Update the code to match the latest HoTT

HoTT commit 3526c344c47d32f5d4d268658031777239ec952b

FiniteSets/implementations/lists.v

@@ -51,7 +51,7 @@ Section ListToSet.
     induction l ; unfold member in * ; simpl in *.
     - reflexivity.
     - rewrite IHl.
-      unfold hor, merely, lor.
+      unfold hor, merely.
       apply path_iff_hprop ; intros z ; strip_truncations ; destruct z as [z1 | z2].
       * apply (tr (inl (tr z1))).
       * apply (tr (inr z2)).

FiniteSets/interfaces/lattice_examples.v

@@ -1,290 +1,3 @@
 (** Some examples of lattices. *)
 Require Export HoTT lattice_interface.
-
-(** [Bool] is a lattice. *)
-Section BoolLattice.
-  Ltac solve_bool :=
-    let x := fresh in
-    repeat (intro x ; destruct x)
-    ; compute
-    ; auto
-    ; try contradiction.
-
-  Instance maximum_bool : Join Bool := orb.
-  Instance minimum_bool : Meet Bool := andb.
-  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. split ; (apply _ || solve_bool). Defined.
-
-  Definition and_true : forall b, andb b true = b.
-  Proof.
-    solve_bool.
-  Defined.
-
-  Definition and_false : forall b, andb b false = false.
-  Proof.
-    solve_bool.
-  Defined.
-
-  Definition dist₁ : forall b₁ b₂ b₃,
-      andb b₁ (orb b₂ b₃) = orb (andb b₁ b₂) (andb b₁ b₃).
-  Proof.
-    solve_bool.
-  Defined.
-
-  Definition dist₂ : forall b₁ b₂ b₃,
-      orb b₁ (andb b₂ b₃) = andb (orb b₁ b₂) (orb b₁ b₃).
-  Proof.
-    solve_bool.
-  Defined.
-
-  Definition max_min : forall b₁ b₂,
-      orb (andb b₁ b₂) b₁ = b₁.
-  Proof.
-    solve_bool.
-  Defined.
-End BoolLattice.
-
-Create HintDb bool_lattice_hints.
-Hint Resolve associativity : bool_lattice_hints.
-(* Hint Resolve (associativity _ _ _)^ : bool_lattice_hints. *)
-Hint Resolve commutativity : bool_lattice_hints.
-Hint Resolve absorption : bool_lattice_hints.
-Hint Resolve idempotency : bool_lattice_hints.
-Hint Resolve left_identity : bool_lattice_hints.
-Hint Resolve right_identity : bool_lattice_hints.
-
-Hint Resolve
-     associativity
-     and_true and_false
-     dist₁ dist₂ max_min
-  : bool_lattice_hints.
-
-(** If [B] is a lattice, then [A -> B] is a lattice. *)
-Section fun_lattice.
-  Context {A B : Type}.
-  Context `{BJoin : Join B}.
-  Context `{BMeet : Meet B}.
-  Context `{@Lattice B BJoin BMeet}.
-  Context `{Funext}.
-  Context `{BBottom : Bottom B}.
-
-  Global Instance bot_fun : Bottom (A -> B)
-    := fun _ => ⊥.
-
-  Global Instance join_fun : Join (A -> B) :=
-    fun (f g : A -> B) (a : A) => (f a) ⊔ (g a).
-
-  Global Instance meet_fun : Meet (A -> B) :=
-    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.
-    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.
-
-(** If [A] is a lattice and [P] is closed under the lattice operations, then [Σ(x:A), P x] is a lattice. *)
-Section sub_lattice.
-  Context {A : Type} {P : A -> hProp}.
-  Context `{Lattice A}.
-  Context `{Bottom A}.
-  Context {Hmax : forall x y, P x -> P y -> P (x ⊔ y)}.
-  Context {Hmin : forall x y, P x -> P y -> P (x ⊓ y)}.
-  Context {Hbot : P ⊥}.
-
-  Definition AP : Type := sig P.
-
-  Instance botAP : Bottom AP.
-  Proof. refine (⊥ ↾ _). apply Hbot. Defined.
-
-  Instance maxAP : Join AP :=
-    fun x y =>
-      match x, y with
-      | (a ; pa), (b ; pb) => (a ⊔ b ; Hmax a b pa pb)
-      end.
-
-  Instance minAP : Meet AP :=
-    fun x y =>
-      match x, y with
-      | (a ; pa), (b ; pb) => (a ⊓ b ; Hmin a b pa pb)
-      end.
-
-  Instance hprop_sub : forall c, IsHProp (P c).
-  Proof.
-    apply _.
-  Defined.
-
-  Ltac solve_sub :=
-    let x := fresh in
-    repeat (intro x ; destruct x)
-    ; simple refine (path_sigma _ _ _ _ _)
-    ; simpl
-    ; try (apply hprop_sub)
-    ; eauto 3 with lattice_hints typeclass_instances.
-
-  Global Instance lattice_sub : Lattice AP.
-  Proof.
-    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 : 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 : Meet hProp := fun X Y => BuildhProp (prod X Y).
-Instance lfalse : Bottom hProp := False_hp.
-
-Notation "A ∧ B" := (land A B) (at level 20, right associativity) : logic_scope.
-Arguments land _%L _%L.
-Open Scope logic_scope.
-
-(** [hProp] is a lattice. *)
-Section hPropLattice.
-  Context `{Univalence}.
-
-  Local Ltac lor_intros :=
-    let x := fresh in intro x
-                      ; repeat (strip_truncations ; destruct x as [x | x]).
-
-  Instance lor_commutative : Commutative lor.
-  Proof.
-    intros X Y.
-    apply path_iff_hprop ; lor_intros
-    ; apply tr ; auto.
-  Defined.
-
-  Instance land_commutative : Commutative land.
-  Proof.
-    intros X Y.
-    apply path_hprop.
-    apply equiv_prod_symm.
-  Defined.
-
-  Instance lor_associative : Associative lor.
-  Proof.
-    intros X Y Z.
-    apply path_iff_hprop ; lor_intros
-    ; apply tr ; auto
-    ; try (left ; apply tr)
-    ; try (right ; apply tr) ; auto.
-  Defined.
-
-  Instance land_associative : Associative land.
-  Proof.
-    intros X Y Z.
-    symmetry.
-    apply path_hprop.
-    symmetry.
-    apply equiv_prod_assoc.
-  Defined.
-
-  Instance lor_idempotent : BinaryIdempotent lor.
-  Proof.
-    intros X.
-    apply path_iff_hprop ; lor_intros
-    ; try(refine (tr(inl _))) ; auto.
-  Defined.
-
-  Instance land_idempotent : BinaryIdempotent land.
-  Proof.
-    intros X.
-    apply path_iff_hprop ; cbn.
-    - intros [a b] ; apply a.
-    - intros a ; apply (pair a a).
-  Defined.
-
-  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 : RightIdentity lor lfalse.
-  Proof.
-    intros X.
-    apply path_iff_hprop ; lor_intros ; try contradiction
-    ; try (refine (tr(inl _))) ; auto.
-  Defined.
-
-  Instance absorption_orb_andb : Absorption lor land.
-  Proof.
-    intros Z1 Z2.
-    apply path_iff_hprop ; cbn.
-    - intros X ; strip_truncations.
-      destruct X as [Xx | [Xy1 Xy2]] ; assumption.
-    - intros X.
-      apply (tr (inl X)).
-  Defined.
-
-  Instance absorption_andb_orb : Absorption land lor.
-  Proof.
-    intros Z1 Z2.
-    apply path_iff_hprop ; cbn.
-    - intros [X Y] ; strip_truncations.
-      assumption.
-    - intros X.
-      split.
-      * assumption.
-      * apply (tr (inl X)).
-  Defined.
-
-  Global Instance lattice_hprop : Lattice hProp.
-  Proof. repeat (split ; try apply _). Defined.
-
-  Global Instance bounded_jsl_hprop : BoundedJoinSemiLattice hProp.
-  Proof. repeat (split ; try apply _). Qed.
-
-  Global Instance top_hprop : Top hProp := Unit_hp.
-  Global Instance bounded_msl_hprop : @BoundedSemiLattice hProp (⊓) ⊤.
-  Proof.
-    repeat (split; try apply _); cbv.
-    - intros X. apply path_trunctype ; apply prod_unit_l.
-    - intros X. apply path_trunctype ; apply prod_unit_r.
-  Defined.
-End hPropLattice.
+From HoTT.Classes.implementations Require Export hprop_lattice bool pointwise.

FiniteSets/interfaces/set_interface.v

@@ -279,7 +279,7 @@ Section properties.
   Defined.
 
   Definition union_isIn : forall (A : Type) (a : A) (X Y : T A),
-    a ∈ (X ∪ Y) = lor (a ∈ X) (a ∈ Y).
+    a ∈ (X ∪ Y) = (a ∈ X) ⊔ (a ∈ Y).
   Proof.
     by (reduce T).
   Defined.

FiniteSets/kuratowski/extensionality.v

@@ -142,4 +142,4 @@ Section ext.
     - symmetry.
       eapply equiv_functor_prod' ; apply subset_union_equiv.
   Defined.
-End ext.
+End ext.

FiniteSets/kuratowski/kuratowski_sets.v

@@ -131,25 +131,19 @@ Defined.
 Section relations.
   Context `{Univalence}.
 
-  (** Membership of finite sets. *) 
+  (** Membership of finite sets. *)
   Global Instance fset_member : forall A, hasMembership (FSet A) A.
   Proof.
     intros A a.
     hrecursion.
-    - apply False_hp.
+    - apply ⊥.
     - 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.
+    - apply (⊔).
+    - eauto 10 with lattice_hints typeclass_instances.
+    - eauto 10 with lattice_hints typeclass_instances.
+    - eauto 10 with lattice_hints typeclass_instances.
+    - eauto 10 with lattice_hints typeclass_instances.
+    - eauto 8 with lattice_hints typeclass_instances.
   Defined.
 
   (** Subset relation of finite sets. *)
@@ -157,17 +151,13 @@ Section relations.
   Proof.
     intros A X Y.
     hrecursion X.
-    - apply Unit_hp.
+    - apply ⊤.
     - apply (fun a => a ∈ Y).
-    - apply land.
-      (* TODO *)
-    - eauto with lattice_hints typeclass_instances. 
-      apply associativity.
-    - eauto with lattice_hints typeclass_instances.
-      apply commutativity.
-    - apply left_identity.
-    - apply right_identity.
-    - eauto with lattice_hints typeclass_instances.
-      intros. apply binary_idempotent.
+    - apply (⊓).
+    - eauto 10 with lattice_hints typeclass_instances.
+    - eauto 10 with lattice_hints typeclass_instances.
+    - eauto 10 with lattice_hints typeclass_instances.
+    - eauto 10 with lattice_hints typeclass_instances.
+    - eauto 8 with lattice_hints typeclass_instances.
   Defined.
 End relations.

FiniteSets/kuratowski/properties.v

@@ -244,7 +244,7 @@ Section properties_membership.
   Definition singleton_isIn (a b: A) : a ∈ {|b|} -> Trunc (-1) (a = b) := idmap.
 
   Definition union_isIn (X Y : FSet A) (a : A)
-    : a ∈ X ∪ Y = a ∈ X ∨ a ∈ Y := idpath.
+    : 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|}.
@@ -261,7 +261,7 @@ Section properties_membership.
       assert (forall c d, ϕ a = c -> ϕ b = d ->
                           a ∈ (if ϕ b then {|b|} else ∅)
                           =
-                          (if ϕ a then BuildhProp (Trunc (-1) (a = b)) else False_hp)) as X.
+                          (if ϕ a then merely (a = b) else False_hp)) as X.
       {
         intros c d Hc Hd.
         destruct c ; destruct d ; rewrite Hc, Hd ; try reflexivity
@@ -286,7 +286,7 @@ Section properties_membership.
   Context {B : Type}.
 
   Lemma singleproduct_isIn (a : A) (b : B) (c : A) : forall (Y : FSet B),
-      (a, b) ∈ (single_product c Y) = land (BuildhProp (Trunc (-1) (a = c))) (b ∈ Y).
+      (a, b) ∈ (single_product c Y) = merely (a = c) ⊓ (b ∈ Y).
   Proof.
     hinduction ; try (intros ; apply path_ishprop).
     - apply path_hprop ; symmetry ; apply prod_empty_r.
@@ -325,7 +325,7 @@ Section properties_membership.
   Defined.
 
   Definition product_isIn (a : A) (b : B) (X : FSet A) (Y : FSet B) :
-    (a,b) ∈ (product X Y) = land (a ∈ X) (b ∈ Y).
+    (a,b) ∈ (product X Y) = (a ∈ X) ⊓ (b ∈ Y).
   Proof.
     hinduction X ; try (intros ; apply path_ishprop).
     - apply path_hprop ; symmetry ; apply prod_empty_l.
@@ -664,8 +664,9 @@ Ltac simplify_isIn_d :=
 
 Ltac toBool :=
   repeat intro;
-  apply ext ; intros ;
-  simplify_isIn_d ; eauto with bool_lattice_hints typeclass_instances.
+  apply ext; intros;
+  simplify_isIn_d;
+  eauto 10 with lattice_hints typeclass_instances.
 
 (** If `A` has decidable equality, then `FSet A` is a lattice *) 
 Section set_lattice.
@@ -680,7 +681,6 @@ Section set_lattice.
     repeat split; try apply _;
       compute[sg_op meet_is_sg_op meet_fset];
       toBool.
-    apply associativity.
   Defined.
 End set_lattice.
 

FiniteSets/misc/dec_fset.v

@@ -64,7 +64,7 @@ Section quantifiers.
     hinduction.
     - apply False_hp.
     - apply P.
-    - apply lor.
+    - apply (⊔).
     - eauto with lattice_hints typeclass_instances.
     (* TODO eauto with .. *)
       apply associativity.

FiniteSets/subobjects/b_finite.v

@@ -189,7 +189,7 @@ Section split.
            (f : {a : A | P a } <~> Fin n + Unit).
 
   Definition split : exists P' : Sub A, exists b : A,
-    prod ({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)).

FiniteSets/subobjects/enumerated.v

@@ -7,7 +7,7 @@ Require Import sub prelude FSets list_representation subobjects.k_finite
 Fixpoint listExt {A} (ls : list A) : Sub A := fun x =>
   match ls with
   | nil => False_hp
-  | cons a ls' => BuildhProp (Trunc (-1) (x = a)) ∨ listExt ls' x
+  | cons a ls' => merely (x = a) ⊔ listExt ls' x
   end.
 
 Fixpoint map {A B} (f : A -> B) (ls : list A) : list B :=
@@ -260,7 +260,7 @@ Section fset_dec_enumerated.
       strip_truncations. apply tr.
       destruct Hls as [ls Hls].
       exists (cons a ls). intros b. cbn.
-      apply (ap (fun z => _ ∨ z) (Hls b)).
+      apply (ap (fun z => _ ⊔ z) (Hls b)).
   Defined.
 
   Definition Kf_enumerated : Kf A -> enumerated A.

FiniteSets/subobjects/sub.v

@@ -10,7 +10,7 @@ Section subobjects.
   Global Instance sub_union : hasUnion Sub := join.
   Global Instance sub_intersection : hasIntersection Sub := meet.
   Global Instance sub_singleton : hasSingleton Sub A
-    := fun a b => BuildhProp (Trunc (-1) (b = a)).
+    := fun a b => merely (b = a).
   Global Instance sub_membership : hasMembership Sub A := fun a X => X a.
   Global Instance sub_comprehension : hasComprehension Sub A
     := fun ϕ X a => BuildhProp (X a * (ϕ a = true)).