(* Copyright (c) 2012-2017, Coq-std++ developers. *) (* This file is distributed under the terms of the BSD license. *) (** This file implements sets as functions into Prop. *) From stdpp Require Export collections. Set Default Proof Using "Type". Record set (A : Type) : Type := mkSet { set_car : A → Prop }. Add Printing Constructor set. Arguments mkSet {_} _ : assert. Arguments set_car {_} _ _ : assert. Notation "{[ x | P ]}" := (mkSet (λ x, P)) (at level 1, format "{[ x | P ]}") : stdpp_scope. Instance set_elem_of {A} : ElemOf A (set A) := λ x X, set_car X x. Instance set_top {A} : Top (set A) := {[ _ | True ]}. Instance set_empty {A} : Empty (set A) := {[ _ | False ]}. Instance set_singleton {A} : Singleton A (set A) := λ y, {[ x | y = x ]}. Instance set_union {A} : Union (set A) := λ X1 X2, {[ x | x ∈ X1 ∨ x ∈ X2 ]}. Instance set_intersection {A} : Intersection (set A) := λ X1 X2, {[ x | x ∈ X1 ∧ x ∈ X2 ]}. Instance set_difference {A} : Difference (set A) := λ X1 X2, {[ x | x ∈ X1 ∧ x ∉ X2 ]}. Instance set_collection : Collection A (set A). Proof. split; [split | |]; by repeat intro. Qed. Lemma elem_of_top {A} (x : A) : x ∈ (⊤ : set A) ↔ True. Proof. done. Qed. Lemma elem_of_mkSet {A} (P : A → Prop) x : x ∈ {[ x | P x ]} ↔ P x. Proof. done. Qed. Lemma not_elem_of_mkSet {A} (P : A → Prop) x : x ∉ {[ x | P x ]} ↔ ¬P x. Proof. done. Qed. Lemma top_subseteq {A} (X : set A) : X ⊆ ⊤. Proof. done. Qed. Hint Resolve top_subseteq. Instance set_ret : MRet set := λ A (x : A), {[ x ]}. Instance set_bind : MBind set := λ A B (f : A → set B) (X : set A), mkSet (λ b, ∃ a, b ∈ f a ∧ a ∈ X). Instance set_fmap : FMap set := λ A B (f : A → B) (X : set A), {[ b | ∃ a, b = f a ∧ a ∈ X ]}. Instance set_join : MJoin set := λ A (XX : set (set A)), {[ a | ∃ X : set A, a ∈ X ∧ X ∈ XX ]}. Instance set_collection_monad : CollectionMonad set. Proof. by split; try apply _. Qed. Instance set_unfold_set_all {A} (x : A) : SetUnfold (x ∈ (⊤ : set A)) True. Proof. by constructor. Qed. Instance set_unfold_mkSet {A} (P : A → Prop) x Q : SetUnfoldSimpl (P x) Q → SetUnfold (x ∈ mkSet P) Q. Proof. intros HPQ. constructor. apply HPQ. Qed. Global Opaque set_elem_of set_top set_empty set_singleton. Global Opaque set_union set_intersection set_difference. Global Opaque set_ret set_bind set_fmap set_join.