Commit 883572e7 authored by Robbert Krebbers's avatar Robbert Krebbers

Add a notion of finite/infinite predicates and define finite/infinite sets using it.

parent ae1dd86c
......@@ -990,17 +990,39 @@ Section set_monad.
End set_monad.
(** Finite sets *)
Definition set_finite `{ElemOf A B} (X : B) := l : list A, x, x X x l.
Definition pred_finite {A} (P : A Prop) := xs : list A, x, P x x xs.
Definition set_finite `{ElemOf A B} (X : B) := pred_finite ( X).
Section finite.
Definition pred_infinite {A} (P : A Prop) := xs : list A, x, P x x xs.
Definition set_infinite `{ElemOf A C} (X : C) := pred_infinite ( X).
Section pred_finite_infinite.
Lemma pred_finite_impl {A} (P Q : A Prop) :
pred_finite P ( x, Q x P x) pred_finite Q.
Proof. unfold pred_finite. set_solver. Qed.
Lemma pred_infinite_impl {A} (P Q : A Prop) :
pred_infinite P ( x, P x Q x) pred_infinite Q.
Proof. unfold pred_infinite. set_solver. Qed.
Lemma pred_not_infinite_finite {A} (P : A Prop) :
pred_infinite P pred_finite P False.
Proof. intros Hinf [xs ?]. destruct (Hinf xs). set_solver. Qed.
End pred_finite_infinite.
Section set_finite_infinite.
Context `{SemiSet A C}.
Implicit Types X Y : C.
Lemma set_not_infinite_finite X : set_infinite X set_finite X False.
Proof. apply pred_not_infinite_finite. Qed.
Global Instance set_finite_subseteq :
Proper (flip () ==> impl) (@set_finite A C _).
Proof. intros X Y HX [l Hl]; exists l; set_solver. Qed.
Proof. intros X Y HX ?. eapply pred_finite_impl; set_solver. Qed.
Global Instance set_finite_proper : Proper (() ==> iff) (@set_finite A C _).
Proof. intros X Y HX; apply exist_proper. by setoid_rewrite HX. Qed.
Lemma empty_finite : set_finite ( : C).
Proof. by exists []; intros ?; rewrite elem_of_empty. Qed.
Lemma singleton_finite (x : A) : set_finite ({[ x ]} : C).
......@@ -1014,7 +1036,18 @@ Section finite.
Proof. intros [l ?]; exists l; set_solver. Qed.
Lemma union_finite_inv_r X Y : set_finite (X Y) set_finite Y.
Proof. intros [l ?]; exists l; set_solver. Qed.
End finite.
Global Instance set_infinite_subseteq :
Proper (() ==> impl) (@set_infinite A C _).
Proof. intros X Y HX ?. eapply pred_infinite_impl; set_solver. Qed.
Global Instance set_infinite_proper : Proper (() ==> iff) (@set_infinite A C _).
Proof. intros X Y HX; apply forall_proper. by setoid_rewrite HX. Qed.
Lemma union_infinite_l X Y : set_infinite X set_infinite (X Y).
Proof. intros Hinf xs. destruct (Hinf xs). set_solver. Qed.
Lemma union_infinite_r X Y : set_infinite Y set_infinite (X Y).
Proof. intros Hinf xs. destruct (Hinf xs). set_solver. Qed.
End set_finite_infinite.
Section more_finite.
Context `{Set_ A C}.
......@@ -1032,6 +1065,10 @@ Section more_finite.
intros [l ?] [k ?]; exists (l ++ k).
intros x ?; destruct (decide (x Y)); rewrite elem_of_app; set_solver.
Qed.
Lemma difference_infinite X Y :
set_infinite X set_finite Y set_infinite (X Y).
Proof. intros Hinf [xs ?] xs'. destruct (Hinf (xs ++ xs')). set_solver. Qed.
End more_finite.
(** Sets of sequences of natural numbers *)
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