Library iris.bi.lib.fractional

From iris.bi Require Export bi.
From iris.proofmode Require Import classes classes_make proofmode.
From iris.prelude Require Import options.

Class Fractional {PROP : bi} (Φ : Qp PROP) :=
  fractional p q : Φ (p + q)%Qp ⊣⊢ Φ p Φ q.
Global Arguments Fractional {_} _%_I : simpl never.
Global Arguments fractional {_ _ _} _ _.

The AsFractional typeclass eta-expands a proposition P into Φ q such that Φ is a fractional predicate. This is needed because higher-order unification cannot be relied upon to guess the right Φ.
AsFractional should generally be used in APIs that work with fractional predicates (instead of Fractional): when the user provides the original resource P, have a premise AsFractional P Φ 1 to convert that into some fractional predicate.
The equivalence in as_fractional should hold definitionally; various typeclass instances assume that Φ q will un-do the eta-expansion performed by AsFractional.
Class AsFractional {PROP : bi} (P : PROP) (Φ : Qp PROP) (q : Qp) := {
  as_fractional : P ⊣⊢ Φ q;
  as_fractional_fractional : Fractional Φ
}.
Global Arguments AsFractional {_} _%_I _%_I _%_Qp.
Global Hint Mode AsFractional - ! - - : typeclass_instances.

The class FrameFractionalQp is used for fractional framing, it subtracts the fractional of the hypothesis from the goal: it computes r := qP - qR. See frame_fractional for how it is used.
Class FrameFractionalQp (qR qP r : Qp) :=
  frame_fractional_qp : qP = (qR + r)%Qp.
Global Hint Mode FrameFractionalQp ! ! - : typeclass_instances.

Section fractional.
  Context {PROP : bi}.
  Implicit Types P Q : PROP.
  Implicit Types Φ : Qp PROP.
  Implicit Types q : Qp.

  Global Instance Fractional_proper :
    Proper (pointwise_relation _ (≡) ==> iff) (@Fractional PROP).
  Proof.
    rewrite /Fractional.
    intros Φ1 Φ2 Hequiv.
    by setoid_rewrite Hequiv.
  Qed.

  Local Instance fractional_as_fractional Φ q :
    Fractional Φ AsFractional (Φ q) Φ q.
  Proof. done. Qed.

This lemma is meant to be used when P is known. But really you should be using iDestruct "H" as "[H1 H2]", which supports splitting at fractions.
  Lemma fractional_split P Φ q1 q2 :
    AsFractional P Φ (q1 + q2)
    P ⊣⊢ Φ q1 Φ q2.
  Proof. by move=>-[-> ->]. Qed.
This lemma is meant to be used when P is known. But really you should be using iDestruct "H" as "[H1 H2]", which supports halving fractions.
  Lemma fractional_half P Φ q :
    AsFractional P Φ q
    P ⊣⊢ Φ (q/2)%Qp Φ (q/2)%Qp.
  Proof. by rewrite -{1}(Qp.div_2 q)=>-[->->]. Qed.
This lemma is meant to be used when P1, P2 are known. But really you should be using iCombine "H1 H2" as "H" (for forwards reasoning) or iSplitL/iSplitR (for goal-oriented reasoning), which support merging fractions.
  Lemma fractional_merge P1 P2 Φ q1 q2 `{!Fractional Φ} :
    AsFractional P1 Φ q1 AsFractional P2 Φ q2
    P1 P2 ⊣⊢ Φ (q1 + q2)%Qp.
  Proof. move=>-[-> _] [-> _]. rewrite fractional //. Qed.

Fractional and logical connectives
  Global Instance persistent_fractional (P : PROP) :
    Persistent P TCOr (Affine P) (Absorbing P) Fractional (λ _, P).
  Proof. intros ?? q q'. apply: bi.persistent_sep_dup. Qed.

We do not have AsFractional instances for and the big operators because the iDestruct tactic already turns P Q into P and Q, [∗ list] kx y :: l, Φ k x into Φ 0 i and [∗ list] kx l, Φ (S k) x, etc. Hence, an AsFractional instance would cause ambiguity because for example l ↦{1} v l' ↦{1} v' could be turned into l ↦{1} v and l' ↦{1} v', or into two times l ↦{1/2} v l' ↦{1/2} v'.
We do provide the Fractional instances so that when one defines a derived connection in terms of or a big operator (and makes that opaque in some way), one could choose to split along the or along the fraction.
  Global Instance fractional_sep Φ Ψ :
    Fractional Φ Fractional Ψ Fractional (λ q, Φ q Ψ q)%I.
  Proof.
    intros ?? q q'. rewrite !fractional -!assoc. f_equiv.
    rewrite !assoc. f_equiv. by rewrite comm.
  Qed.

  Global Instance fractional_embed `{!BiEmbed PROP PROP'} Φ :
    Fractional Φ Fractional (λ q, Φ q : PROP')%I.
  Proof. intros ???. by rewrite fractional embed_sep. Qed.

  Global Instance as_fractional_embed `{!BiEmbed PROP PROP'} P Φ q :
    AsFractional P Φ q AsFractional ( P ) (λ q, Φ q )%I q.
  Proof. intros [??]; split; [by f_equiv|apply _]. Qed.

  Global Instance fractional_big_sepL {A} (l : list A) Ψ :
    ( k x, Fractional (Ψ k x))
    Fractional (PROP:=PROP) (λ q, [∗ list] kx l, Ψ k x q)%I.
  Proof. intros ? q q'. rewrite -big_sepL_sep. by setoid_rewrite fractional. Qed.

  Global Instance fractional_big_sepL2 {A B} (l1 : list A) (l2 : list B) Ψ :
    ( k x1 x2, Fractional (Ψ k x1 x2))
    Fractional (PROP:=PROP) (λ q, [∗ list] kx1; x2 l1; l2, Ψ k x1 x2 q)%I.
  Proof. intros ? q q'. rewrite -big_sepL2_sep. by setoid_rewrite fractional. Qed.

  Global Instance fractional_big_sepM `{Countable K} {A} (m : gmap K A) Ψ :
    ( k x, Fractional (Ψ k x))
    Fractional (PROP:=PROP) (λ q, [∗ map] kx m, Ψ k x q)%I.
  Proof. intros ? q q'. rewrite -big_sepM_sep. by setoid_rewrite fractional. Qed.

  Global Instance fractional_big_sepS `{Countable A} (X : gset A) Ψ :
    ( x, Fractional (Ψ x))
    Fractional (PROP:=PROP) (λ q, [∗ set] x X, Ψ x q)%I.
  Proof. intros ? q q'. rewrite -big_sepS_sep. by setoid_rewrite fractional. Qed.

  Global Instance fractional_big_sepMS `{Countable A} (X : gmultiset A) Ψ :
    ( x, Fractional (Ψ x))
    Fractional (PROP:=PROP) (λ q, [∗ mset] x X, Ψ x q)%I.
  Proof. intros ? q q'. rewrite -big_sepMS_sep. by setoid_rewrite fractional. Qed.

Proof mode instances
  Global Instance from_sep_fractional P Φ q1 q2 :
    AsFractional P Φ (q1 + q2) FromSep P (Φ q1) (Φ q2).
  Proof. rewrite /FromSep=>-[-> ->] //. Qed.
  Global Instance combine_sep_as_fractional P1 P2 Φ q1 q2 :
    AsFractional P1 Φ q1 AsFractional P2 Φ q2
    CombineSepAs P1 P2 (Φ (q1 + q2)%Qp) | 50.
  Proof. rewrite /CombineSepAs =>-[-> _] [-> <-] //. Qed.

  Global Instance from_sep_fractional_half P Φ q :
    AsFractional P Φ q FromSep P (Φ (q / 2)%Qp) (Φ (q / 2)%Qp) | 10.
  Proof. rewrite /FromSep -{1}(Qp.div_2 q). intros [-> <-]. rewrite Qp.div_2 //. Qed.
  Global Instance combine_sep_as_fractional_half P Φ q :
    AsFractional P Φ (q/2) CombineSepAs P P (Φ q) | 50.
  Proof. rewrite /CombineSepAs=>-[-> <-]. by rewrite Qp.div_2. Qed.

  Global Instance into_sep_fractional P Φ q1 q2 :
    AsFractional P Φ (q1 + q2) IntoSep P (Φ q1) (Φ q2).
  Proof. intros [??]. rewrite /IntoSep [P]fractional_split //. Qed.

  Global Instance into_sep_fractional_half P Φ q :
    AsFractional P Φ q IntoSep P (Φ (q / 2)%Qp) (Φ (q / 2)%Qp) | 100.
  Proof. intros [??]. rewrite /IntoSep [P]fractional_half //. Qed.

  Global Instance frame_fractional_qp_add_l q q' : FrameFractionalQp q (q + q') q'.
  Proof. by rewrite /FrameFractionalQp. Qed.
  Global Instance frame_fractional_qp_add_r q q' : FrameFractionalQp q' (q + q') q.
  Proof. by rewrite /FrameFractionalQp Qp.add_comm. Qed.
  Global Instance frame_fractional_qp_half q : FrameFractionalQp (q/2) q (q/2).
  Proof. by rewrite /FrameFractionalQp Qp.div_2. Qed.

  Lemma frame_fractional Φ p R P qR qP r :
    AsFractional R Φ qR
    AsFractional P Φ qP
    FrameFractionalQp qR qP r
    Frame p R P (Φ r).
  Proof.
    rewrite /Frame /FrameFractionalQp⇒ -[-> _] [-> ?] →.
    by rewrite bi.intuitionistically_if_elim fractional.
  Qed.
End fractional.

Marked tc_opaque instead Typeclasses Opaque so that you can use iDestruct to eliminate and iModIntro to introduce internal_fractional, while still preventing iFrame and iNext from unfolding.
Definition internal_fractional {PROP : bi} (Φ : Qp PROP) : PROP :=
  tc_opaque ( p q, Φ (p + q)%Qp ∗-∗ Φ p Φ q)%I.
Global Instance: Params (@internal_fractional) 1 := {}.

Section internal_fractional.
  Context {PROP : bi}.
  Implicit Types Φ Ψ : Qp PROP.
  Implicit Types q : Qp.

  Global Instance internal_fractional_ne n :
    Proper (pointwise_relation _ (dist n) ==> dist n) (@internal_fractional PROP).
  Proof. solve_proper. Qed.
  Global Instance internal_fractional_proper :
    Proper (pointwise_relation _ (≡) ==> (≡)) (@internal_fractional PROP).
  Proof. solve_proper. Qed.

  Global Instance internal_fractional_affine Φ : Affine (internal_fractional Φ).
  Proof. rewrite /internal_fractional /=. apply _. Qed.
  Global Instance internal_fractional_persistent Φ :
    Persistent (internal_fractional Φ).
  Proof. rewrite /internal_fractional /=. apply _. Qed.

  Lemma fractional_internal_fractional Φ :
    Fractional Φ internal_fractional Φ.
  Proof.
    intros. iIntros "!>" (q1 q2).
    rewrite [Φ (q1 + q2)%Qp]fractional //=; auto.
  Qed.

  Lemma internal_fractional_iff Φ Ψ:
     ( q, Φ q ∗-∗ Ψ q) internal_fractional Φ -∗ internal_fractional Ψ.
  Proof.
    iIntros "#Hiff #HΦdup !>" (p q).
    iSplit.
    - iIntros "H".
      iDestruct ("Hiff" with "H") as "HΦ".
      iDestruct ("HΦdup" with "HΦ") as "[H1 ?]".
      iSplitL "H1"; iApply "Hiff"; iFrame.
    - iIntros "[H1 H2]".
      iDestruct ("Hiff" with "H1") as "HΦ1".
      iDestruct ("Hiff" with "H2") as "HΦ2".
      iApply "Hiff".
      iApply "HΦdup".
      iFrame.
  Qed.
End internal_fractional.