(** This file provides a version of the fractional camera whose elements are in the internal (0,1] of the rational numbers. Notice that this camera could in principle be obtained by restricting the validity of the unbounded fractional camera [ufrac]. *) From Coq.QArith Require Import Qcanon. From iris.algebra Require Export cmra. From iris.algebra Require Import proofmode_classes. Set Default Proof Using "Type". (** Since the standard (0,1] fractional camera is used more often, we define [frac] through a [Notation] instead of a [Definition]. That way, Coq infers the [frac] camera by default when using the [Qp] type. *) Notation frac := Qp (only parsing). Section frac. Canonical Structure fracC := leibnizC frac. Instance frac_valid : Valid frac := λ x, (x ≤ 1)%Qc. Instance frac_pcore : PCore frac := λ _, None. Instance frac_op : Op frac := λ x y, (x + y)%Qp. Lemma frac_included (x y : frac) : x ≼ y ↔ (x < y)%Qc. Proof. by rewrite Qp_lt_sum. Qed. Corollary frac_included_weak (x y : frac) : x ≼ y → (x ≤ y)%Qc. Proof. intros ?%frac_included. auto using Qclt_le_weak. Qed. Definition frac_ra_mixin : RAMixin frac. Proof. split; try apply _; try done. unfold valid, op, frac_op, frac_valid. intros x y. trans (x+y)%Qp; last done. rewrite -{1}(Qcplus_0_r x) -Qcplus_le_mono_l; auto using Qclt_le_weak. Qed. Canonical Structure fracR := discreteR frac frac_ra_mixin. Global Instance frac_cmra_discrete : CmraDiscrete fracR. Proof. apply discrete_cmra_discrete. Qed. End frac. Global Instance frac_full_exclusive : Exclusive 1%Qp. Proof. move=> y /Qcle_not_lt [] /=. by rewrite -{1}(Qcplus_0_r 1) -Qcplus_lt_mono_l. Qed. Global Instance frac_cancelable (q : frac) : Cancelable q. Proof. intros ?????. by apply Qp_eq, (inj (Qcplus q)), (Qp_eq (q+y) (q+z))%Qp. Qed. Global Instance frac_id_free (q : frac) : IdFree q. Proof. intros [q0 Hq0] ? EQ%Qp_eq. rewrite -{1}(Qcplus_0_r q) in EQ. eapply Qclt_not_eq; first done. by apply (inj (Qcplus q)). Qed. Lemma frac_op' (q p : Qp) : (p ⋅ q) = (p + q)%Qp. Proof. done. Qed. Lemma frac_valid' (p : Qp) : ✓ p ↔ (p ≤ 1%Qp)%Qc. Proof. done. Qed. Global Instance is_op_frac q : IsOp' q (q/2)%Qp (q/2)%Qp. Proof. by rewrite /IsOp' /IsOp frac_op' Qp_div_2. Qed.