Simplified proof of extensionalty and proofs in interface.v

FiniteSets/_CoqProject

@@ -12,7 +12,6 @@ fsets/isomorphism.v
 fsets/operations.v
 fsets/operations_decidable.v
 fsets/extensionality.v
-fsets/extensionality_alt.v
 fsets/properties.v
 fsets/properties_decidable.v
 fsets/length.v

FiniteSets/fsets/extensionality.v

@@ -6,29 +6,86 @@ Section ext.
   Context {A : Type}.
   Context `{Univalence}.
 
+  Lemma equiv_subset1_l (X Y : FSet A) (H1 : Y ∪ X = X) (a : A) (Ya : a ∈ Y) : a ∈ X.
+  Proof.
+    apply (transport (fun Z => a ∈ Z) H1 (tr(inl Ya))).
+  Defined.
+  
+  Lemma equiv_subset1_r X : forall (Y : FSet A), (forall a, a ∈ Y -> a ∈ X) -> Y ∪ X = X.
+  Proof.
+    hinduction ; try (intros ; apply path_ishprop).
+    - intros.
+      apply nl.
+    - intros b sub.
+      specialize (sub b (tr idpath)).
+      revert sub.
+      hinduction X ; try (intros ; apply path_ishprop).
+      * contradiction.
+      * intros.
+        strip_truncations.
+        rewrite sub.
+        apply union_idem.
+      * intros X Y subX subY mem.
+        strip_truncations.
+        destruct mem as [t | t].
+        ** rewrite assoc, (subX t).
+           reflexivity.
+        ** rewrite (comm X), assoc, (subY t).
+           reflexivity.
+    - intros Y1 Y2 H1 H2 H3.
+      rewrite <- assoc.
+      rewrite (H2 (fun a HY => H3 a (tr(inr HY)))).
+      apply (H1 (fun a HY => H3 a (tr(inl HY)))).
+  Defined.
+
+  Lemma eq_subset1 X Y : (Y ∪ X = X) * (X ∪ Y = Y) <~> forall (a : A), a ∈ X = a ∈ Y.
+  Proof.    
+    eapply equiv_iff_hprop_uncurried ; split.
+    - intros [H1 H2] a.
+      apply path_iff_hprop ; apply equiv_subset1_l ; assumption.
+    - intros H1.
+      split ; apply equiv_subset1_r ; intros.
+      * rewrite H1 ; assumption.
+      * rewrite <- H1 ; assumption.
+  Defined.
+
+  Lemma eq_subset2 (X Y : FSet A) : X = Y <~> (Y ∪ X = X) * (X ∪ Y = Y).
+  Proof.
+    eapply equiv_iff_hprop_uncurried ; split.
+    - intro Heq.
+      split.
+      * apply (ap (fun Z => Z ∪ X) Heq^ @ union_idem X).
+      * apply (ap (fun Z => Z ∪ Y) Heq @ union_idem Y).
+    - intros [H1 H2].
+      apply (H1^ @ comm Y X @ H2).
+  Defined.
+  
+  Theorem fset_ext (X Y : FSet A) :
+    X = Y <~> forall (a : A), a ∈ X = a ∈ Y.
+  Proof.
+    apply (equiv_compose' (eq_subset1 X Y) (eq_subset2 X Y)).
+  Defined.
+
   Lemma subset_union (X Y : FSet A) :
     X ⊆ Y -> X ∪ Y = Y.
   Proof.
-    hinduction X ; try (intros; apply path_forall; intro; apply set_path2).
+    hinduction X ; try (intros ; apply path_ishprop). 
-    - intros. apply nl.
+    - intros.
+      apply nl.
     - intros a.
-      hinduction Y ; try (intros; apply path_forall; intro; apply set_path2).
+      hinduction Y ; try (intros ; apply path_ishprop).
       + intro.
         contradiction.
-      + intro a0.
+      + intros b p.
-        simple refine (Trunc_ind _ _).
+        strip_truncations.
-        intro p ; simpl.
+        rewrite p.
-        rewrite p; apply idem.
+        apply idem.
-      + intros X1 X2 IH1 IH2.
+      + intros X1 X2 IH1 IH2 t.
-        simple refine (Trunc_ind _ _).
+        strip_truncations.
-        intros [e1 | e2].
+        destruct t as [t | t].
-        ++ rewrite assoc.
+        ++ rewrite assoc, (IH1 t).
-           rewrite (IH1 e1).
            reflexivity.
-        ++ rewrite comm.
+        ++ rewrite comm, <- assoc, (comm X2), (IH2 t).
-           rewrite <- assoc.
-           rewrite (comm X2).
-           rewrite (IH2 e2).
            reflexivity.
     - intros X1 X2 IH1 IH2 [G1 G2].
       rewrite <- assoc.
@@ -39,15 +96,16 @@ Section ext.
   Lemma subset_union_l (X : FSet A) :
     forall Y, X ⊆ X ∪ Y.
   Proof.
-    hinduction X ; try (repeat (intro; intros; apply path_forall);
+    hinduction X ; try (intros ; apply path_ishprop).
-                        intro ; apply path_ishprop).
     - apply (fun _ => tt).
-    - intros a Y.
+    - intros.
       apply (tr(inl(tr idpath))).
     - intros X1 X2 HX1 HX2 Y.
       split ; unfold subset in *.
-      * rewrite <- assoc. apply HX1.
+      * rewrite <- assoc.
-      * rewrite (comm X1 X2). rewrite <- assoc. apply HX2.
+        apply HX1.
+      * rewrite (comm X1 X2), <- assoc.
+        apply HX2.
   Defined.
 
   Lemma subset_union_equiv
@@ -63,90 +121,21 @@ Section ext.
   Defined.
   
   Lemma subset_isIn (X Y : FSet A) :
-    (forall (a : A), a ∈ X -> a ∈ Y)
+    X ⊆ Y <~> forall (a : A), a ∈ X -> a ∈ Y.
-      <~> X ⊆ Y.
   Proof.
-    eapply equiv_iff_hprop_uncurried.
+    etransitivity.
-    split.
+    - apply subset_union_equiv.
-    - hinduction X ;
+    - eapply equiv_iff_hprop_uncurried ; split.
-        try (intros; repeat (apply path_forall; intro); apply path_ishprop).
+      * apply equiv_subset1_l.
-      + intros ; reflexivity.
+      * apply equiv_subset1_r.
-      + intros a f.
-        apply f.
-        apply tr ; reflexivity.
-      + intros X1 X2 H1 H2 f.
-        enough (X1 ⊆ Y).
-        enough (X2 ⊆ Y).
-        { split. apply X. apply X0. }
-        * apply H2.
-          intros a Ha.
-          refine (f _ (tr _)).
-          right.
-          apply Ha.
-        * apply H1.
-          intros a Ha.
-          refine (f _ (tr _)).
-          left.
-          apply Ha.
-    - hinduction X ;
-        try (intros; repeat (apply path_forall; intro); apply path_ishprop).
-      + intros. contradiction.
-      + intros b f a.
-        simple refine (Trunc_ind _ _) ; cbn.
-        intro p.
-        rewrite p^ in f.
-        apply f.
-      + intros X1 X2 IH1 IH2 [H1 H2] a.
-        simple refine (Trunc_ind _ _) ; cbn.
-        intros [C1 | C2].
-        ++ apply (IH1 H1 a C1).
-        ++ apply (IH2 H2 a C2).
-  Defined.
-
-  (** ** Extensionality proof *)
-
-  Lemma eq_subset' (X Y : FSet A) : X = Y <~> (Y ∪ X = X) * (X ∪ Y = Y).
-  Proof.
-    unshelve eapply BuildEquiv.
-    { intro H'. rewrite H'. split; apply union_idem. }
-    unshelve esplit.
-    { intros [H1 H2]. etransitivity. apply H1^.
-      rewrite comm. apply H2. }
-    intro; apply path_prod; apply set_path2.
-    all: intro; apply set_path2.
   Defined.
 
   Lemma eq_subset (X Y : FSet A) :
     X = Y <~> (Y ⊆ X * X ⊆ Y).
   Proof.
-    transitivity ((Y ∪ X = X) * (X ∪ Y = Y)).
+    etransitivity ((Y ∪ X = X) * (X ∪ Y = Y)).
-    apply eq_subset'.
+    - apply eq_subset2.
-    symmetry.
+    - symmetry.
       eapply equiv_functor_prod' ; apply subset_union_equiv.
   Defined.
-
-  Theorem fset_ext (X Y : FSet A) :
-    X = Y <~> (forall (a : A), a ∈ X = a ∈ Y).
-  Proof.
-    refine (@equiv_compose' _ _ _ _ _) ; [ | apply eq_subset ].
-    refine (@equiv_compose' _ ((forall a, a ∈ Y -> a ∈ X)
-                               *(forall a, a ∈ X -> a ∈ Y)) _ _ _).
-    - apply equiv_iff_hprop_uncurried.
-      split.
-      * intros [H1 H2 a].
-        specialize (H1 a) ; specialize (H2 a).
-        apply path_iff_hprop.
-        apply H2.
-        apply H1.
-      * intros H1.
-        split ; intro a ; intro H2.
-      + rewrite (H1 a).
-        apply H2.
-      + rewrite <- (H1 a).
-        apply H2.
-    - eapply equiv_functor_prod' ;
-        apply equiv_iff_hprop_uncurried ;
-        split ; apply subset_isIn.
-  Defined.
-
 End ext.

FiniteSets/fsets/extensionality_alt.v

@@ -1,68 +0,0 @@
-(** Extensionality of the FSets *)
-Require Import HoTT HitTactics.
-Require Import representations.definition fsets.operations.
-
-Section ext.
-  Context {A : Type}.
-  Context `{Univalence}.
-
-  Lemma equiv_subset1_l (X Y : FSet A) (H1 : Y ∪ X = X) (a : A) (Ya : a ∈ Y) : a ∈ X.
-  Proof.
-    apply (transport (fun Z => a ∈ Z) H1 (tr(inl Ya))).
-  Defined.
-  
-  Lemma equiv_subset1_r X : forall (Y : FSet A), (forall a, a ∈ Y -> a ∈ X) -> Y ∪ X = X.
-  Proof.
-    hinduction ; try (intros ; apply path_ishprop).
-    - intros.
-      apply nl.
-    - intros b sub.
-      specialize (sub b (tr idpath)).
-      revert sub.
-      hinduction X ; try (intros ; apply path_ishprop).
-      * contradiction.
-      * intros.
-        strip_truncations.
-        rewrite sub.
-        apply union_idem.
-      * intros X Y subX subY mem.
-        strip_truncations.
-        destruct mem as [t | t].
-        ** rewrite assoc, (subX t).
-           reflexivity.
-        ** rewrite (comm X), assoc, (subY t).
-           reflexivity.
-    - intros Y1 Y2 H1 H2 H3.
-      rewrite <- assoc.
-      rewrite (H2 (fun a HY => H3 a (tr(inr HY)))).
-      apply (H1 (fun a HY => H3 a (tr(inl HY)))).
-  Defined.
-
-  Lemma eq_subset1 X Y : (Y ∪ X = X) * (X ∪ Y = Y) <~> forall (a : A), a ∈ X = a ∈ Y.
-  Proof.    
-    eapply equiv_iff_hprop_uncurried ; split.
-    - intros [H1 H2] a.
-      apply path_iff_hprop ; apply equiv_subset1_l ; assumption.
-    - intros H1.
-      split ; apply equiv_subset1_r ; intros.
-      * rewrite H1 ; assumption.
-      * rewrite <- H1 ; assumption.
-  Defined.
-
-  Lemma eq_subset2 (X Y : FSet A) : X = Y <~> (Y ∪ X = X) * (X ∪ Y = Y).
-  Proof.
-    eapply equiv_iff_hprop_uncurried ; split.
-    - intro Heq.
-      split.
-      * apply (ap (fun Z => Z ∪ X) Heq^ @ union_idem X).
-      * apply (ap (fun Z => Z ∪ Y) Heq @ union_idem Y).
-    - intros [H1 H2].
-      apply (H1^ @ comm Y X @ H2).
-  Defined.
-  
-  Theorem fset_ext (X Y : FSet A) :
-    X = Y <~> forall (a : A), a ∈ X = a ∈ Y.
-  Proof.
-    apply (equiv_compose' (eq_subset1 X Y) (eq_subset2 X Y)).
-  Defined.
-End ext.

FiniteSets/implementations/interface.v

@@ -77,49 +77,46 @@ Section properties.
 
   Ltac simplify := intros ; autounfold in * ; apply reflect_eq ; reduce.
 
-  Definition well_defined_union : forall (A : Type) (X1 X2 Y1 Y2 : T A),
+  Definition well_defined_union (A : Type) (X1 X2 Y1 Y2 : T A) :
     set_eq A X1 Y1 -> set_eq A X2 Y2 -> set_eq A (union X1 X2) (union Y1 Y2).
   Proof.
-    intros A X1 X2 Y1 Y2 HXY1 HXY2.
+    intros HXY1 HXY2.
     simplify.
     by rewrite HXY1, HXY2.
   Defined.
 
-  Definition well_defined_filter : forall (A : Type) (ϕ : A -> Bool) (X Y : T A),
+  Definition well_defined_filter (A : Type) (ϕ : A -> Bool) (X Y : T A) :
     set_eq A X Y -> set_eq A (filter ϕ X) (filter ϕ Y).
   Proof.
-    intros A ϕ X Y HXY.
+    intros HXY.
     simplify.
     by rewrite HXY.
   Defined.
 
+  Ltac reflect_equality := simplify ; eauto with lattice_hints typeclass_instances.
+
   Lemma union_comm : forall A (X Y : T A),
       set_eq A (X ∪ Y) (Y ∪ X).
   Proof.
-    simplify.
+    reflect_equality.
-    apply comm.
   Defined.
 
   Lemma union_assoc : forall A (X Y Z : T A),
     set_eq A ((X ∪ Y) ∪ Z) (X ∪ (Y ∪ Z)).
   Proof.
-    simplify.
+    reflect_equality.
-    symmetry.
-    apply assoc.
   Defined.
 
   Lemma union_idem : forall A (X : T A),
     set_eq A (X ∪ X) X.
   Proof.
-    simplify.
+    reflect_equality.
-    apply union_idem.
   Defined.
 
   Lemma union_neutral : forall A (X : T A),
     set_eq A (∅ ∪ X) X.
   Proof.
-    simplify.
+    reflect_equality.
-    apply nl.
   Defined.
 
 End properties.
@@ -258,8 +255,7 @@ Proof.
   - intros. apply path_forall; intro. apply set_path2.
 Defined.
 
-Ltac buggeroff := intros;
+Ltac buggeroff := intros; apply path_ishprop.
-  (repeat (apply path_forall; intro)); apply set_path2.
 
 Instance View_max_assoc A: Associative (@max_L (View A) _).
 Proof.