mirror of https://github.com/nmvdw/HITs-Examples
Further work on associativity of union.
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Niels
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121
FSet.v
121
FSet.v
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@ -322,25 +322,53 @@ Axiom FSet_ind_beta_idem : forall
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End FSet.
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Parameter A : Type.
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Parameter eq : A -> A -> Bool.
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Parameter eq_refl: forall a:A, eq a a = true.
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Parameter A_eqdec : forall (x y : A), Decidable (x = y).
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Definition deceq (x y : A) :=
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if dec (x = y) then true else false.
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Theorem deceq_sym : forall x y, deceq x y = deceq y x.
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Proof.
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intros x y.
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unfold deceq.
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destruct (dec (x = y)) ; destruct (dec (y = x)) ; cbn.
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- reflexivity.
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- pose (n (p^)) ; contradiction.
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- pose (n (p^)) ; contradiction.
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- reflexivity.
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Defined.
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Arguments E {_}.
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Arguments U {_} _ _.
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Arguments L {_} _.
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Theorem idemU : forall x : FSet A, U x x = x.
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Proof.
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simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2) ; cbn.
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- apply nl.
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- apply idem.
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- intros x y P Q.
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rewrite assoc.
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rewrite (comm A x y).
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rewrite <- (assoc A y x x).
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rewrite P.
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rewrite (comm A y x).
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rewrite <- (assoc A x y y).
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rewrite Q.
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reflexivity.
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Defined.
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Definition isIn : A -> FSet A -> Bool.
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Proof.
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intros a.
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simple refine (FSet_rec A _ _ _ _ _ _ _ _ _ _).
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- exact false.
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- intro a'. apply (eq a a').
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- intro a'. apply (deceq a a').
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- apply orb.
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- intros x y z. destruct x; reflexivity.
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- intros x y. destruct x, y; reflexivity.
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- intros x. reflexivity.
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- intros x. destruct x; reflexivity.
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- intros a'. destruct (eq a a'); reflexivity.
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- intros a'. destruct (deceq a a'); reflexivity.
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Defined.
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Set Implicit Arguments.
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@ -393,7 +421,12 @@ intro a.
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simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2).
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- reflexivity.
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- intro b.
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admit.
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cbn.
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rewrite deceq_sym.
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unfold deceq.
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destruct (dec (a = b)).
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* rewrite p ; reflexivity.
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* reflexivity.
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- unfold intersection.
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intros x y P Q.
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cbn.
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@ -404,7 +437,7 @@ simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2).
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* apply nr.
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* apply nl.
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* apply nl.
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Admitted.
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Defined.
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Theorem comprehension_or : forall ϕ ψ x,
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comprehension (fun a => orb (ϕ a) (ψ a)) x = U (comprehension ϕ x) (comprehension ψ x).
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@ -422,35 +455,81 @@ simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2).
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cbn.
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rewrite P.
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rewrite Q.
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rewrite <- assoc.
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rewrite (assoc A (comprehension ψ x)).
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rewrite (comm A (comprehension ψ x) (comprehension ϕ y)).
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rewrite <- assoc.
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rewrite <- assoc.
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reflexivity.
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Defined.
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Theorem intersection_isIn : forall a x y,
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isIn a (intersection x y) = andb (isIn a x) (isIn a y).
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Proof.
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intros a.
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simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; cbn.
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- intros y.
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rewrite intersection_E.
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reflexivity.
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- intros b y.
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rewrite intersection_La.
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unfold deceq.
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destruct (dec (a = b)) ; cbn.
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* rewrite p.
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destruct (isIn b y).
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+ cbn.
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unfold deceq.
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destruct (dec (b = b)).
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{ reflexivity. }
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{ pose (n idpath). contradiction. }
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+ reflexivity.
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* destruct (isIn b y).
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+ cbn.
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unfold deceq.
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destruct (dec (a = b)).
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{ pose (n p). contradiction. }
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{ reflexivity. }
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+ reflexivity.
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- intros x y P Q z.
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enough (intersection (U x y) z = U (intersection x z) (intersection y z)).
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rewrite X.
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cbn.
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rewrite P.
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rewrite Q.
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destruct (isIn a x) ; destruct (isIn a y) ; destruct (isIn a z) ; reflexivity.
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admit.
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Admitted.
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Theorem intersection_assoc : forall x y z,
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intersection x (intersection y z) = intersection (intersection x y) z.
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Proof.
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simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2).
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- cbn.
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intros y z.
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- intros y z.
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cbn.
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rewrite intersection_E.
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rewrite intersection_E.
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rewrite intersection_E.
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reflexivity.
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- intro a.
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- intros a y z.
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cbn.
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intros y z.
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(* simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _ y) ; try (intros ; apply set_path2). *)
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admit.
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- unfold intersection.
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intros x y P Q z z'.
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cbn.
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rewrite Q.
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rewrite intersection_La.
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rewrite intersection_La.
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rewrite intersection_isIn.
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destruct (isIn a y).
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* rewrite intersection_La.
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destruct (isIn a (intersection y z)).
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destruct (isIn a z).
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+ reflexivity.
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+
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*
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destruct (isIn a (intersection y z)).
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+ reflexivity.
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* rewrite intersection_E.
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reflexivity.
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- unfold intersection. cbn.
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intros x y P Q z z'.
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rewrite comprehension_or.
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rewrite P.
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rewrite Q.
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rewrite comprehension_or.
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cbn.
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rewrite comprehension_or.
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reflexivity.
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