Some cleaning

FiniteSets/fsets/properties.v

@@ -9,8 +9,7 @@ Section operations_isIn.
 
   Lemma union_idem : forall x: FSet A, U x x = x.
   Proof.
-    hinduction; 
+    hinduction ; try (intros ; apply set_path2).
-      try (intros ; apply set_path2).
     - apply nl.
     - apply idem.
     - intros x y P Q.
@@ -21,16 +20,16 @@ Section operations_isIn.
       rewrite (comm y x).
       rewrite <- (assoc x y y).
       f_ap. 
-  Qed.
+  Defined.
 
   (** ** Properties about subset relation. *)
   Lemma subset_union (X Y : FSet A) : 
     subset X Y -> U X Y = Y.
   Proof.
-    hinduction X; try (intros; apply path_forall; intro; apply set_path2).
+    hinduction X ; try (intros; apply path_forall; intro; apply set_path2).
     - intros. apply nl.
-    - intros a. hinduction Y;
+    - intros a.
-                  try (intros; apply path_forall; intro; apply set_path2).
+      hinduction Y ; try (intros; apply path_forall; intro; apply set_path2).
       + intro.
         contradiction.
       + intro a0.
@@ -57,30 +56,24 @@ Section operations_isIn.
   Lemma subset_union_l (X : FSet A) :
     forall Y, subset X (U X Y).
   Proof.
-    hinduction X ;
+    hinduction X ; try (repeat (intro; intros; apply path_forall);
-      try (repeat (intro; intros; apply path_forall); intro; apply equiv_hprop_allpath ; apply _).
+                        intro ; apply equiv_hprop_allpath ; apply _).
     - apply (fun _ => tt).
     - intros a Y.
-      apply tr ; left ; apply tr ; reflexivity.
+      apply (tr(inl(tr idpath))).
     - intros X1 X2 HX1 HX2 Y.
       split. 
       * rewrite <- assoc. apply HX1.
       * rewrite (comm X1 X2). rewrite <- assoc. apply HX2.
   Defined.
 
-  (* Union and membership *)
+  (* simplify it using extensionality *)
-  Lemma union_isIn (X Y : FSet A) (a : A) :
-    isIn a (U X Y) = isIn a X ∨ isIn a Y.
-  Proof.
-    reflexivity.
-  Defined.
-
   Lemma comprehension_or : forall ϕ ψ (x: FSet A),
       comprehension (fun a => orb (ϕ a) (ψ a)) x = U (comprehension ϕ x) 
                                                      (comprehension ψ x).
   Proof.
     intros ϕ ψ.
-    hinduction; try (intros; apply set_path2). 
+    hinduction ; try (intros; apply set_path2). 
     - apply (union_idem _)^.
     - intros.
       destruct (ϕ a) ; destruct (ψ a) ; symmetry.
@@ -108,78 +101,12 @@ Section properties.
   Context `{Univalence}.
 
   (** isIn properties *)
-  Lemma singleton_isIn (a b: A) : isIn a (L b) -> Trunc (-1) (a = b).
+  Definition empty_isIn (a: A) : isIn a E -> Empty := idmap.
-  Proof.
-    apply idmap.
-  Defined.
   
-  Lemma empty_isIn (a: A) : isIn a E -> Empty.
+  Definition singleton_isIn (a b: A) : isIn a (L b) -> Trunc (-1) (a = b) := idmap.
-  Proof. 
-    apply idmap.
-  Defined.
 
-  (** comprehension properties *)
+  Definition union_isIn (X Y : FSet A) (a : A)
-  Lemma comprehension_false Y : comprehension (fun (_ : A) => false) Y = E.
+    : isIn a (U X Y) = isIn a X ∨ isIn a Y := idpath.
-  Proof.
-    hrecursion Y; try (intros; apply set_path2).
-    - reflexivity.
-    - reflexivity.
-    - intros x y IHa IHb.
-      rewrite IHa.
-      rewrite IHb.
-      apply union_idem.
-  Defined.
-
-  Lemma comprehension_subset : forall ϕ (X : FSet A),
-      U (comprehension ϕ X) X = X.
-  Proof.
-    intros ϕ.
-    hrecursion; try (intros ; apply set_path2) ; cbn.
-    - apply union_idem.
-    - intro a.
-      destruct (ϕ a).
-      * apply union_idem.
-      * apply nl.
-    - intros X Y P Q.
-      rewrite assoc.
-      rewrite <- (assoc (comprehension ϕ X) (comprehension ϕ Y) X).
-      rewrite (comm (comprehension ϕ Y) X).
-      rewrite assoc.
-      rewrite P.
-      rewrite <- assoc.
-      rewrite Q.
-      reflexivity.
-  Defined.
-  
-  Lemma merely_choice : forall X : FSet A, hor (X = E) (hexists (fun a => isIn a X)).
-  Proof.
-    hinduction; try (intros; apply equiv_hprop_allpath ; apply _).
-    - apply (tr (inl idpath)).
-    - intro a.
-      refine (tr (inr (tr (a ; tr idpath)))).
-    - intros X Y TX TY.
-      strip_truncations.
-      destruct TX as [XE | HX] ; destruct TY as [YE | HY] ; try(strip_truncations ; apply tr).
-      * apply tr ; left.
-        rewrite XE, YE.
-        apply (union_idem E).
-      * right.
-        destruct HY as [a Ya].
-        apply tr.
-        exists a.
-        apply (tr (inr Ya)).
-      * right.
-        destruct HX as [a Xa].
-        apply tr.
-        exists a.
-        apply (tr (inl Xa)).
-      * right.
-        destruct HX as [a Xa].
-        destruct HY as [b Yb].
-        apply tr.
-        exists a.
-        apply (tr (inl Xa)).
-  Defined.
 
   Lemma comprehension_isIn (ϕ : A -> Bool) (a : A) : forall X : FSet A,
       isIn a (comprehension ϕ X) = if ϕ a then isIn a X else False_hp.
@@ -210,4 +137,64 @@ Section properties.
         ** intros ; apply tr ; right ; assumption.
   Defined.
 
+  (* The proof can be simplified using extensionality *)
+  (** comprehension properties *)
+  Lemma comprehension_false Y : comprehension (fun (_ : A) => false) Y = E.
+  Proof.
+    hrecursion Y; try (intros; apply set_path2).
+    - reflexivity.
+    - reflexivity.
+    - intros x y IHa IHb.
+      rewrite IHa, IHb.
+      apply union_idem.
+  Defined.
+
+  (* Can be simplified using extensionality *)
+  Lemma comprehension_subset : forall ϕ (X : FSet A),
+      U (comprehension ϕ X) X = X.
+  Proof.
+    intros ϕ.
+    hrecursion; try (intros ; apply set_path2) ; cbn.
+    - apply union_idem.
+    - intro a.
+      destruct (ϕ a).
+      * apply union_idem.
+      * apply nl.
+    - intros X Y P Q.
+      rewrite assoc.
+      rewrite <- (assoc (comprehension ϕ X) (comprehension ϕ Y) X).
+      rewrite (comm (comprehension ϕ Y) X).
+      rewrite assoc.
+      rewrite P.
+      rewrite <- assoc.
+      rewrite Q.
+      reflexivity.
+  Defined.
+  
+  Lemma merely_choice : forall X : FSet A, hor (X = E) (hexists (fun a => isIn a X)).
+  Proof.
+    hinduction; try (intros; apply equiv_hprop_allpath ; apply _).
+    - apply (tr (inl idpath)).
+    - intro a.
+      refine (tr (inr (tr (a ; tr idpath)))).
+    - intros X Y TX TY.
+      strip_truncations.
+      destruct TX as [XE | HX] ; destruct TY as [YE | HY] ; try(strip_truncations ; apply tr).
+      * refine (tr (inl _)).
+        rewrite XE, YE.
+        apply (union_idem E).
+      * destruct HY as [a Ya].
+        refine (inr (tr _)).
+        exists a.
+        apply (tr (inr Ya)).
+      * destruct HX as [a Xa].
+        refine (inr (tr _)).
+        exists a.
+        apply (tr (inl Xa)).
+      * destruct (HX, HY) as [[a Xa] [b Yb]].
+        refine (inr (tr _)).
+        exists a.
+        apply (tr (inl Xa)).
+  Defined.
+  
 End properties.

FiniteSets/representations/bad.v

@@ -42,19 +42,22 @@ Module Export FSet.
   Arguments nl {_} _.
   Arguments nr {_} _.
   Arguments idem {_} _.
+  Notation "{| x |}" :=  (L x).
+  Infix "∪" := U (at level 8, right associativity).
+  Notation "∅" := E.
 
   Section FSet_induction.
     Variable A: Type.
     Variable  (P : FSet A -> Type).
-    Variable  (eP : P E).
+    Variable  (eP : P ∅).
-    Variable  (lP : forall a: A, P (L a)).
+    Variable  (lP : forall a: A, P {|a |}).
-    Variable  (uP : forall (x y: FSet A), P x -> P y -> P (U x y)).
+    Variable  (uP : forall (x y: FSet A), P x -> P y -> P (x ∪ y)).
     Variable  (assocP : forall (x y z : FSet A) 
                                (px: P x) (py: P y) (pz: P z),
                   assoc x y z #
-                        (uP      x    (U y z)          px        (uP y z py pz)) 
+                        (uP x (y ∪ z) px (uP y z py pz)) 
                   = 
-                  (uP   (U x y)    z       (uP x y px py)          pz)).
+                  (uP (x ∪ y) z (uP x y px py) pz)).
     Variable  (commP : forall (x y: FSet A) (px: P x) (py: P y),
                   comm x y #
                        uP x y px py = uP y x py px).
@@ -71,11 +74,9 @@ Module Export FSet.
              {struct x}
       : P x
       := (match x return _ -> _ -> _ -> _ -> _ -> P x with
-          | E => fun _ => fun _ => fun _ => fun _ => fun _ => eP
+          | ∅ => fun _ _ _ _ _ => eP
-          | L a => fun _ => fun _ => fun _ => fun _ => fun _ => lP a
+          | {|a|} => fun _ _ _ _ _ => lP a
-          | U y z => fun _ => fun _ => fun _ => fun _ => fun _ => uP y z
+          | y ∪ z => fun _ _ _ _ _ => uP y z (FSet_ind y) (FSet_ind z)
-                                                                     (FSet_ind y)
-                                                                     (FSet_ind z)
           end) assocP commP nlP nrP idemP.
 
 
@@ -84,13 +85,13 @@ Module Export FSet.
         (assocP x y z (FSet_ind x) (FSet_ind y) (FSet_ind z)).
 
     Axiom FSet_ind_beta_comm : forall (x y : FSet A),
-        apD FSet_ind (comm x y) = (commP x y (FSet_ind x) (FSet_ind y)).
+        apD FSet_ind (comm x y) = commP x y (FSet_ind x) (FSet_ind y).
 
     Axiom FSet_ind_beta_nl : forall (x : FSet A),
-        apD FSet_ind (nl x) = (nlP x (FSet_ind x)).
+        apD FSet_ind (nl x) = nlP x (FSet_ind x).
 
     Axiom FSet_ind_beta_nr : forall (x : FSet A),
-        apD FSet_ind (nr x) = (nrP x (FSet_ind x)).
+        apD FSet_ind (nr x) = nrP x (FSet_ind x).
 
     Axiom FSet_ind_beta_idem : forall (x : A), apD FSet_ind (idem x) = idemP x.
   End FSet_induction.
@@ -183,9 +184,11 @@ Module Export FSet.
     
   End FSet_recursion.
 
-  Instance FSet_recursion A : HitRecursion (FSet A) := {
+  Instance FSet_recursion A : HitRecursion (FSet A) :=
+    {
       indTy := _; recTy := _; 
-                                                        H_inductor := FSet_ind A; H_recursor := FSet_rec A }.
+      H_inductor := FSet_ind A; H_recursor := FSet_rec A
+    }.
 
 End FSet.
 
@@ -195,10 +198,12 @@ Notation "∅" := E.
 
 Section set_sphere.
   From HoTT.HIT Require Import Circle.
-  From HoTT Require Import UnivalenceAxiom.
+  Context `{Univalence}.
-  Instance S1_recursion : HitRecursion S1 := {
+  Instance S1_recursion : HitRecursion S1 :=
+    {
       indTy := _; recTy := _; 
-                                              H_inductor := S1_ind; H_recursor := S1_rec }.
+      H_inductor := S1_ind; H_recursor := S1_rec
+    }.
 
   Variable A : Type.
 
@@ -206,8 +211,7 @@ Section set_sphere.
   Proof.
     hrecursion x.
     - exact loop.
-    - etransitivity. 
+    - refine (transport_paths_FlFr _ _ @ _). 
-      eapply (@transport_paths_FlFr S1 S1 idmap idmap).
       hott_simpl.
   Defined.
 
@@ -225,11 +229,10 @@ Section set_sphere.
   Proof.
     hrecursion x.
     - exact loop.
-    - etransitivity.
+    - refine (transport_paths_FlFr loop _ @ _).
-      apply (@transport_paths_FlFr _ _ (fun x => S1op base x) idmap _ _ loop loop).
       hott_simpl.
       apply moveR_pM. apply moveR_pM. hott_simpl.
-      etransitivity. apply (ap_V (S1op base) loop).
+      refine (ap_V _ _ @ _).
       f_ap. apply S1_rec_beta_loop.
   Defined.
 
@@ -237,8 +240,7 @@ Section set_sphere.
   Proof.
     hrecursion z.
     - reflexivity.
-    - etransitivity.
+    - refine (transport_paths_FlFr loop _ @ _).
-      apply (@transport_paths_FlFr _ _ (fun z => S1op x (S1op y z)) (S1op (S1op x y)) _ _ loop idpath). 
       hott_simpl.
       apply moveR_Mp. hott_simpl.
       rewrite S1_rec_beta_loop.
@@ -274,7 +276,7 @@ Section set_sphere.
 
   Lemma FSet_S_ap : (nl (@E A)) = (nr E) -> idpath = loop.
   Proof.
-    intros H.
+    intros H1.
     enough (ap FSet_to_S (nl E) = ap FSet_to_S (nr E)) as H'.
     - rewrite FSet_rec_beta_nl in H'. 
       rewrite FSet_rec_beta_nr in H'.
@@ -285,7 +287,7 @@ Section set_sphere.
 
   Lemma FSet_not_hset : IsHSet (FSet A) -> False.
   Proof.
-    intros H.
+    intros H1.
     enough (idpath = loop). 
     - assert (S1_encode _ idpath = S1_encode _ (loopexp loop (pos Int.one))) as H' by f_ap.
       rewrite S1_encode_loopexp in H'. simpl in H'. symmetry in H'.