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 {_ _ _} _ _.
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.
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 substracts
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.
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.
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.
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.
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.
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] k↦x ∈ y :: l, Φ k x into Φ 0 i and [∗ list] k↦x ∈ 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] k↦x ∈ 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] k↦x1; 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] k↦x ∈ 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.
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] k↦x ∈ 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] k↦x1; 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] k↦x ∈ 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.
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.
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.