Split the development into different directories

FiniteSets/fsets/properties.v

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+Require Import HoTT HitTactics.
+Require Export representations.definition disjunction fsets.operations.
+
+(* Lemmas relating operations to the membership predicate *)
+Section operations_isIn.
+
+Context {A : Type}.
+Context `{Univalence}.
+
+
+
+Lemma union_idem : forall x: FSet A, U x x = x.
+Proof.
+hinduction; 
+try (intros ; apply set_path2) ; cbn.
+- apply nl.
+- apply idem.
+- intros x y P Q.
+  rewrite assoc.
+  rewrite (comm x y).
+  rewrite <- (assoc y x x).
+  rewrite P.
+  rewrite (comm y x).
+  rewrite <- (assoc x y y).
+  f_ap. 
+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).
+- intros. apply nl.
+- intros a. hinduction Y;
+  try (intros; apply path_forall; intro; apply set_path2).
+  + intro.
+    contradiction.
+  + intro a0.
+    simple refine (Trunc_ind _ _).
+    intro p ; cbn. 
+    rewrite p; apply idem.
+  + intros X1 X2 IH1 IH2.
+    simple refine (Trunc_ind _ _).
+    intros [e1 | e2].
+    ++ rewrite assoc.
+       rewrite (IH1 e1).
+       reflexivity.
+    ++ rewrite comm.
+       rewrite <- assoc.
+       rewrite (comm X2).
+       rewrite (IH2 e2).
+       reflexivity.
+- intros X1 X2 IH1 IH2 [G1 G2].
+  rewrite <- assoc.
+  rewrite (IH2 G2).
+  apply (IH1 G1).
+Defined.
+
+Lemma subset_union_l (X : FSet A) :
+  forall Y, subset X (U X Y).
+Proof.
+hinduction X ;
+  try (repeat (intro; intros; apply path_forall); intro; apply equiv_hprop_allpath ; apply _).
+- apply (fun _ => tt).
+- intros a Y.
+  apply tr ; left ; apply tr ; reflexivity.
+- intros X1 X2 HX1 HX2 Y.
+  split. 
+  * rewrite <- assoc. apply HX1.
+  * rewrite (comm X1 X2). rewrite <- assoc. apply HX2.
+Defined.
+
+(* Union and membership *)
+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). 
+- cbn. apply (union_idem _)^.
+- cbn. intros.
+  destruct (ϕ a) ; destruct (ψ a) ; symmetry.
+  * apply union_idem.
+  * apply nr. 
+  * apply nl.
+  * apply union_idem.
+- simpl. intros x y P Q.
+  rewrite P.
+  rewrite Q.
+  rewrite <- assoc.
+  rewrite (assoc  (comprehension ψ x)).
+  rewrite (comm  (comprehension ψ x) (comprehension ϕ y)).
+  rewrite <- assoc.
+  rewrite <- assoc.
+  reflexivity.
+Defined.
+
+End operations_isIn.
+
+(* Other properties *)
+Section properties.
+
+Context {A : Type}.
+Context `{Univalence}.
+
+(** isIn properties *)
+Lemma singleton_isIn (a b: A) : isIn a (L b) -> Trunc (-1) (a = b).
+Proof.
+  apply idmap.
+Defined.
+
+Lemma empty_isIn (a: A) : isIn a E -> Empty.
+Proof. 
+  apply idmap.
+Defined.
+
+(** 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.
+  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.
+
+End properties.