inr tt).
-      unfold RTop ; simpl.
-      induction x as [a1 | b1] ; induction y as [a2 | b2] ; try (apply (inl idpath)).
-      * destruct (TotalOrder_Total a1 a2).
-        ** left ; f_ap ; assumption.
-        ** right ; assumption.
-      * apply (inr(inl tt)).
-      * apply (inr(inr tt)).
-      * left ; induction b1 ; induction b2 ; reflexivity.
   Defined.
 
-  Global Instance top_a_top : IsTop Top RTop top.
+  Global Instance is_top_topped : IsTop RTop.
-  Proof.
+  Proof. intros [?|?]; cbv; apply tt. Defined.
-    intro x ; destruct x ; apply tt.
-  Defined.
 End add_top.
 
 (** If [A] has a total order, then a nonempty finite set has a minimum element. *)
 Section min_set.
   Variable (A : Type).
   Context `{TotalOrder A}.
-  Context `{is_mere_relation A R}.
   Context `{Univalence} `{IsHSet A}.
 
-  Definition min_set : FSet A -> Top A.
+  Definition min_set : FSet A -> Topped A.
   Proof.
     hrecursion.
-    - apply (top A).
+    - apply ⊤.
     - apply inl.
-    - apply min.
+    - apply min with (RTop A). apply _.
     - intros ; symmetry ; apply min_assoc.
     - apply min_comm.
     - apply min_nl. apply _.
@@ -217,7 +168,7 @@ Section min_set.
     - intros ; apply min_idem.
   Defined.
 
-  Definition empty_min : forall (X : FSet A), min_set X = top A -> X = ∅.
+  Definition empty_min : forall (X : FSet A), min_set X = ⊤ -> X = ∅.
   Proof.
     simple refine (FSet_ind _ _ _ _ _ _ _ _ _ _ _)
     ; try (intros ; apply path_forall ; intro q ; apply set_path2)
@@ -230,21 +181,19 @@ Section min_set.
       refine (not_is_inl_and_inr' (inl a) _ _).
       * apply tt.
       * rewrite X ; apply tt.
-    - intros.
+    - intros x y ? ? ?.
-      assert (min_set x = top A).
+      assert (min_set x = ⊤).
-      {
+      { simple refine (min_top_l _ _ (min_set y) _) ; assumption. }
-        simple refine (min_top_l _ _ (min_set y) _) ; assumption.
-      }
       rewrite (X X2).
       rewrite nl.
-      assert (min_set y = top A).
+      assert (min_set y = ⊤).
       { simple refine (min_top_r _ (min_set x) _ _) ; assumption. }
       rewrite (X0 X3).
       reflexivity.
   Defined.
   
   Definition min_set_spec (a : A) : forall (X : FSet A),
-      a ∈ X -> RTop A (min_set X) (inl a).
+      a ∈ X -> min_set X ≤ inl a.
   Proof.
     simple refine (FSet_ind _ _ _ _ _ _ _ _ _ _ _)
     ; try (intros ; apply path_ishprop)
@@ -259,13 +208,13 @@ Section min_set.
       unfold member in X, X0.
       destruct X1.
       * specialize (X t).
-        assert (RTop A (min (min_set x) (min_set y)) (min_set x)) as X1.
+        assert (RTop A (min _ (min_set x) (min_set y)) (min_set x)) as X1.
         { apply min_spec1. }
         etransitivity.
         { apply X1. }
         assumption.
       * specialize (X0 t).
-        assert (RTop A (min (min_set x) (min_set y)) (min_set y)) as X1.
+        assert (RTop A (min _ (min_set x) (min_set y)) (min_set y)) as X1.
         { rewrite min_comm ; apply min_spec1. }
         etransitivity.
         { apply X1. }

FiniteSets/subobjects/b_finite.v

@@ -8,9 +8,9 @@ Section finite_hott.
   Context `{Univalence}.
 
   (* A subobject is B-finite if its extension is B-finite as a type *)
-  Definition Bfin (X : Sub A) : hProp := BuildhProp (Finite {a : A & a ∈ X}).
+  Definition Bfin (X : Sub A) : hProp := BuildhProp (Finite {a : A | a ∈ X}).
 
-  Global Instance singleton_contr a `{IsHSet A} : Contr {b : A & b ∈ {|a|}}.
+  Global Instance singleton_contr a `{IsHSet A} : Contr {b : A | b ∈ {|a|}}.
   Proof.
     exists (a; tr idpath).
     intros [b p].
@@ -20,7 +20,7 @@ Section finite_hott.
     apply p^.
   Defined.
   
-  Definition singleton_fin_equiv' a : Fin 1 -> {b : A & b ∈ {|a|}}.
+  Definition singleton_fin_equiv' a : Fin 1 -> {b : A | b ∈ {|a|}}.
   Proof.  
     intros _. apply (a; tr idpath).
   Defined.
@@ -66,7 +66,8 @@ Section finite_hott.
     : DecidablePaths A.
   Proof.
     intros a b.
-    destruct (U {|a|} {|b|} (singleton a) (singleton b)) as [n pn].
+    specialize (U {|a|} {|b|} (singleton a) (singleton b)).
+    destruct U as [n pn].
     strip_truncations.
     unfold Sect in *.
     destruct pn as [f [g fg gf _]], n as [ | [ | n]].
@@ -75,9 +76,9 @@ Section finite_hott.
       apply (tr(inl(tr idpath))).
     - refine (inl _).
       pose (s1 := (a;tr(inl(tr idpath)))
-                  : {c : A & Trunc (-1) (Trunc (-1) (c = a) + Trunc (-1) (c = b))}).
+                  : {c : A | Trunc (-1) (Trunc (-1) (c = a) + Trunc (-1) (c = b))}).
       pose (s2 := (b;tr(inr(tr idpath)))
-                  : {c : A & Trunc (-1) (Trunc (-1) (c = a) + Trunc (-1) (c = b))}).
+                  : {c : A | Trunc (-1) (Trunc (-1) (c = a) + Trunc (-1) (c = b))}).
       refine (ap (fun x => x.1) (gf s1)^ @ _ @ (ap (fun x => x.1) (gf s2))).
       assert (fs_eq : f s1 = f s2).
       { by apply path_ishprop. }
@@ -116,9 +117,9 @@ Section singleton_set.
   Variable (A : Type).
   Context `{Univalence}.
 
-  Variable (HA : forall a, {b : A & b ∈ {|a|}} <~> Fin 1).
+  Variable (HA : forall a, {b : A | b ∈ {|a|}} <~> Fin 1).
 
-  Definition el x : {b : A & b ∈ {|x|}} := (x;tr idpath).
+  Definition el x : {b : A | b ∈ {|x|}} := (x;tr idpath).
   
   Theorem single_bfin_set : forall (x : A) (p : x = x), p = idpath.
   Proof.
@@ -132,7 +133,7 @@ Section singleton_set.
       rewrite <- ap_compose.
       enough(forall r,
                 (eissect HA X)^
-                @ ap (fun x0 : {b : A & b ∈ {|x|}} => HA^-1 (HA x0)) r
+                @ ap (fun x0 : {b : A | b ∈ {|x|}} => HA^-1 (HA x0)) r
                   @ eissect HA X = r
             ) as H2.
       {
@@ -166,7 +167,7 @@ End singleton_set.
 Section empty.
   Variable (A : Type).
   Variable (X : A -> hProp)
-           (Xequiv : {a : A & a ∈ X} <~> Fin 0).
+           (Xequiv : {a : A | a ∈ X} <~> Fin 0).
   Context `{Univalence}.
 
   Lemma X_empty : X = ∅.
@@ -185,10 +186,10 @@ Section split.
   Variable (A : Type).
   Variable (P : A -> hProp)
            (n : nat)
-           (f : {a : A & P a } <~> Fin n + Unit).
+           (f : {a : A | P a } <~> Fin n + Unit).
 
   Definition split : exists P' : Sub A, exists b : A,
-        ({a : A & P' a} <~> Fin n) * (forall x, P x = (P' x ∨ merely (x = b))).
+    prod ({a : A | P' a} <~> Fin n) (forall x, P x = (P' x ∨ merely (x = b))).
   Proof.
     pose (fun x : A => sig (fun y : Fin n => x = (f^-1 (inl y)).1)) as P'.
     assert (forall x, IsHProp (P' x)).
@@ -267,7 +268,7 @@ Arguments Bfin {_} _.
 Arguments split {_} {_} _ _ _.
 
 Section Bfin_no_singletons.
-  Definition S1toSig (x : S1) : {x : S1 & merely(x = base)}.
+  Definition S1toSig (x : S1) : {x : S1 | merely(x = base)}.
   Proof.
     exists x.
     simple refine (S1_ind (fun z => merely(z = base)) (tr idpath) _ x).
@@ -382,7 +383,7 @@ Section kfin_bfin.
 
   Lemma notIn_ext_union_singleton (b : A) (Y : Sub A) :
     ~ (b ∈ Y) ->
-    {a : A & a ∈ ({|b|} ∪ Y)} <~> {a : A & a ∈ Y} + Unit.
+    {a : A | a ∈ ({|b|} ∪ Y)} <~> {a : A | a ∈ Y} + Unit.
   Proof.
     intros HYb.
     unshelve eapply BuildEquiv.
@@ -422,7 +423,7 @@ Section kfin_bfin.
     strip_truncations.
     revert fX. revert X.
     induction n; intros X fX.
-    - rewrite (X_empty _ X fX), (neutralL_max (Sub A)).
+    - rewrite (X_empty _ X fX). rewrite left_identity.
       apply HY.
     - destruct (split X n fX) as
         (X' & b & HX' & HX).
@@ -431,10 +432,10 @@ Section kfin_bfin.
       + cut (X ∪ Y = X' ∪ Y).
         { intros HXY. rewrite HXY. assumption. }
         apply path_forall. intro a.
-        unfold union, sub_union, max_fun.
+        unfold union, sub_union, join, join_fun.
         rewrite HX.
         rewrite (commutativity (X' a)).
-        rewrite (associativity _ (X' a)).
+        rewrite <- (associativity _ (X' a)).
         apply path_iff_hprop.
         * intros Ha.
           strip_truncations.
@@ -448,15 +449,14 @@ Section kfin_bfin.
         strip_truncations.
         exists (n'.+1).
         apply tr.
-        transitivity ({a : A & a ∈ (fun a => merely (a = b)) ∪ (X' ∪ Y)}).
+        transitivity ({a : A | a ∈ (fun a => merely (a = b)) ∪ (X' ∪ Y)}).
         * apply equiv_functor_sigma_id.
           intro a.
-          rewrite <- (associative_max (Sub A)).
+          rewrite (associativity (fun a => merely (a = b)) X' Y).
           assert (X = X' ∪ (fun a => merely (a = b))) as HX_.
           ** apply path_forall. intros ?.
-             unfold union, sub_union, max_fun.
              apply HX.
-          ** rewrite HX_, <- (commutative_max (Sub A) X').
+          ** rewrite HX_, <- (commutativity X').
              reflexivity.
         * etransitivity.
           { apply (notIn_ext_union_singleton b _ HX'Yb). }
@@ -466,7 +466,7 @@ Section kfin_bfin.
   Definition FSet_to_Bfin : forall (X : FSet A), Bfin (map X).
   Proof.
     hinduction; try (intros; apply path_ishprop).
-    - exists 0.
+    - exists 0%nat.
       apply tr.
       simple refine (BuildEquiv _ _ _ _).
       destruct 1 as [? []].

FiniteSets/subobjects/enumerated.v

@@ -300,7 +300,7 @@ Section subobjects.
     simpl. apply path_ishprop.
   Defined.
 
-  Fixpoint weaken_list_r (P Q : Sub A) (ls : list (sigT Q)) : list (sigT (max_L P Q)).
+  Fixpoint weaken_list_r (P Q : Sub A) (ls : list (sigT Q)) : list (sigT (P ⊔ Q)).
   Proof.
     destruct ls as [|[x Hx] ls].
     - exact nil.
@@ -323,7 +323,7 @@ Section subobjects.
   Defined.
 
   Fixpoint concatD {P Q : Sub A}
-    (ls : list (sigT P)) (ls' : list (sigT Q)) : list (sigT (max_L P Q)).
+    (ls : list (sigT P)) (ls' : list (sigT Q)) : list (sigT (P ⊔ Q)).
   Proof.
     destruct ls as [|[y Hy] ls].
     - apply weaken_list_r. exact ls'.
@@ -356,7 +356,7 @@ Section subobjects.
   Defined.
 
   Lemma enumeratedS_union (P Q : Sub A) :
-    enumeratedS P -> enumeratedS Q -> enumeratedS (max_L P Q).
+    enumeratedS P -> enumeratedS Q -> enumeratedS (P ⊔ Q).
   Proof.
     intros HP HQ.
     strip_truncations; apply tr.
@@ -388,7 +388,7 @@ Section subobjects.
   Transparent enumeratedS.
 
   Instance hprop_sub_fset (P : Sub A) :
-    IsHProp {X : FSet A & k_finite.map X = P}.
+    IsHProp {X : FSet A | k_finite.map X = P}.
   Proof.
     apply hprop_allpath. intros [X HX] [Y HY].
     assert (X = Y) as HXY.
@@ -433,7 +433,7 @@ Section subobjects.
 
   Definition enumeratedS_to_FSet :
     forall (P : Sub A), enumeratedS P ->
-    {X : FSet A & k_finite.map X = P}.
+    {X : FSet A | k_finite.map X = P}.
   Proof.
     intros P HP.
     strip_truncations.

FiniteSets/subobjects/k_finite.v

@@ -33,7 +33,7 @@ Section k_finite.
 
   Definition Kf : hProp := Kf_sub (fun x => True).
 
-  Instance: IsHProp {X : FSet A & forall a : A, map X a}.
+  Instance: IsHProp {X : FSet A | forall a : A, map X a}.
   Proof.
     apply hprop_allpath.
     intros [X PX] [Y PY].
@@ -77,19 +77,16 @@ Section structure_k_finite.
   Context (A : Type).
   Context `{Univalence}.
 
-  Lemma map_union : forall X Y : FSet A, map (X ∪ Y) = max_fun (map X) (map Y).
+  Lemma map_union : forall X Y : FSet A, map (X ∪ Y) = (map X) ⊔ (map Y).
   Proof.
     intros.     
-    unfold map, max_fun.
     reflexivity. 
   Defined.
 
   Lemma k_finite_union : closedUnion (Kf_sub A).
   Proof.
     unfold closedUnion, Kf_sub, Kf_sub_intern.
-    intros.
+    intros X Y [SX XP] [SY YP].
-    destruct X0 as [SX XP].
-    destruct X1 as [SY YP].
     exists (SX ∪ SY).
     rewrite map_union.
     rewrite XP, YP.
@@ -179,9 +176,9 @@ Section k_properties.
                      | inl a => ({|a|} : FSet A)
                      | inr b => ∅
                      end).
-    exists (bind _ (fset_fmap f X)).
+    exists (mjoin (fset_fmap f X)).
     intro a.
-    apply bind_isIn.
+    apply mjoin_isIn.
     specialize (HX (inl a)).
     exists {|a|}. split; [ | apply tr; reflexivity ].
     apply (fmap_isIn f (inl a) X).
@@ -259,33 +256,35 @@ Section alternative_definition.
   Local Ltac help_solve :=
     repeat (let x := fresh in intro x ; destruct x) ; intros
     ; try (simple refine (path_sigma _ _ _ _ _)) ; try (apply path_ishprop) ; simpl
-    ; unfold union, sub_union, max_fun
+    ; unfold union, sub_union, join, join_fun
     ; apply path_forall
     ; intro z
     ; eauto with lattice_hints typeclass_instances.
   
-  Definition fset_to_k : FSet A -> {P : A -> hProp & kf_sub P}.
+  Definition fset_to_k : FSet A -> {P : A -> hProp | kf_sub P}.
   Proof.
-    hinduction.
+    assert (IsHSet {P : A -> hProp | kf_sub P}) as Hs.
+    { apply trunc_sigma. }
+    simple refine (FSet_rec A {P : A -> hProp | kf_sub P} Hs _ _ _ _ _ _ _ _).
     - exists ∅.
-      auto.
+      simpl. auto.
     - intros a.
       exists {|a|}.
-      auto.
+      simpl. auto.
     - intros [P1 HP1] [P2 HP2].
       exists (P1 ∪ P2).
       intros ? ? ? HP.
       apply HP.
       * apply HP1 ; assumption.
       * apply HP2 ; assumption.
-    - help_solve.
+    - help_solve. (* TODO: eauto *) apply associativity.
-    - help_solve.
+    - help_solve. apply commutativity.
-    - help_solve.
+    - help_solve. apply left_identity.
-    - help_solve.
+    - help_solve. apply right_identity.
-    - help_solve.
+    - help_solve. apply binary_idempotent.
   Defined.
 
-  Definition k_to_fset : {P : A -> hProp & kf_sub P} -> FSet A.
+  Definition k_to_fset : {P : A -> hProp | kf_sub P} -> FSet A.
   Proof.
     intros [P HP].
     destruct (HP (Kf_sub _) (k_finite_empty _) (k_finite_singleton _) (k_finite_union _)).
@@ -299,7 +298,7 @@ Section alternative_definition.
     refine ((ap (fun z => _ ∪ z) HX2^)^ @ (ap (fun z => z ∪ X2) HX1^)^).
   Defined.
     
-  Lemma k_to_fset_to_k (X : {P : A -> hProp & kf_sub P}) : fset_to_k(k_to_fset X) = X.
+  Lemma k_to_fset_to_k (X : {P : A -> hProp | kf_sub P}) : fset_to_k(k_to_fset X) = X.
   Proof.
     simple refine (path_sigma _ _ _ _ _) ; try (apply path_ishprop).
     apply path_forall.

FiniteSets/subobjects/sub.v

@@ -6,9 +6,9 @@ Section subobjects.
 
   Definition Sub := A -> hProp.
 
-  Global Instance sub_empty : hasEmpty Sub := fun _ => False_hp.
+  Global Instance sub_empty : hasEmpty Sub := fun _ => ⊥.
-  Global Instance sub_union : hasUnion Sub := max_fun.
+  Global Instance sub_union : hasUnion Sub := join.
-  Global Instance sub_intersection : hasIntersection Sub := min_fun.
+  Global Instance sub_intersection : hasIntersection Sub := meet.
   Global Instance sub_singleton : hasSingleton Sub A
     := fun a b => BuildhProp (Trunc (-1) (b = a)).
   Global Instance sub_membership : hasMembership Sub A := fun a X => X a.