(* Copyright (c) 2012-2015, Robbert Krebbers. *) (* This file is distributed under the terms of the BSD license. *) (** This file implements bsets as functions into Prop. *) From iris.prelude Require Export prelude. Record bset (A : Type) : Type := mkBSet { bset_car : A → bool }. Arguments mkBSet {_} _. Arguments bset_car {_} _ _. Instance bset_top {A} : Top (bset A) := mkBSet (λ _, true). Instance bset_empty {A} : Empty (bset A) := mkBSet (λ _, false). Instance bset_singleton {A} `{∀ x y : A, Decision (x = y)} : Singleton A (bset A) := λ x, mkBSet (λ y, bool_decide (y = x)). Instance bset_elem_of {A} : ElemOf A (bset A) := λ x X, bset_car X x. Instance bset_union {A} : Union (bset A) := λ X1 X2, mkBSet (λ x, bset_car X1 x || bset_car X2 x). Instance bset_intersection {A} : Intersection (bset A) := λ X1 X2, mkBSet (λ x, bset_car X1 x && bset_car X2 x). Instance bset_difference {A} : Difference (bset A) := λ X1 X2, mkBSet (λ x, bset_car X1 x && negb (bset_car X2 x)). Instance bset_collection {A} `{∀ x y : A, Decision (x = y)} : Collection A (bset A). Proof. split; [split| |]. - by intros x ?. - by intros x y; rewrite <-(bool_decide_spec (x = y)). - split. apply orb_prop_elim. apply orb_prop_intro. - split. apply andb_prop_elim. apply andb_prop_intro. - intros X Y x; unfold elem_of, bset_elem_of; simpl. destruct (bset_car X x), (bset_car Y x); simpl; tauto. Qed. Instance bset_elem_of_dec {A} x (X : bset A) : Decision (x ∈ X) := _. Typeclasses Opaque bset_elem_of. Global Opaque bset_empty bset_singleton bset_union bset_intersection bset_difference.