mirror of https://github.com/nmvdw/HITs-Examples
Associativity of intersection.
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Niels
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@ -21,10 +21,10 @@ 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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Theorem deceq_sym : forall x y, operations.deceq A x y = operations.deceq A y x.
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Proof.
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intros x y.
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unfold deceq.
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unfold operations.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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@ -92,47 +92,6 @@ simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2).
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reflexivity.
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Defined.
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Theorem intersection_isIn : forall a (x y: FSet A),
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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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Admitted.
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(*
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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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*)
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Theorem union_idem : forall x: FSet A, U x x = x.
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Proof.
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simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ;
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@ -167,8 +126,68 @@ Defined.
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Definition intersection_0r (X: FSet A): intersection X E = E := idpath.
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Theorem absorbtion_1 : forall (X1 X2 Y : FSet A),
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intersection (U X1 X2) Y = U (intersection X1 Y) (intersection X2 Y).
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Admitted.
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Theorem intersection_La : forall (a : A) (X : FSet A),
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intersection (L a) X = if isIn a X then L a else E.
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Proof.
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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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cbn.
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rewrite deceq_sym.
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unfold operations.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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rewrite P.
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rewrite Q.
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destruct (isIn a x) ; destruct (isIn a y).
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* apply idem.
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* apply nr.
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* apply nl.
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* apply nl.
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Defined.
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Theorem intersection_isIn : forall a (x y: FSet A),
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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 x y.
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simple refine (FSet_ind A (fun z => isIn a (intersection z y) = andb (isIn a z) (isIn a y)) _ _ _ _ _ _ _ _ _ x) ; try (intros ; apply set_path2) ; cbn.
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- rewrite intersection_0l.
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reflexivity.
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- intro b.
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rewrite intersection_La.
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unfold operations.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 operations.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 operations.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 X1 X2 P Q.
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rewrite absorbtion_1.
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cbn.
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rewrite P.
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rewrite Q.
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destruct (isIn a X1) ; destruct (isIn a X2) ; destruct (isIn a y) ; reflexivity.
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Defined.
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Lemma intersection_comm (X Y: FSet A): intersection X Y = intersection Y X.
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Proof.
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@ -233,31 +252,32 @@ hrecursion X; try (intros; apply set_path2).
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apply comm.
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Defined.*)
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Theorem intersection_assoc : forall (x y z: FSet A),
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intersection x (intersection y z) = intersection (intersection x y) z.
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Admitted.
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(*
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simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2).
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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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Theorem intersection_assoc (X Y Z: FSet A) :
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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
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(FSet_ind A
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(fun z => intersection z (intersection Y Z) = intersection (intersection z Y) Z)
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_ _ _ _ _ _ _ _ _ X) ; try (intros ; apply set_path2).
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- cbn.
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rewrite intersection_0l.
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rewrite intersection_0l.
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rewrite intersection_0l.
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reflexivity.
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- intros a y z.
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- intros a.
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cbn.
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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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destruct (isIn a Y).
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* rewrite intersection_La.
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destruct (isIn a z).
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destruct (isIn a Z).
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+ reflexivity.
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+ reflexivity.
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* rewrite intersection_E.
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* rewrite intersection_0l.
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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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intros X1 X2 P Q.
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rewrite comprehension_or.
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rewrite P.
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rewrite Q.
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@ -265,7 +285,7 @@ simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2).
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cbn.
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rewrite comprehension_or.
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reflexivity.
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*)
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Defined.
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Theorem comprehension_subset : forall ϕ (X : FSet A),
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U (comprehension ϕ X) X = X.
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