clutch.eris.array_laws
This file extends the Clutch program logic with some derived laws (not
using the lifting lemmas) about arrays
For utility functions on arrays (e.g., freeing/copying an array), see
lib.array.
From stdpp Require Import fin_maps.
From iris.bi Require Import lib.fractional.
From iris.proofmode Require Import proofmode.
From clutch.prob_lang Require Import tactics lang notation.
From clutch.eris Require Export primitive_laws.
From iris.prelude Require Import options.
The array connective is a version of pointsto that works
with lists of values.
Definition array `{!erisGS Σ} (l : loc) (dq : dfrac) (vs : list val) : iProp Σ :=
[∗ list] i ↦ v ∈ vs, (l +ₗ i) ↦{dq} v.
(*
Notation "l ↦∗{ dq } vs" := (array l dq vs)
(at level 20, dq custom dfrac at level 1, format "l ↦∗{ dq } vs") : bi_scope.
*)
Notation "l ↦∗{ dq } vs" := (array l dq vs)
(at level 20, format "l ↦∗{ dq } vs") : bi_scope.
Notation "l ↦∗ v" := (l ↦∗{ DfracOwn 1 } v)%I
(at level 20, format "l ↦∗ v") : bi_scope.
We have FromSep and IntoSep instances to split the fraction (via the
AsFractional instance below), but not for splitting the list, as that would
lead to overlapping instances.
Section array_lemmas.
Context `{!erisGS Σ}.
Implicit Types P Q : iProp Σ.
Implicit Types Φ Ψ : val → iProp Σ.
Implicit Types σ : state.
Implicit Types v : val.
Implicit Types l : loc.
Implicit Types vs : list val.
Implicit Types sz off : nat.
(*
Global Instance array_timeless l q vs : Timeless (array l q vs) := _.
Global Instance array_fractional l vs : Fractional (λ q, l ↦∗{q} vs)%I := _. Global Instance array_as_fractional l q vs : AsFractional (l ↦∗{q} vs) (λ q, l ↦∗{q} vs)%I q. Proof. split; done || apply _. Qed. *)
Lemma array_nil l dq : l ↦∗{dq} [] ⊣⊢ emp.
Proof. by rewrite /array. Qed.
Lemma array_singleton l dq v : l ↦∗{dq} [v] ⊣⊢ l ↦{dq} v.
Proof. by rewrite /array /= right_id loc_add_0. Qed.
Lemma array_app l dq vs ws :
l ↦∗{dq} (vs ++ ws) ⊣⊢ l ↦∗{dq} vs ∗ (l +ₗ length vs) ↦∗{dq} ws.
Proof.
rewrite /array big_sepL_app.
setoid_rewrite Nat2Z.inj_add.
by setoid_rewrite loc_add_assoc.
Qed.
Lemma array_cons l dq v vs : l ↦∗{dq} (v :: vs) ⊣⊢ l ↦{dq} v ∗ (l +ₗ 1) ↦∗{dq} vs.
Proof.
rewrite /array big_sepL_cons loc_add_0.
setoid_rewrite loc_add_assoc.
setoid_rewrite Nat2Z.inj_succ.
by setoid_rewrite Z.add_1_l.
Qed.
Global Instance array_cons_frame l dq v vs R Q :
Frame false R (l ↦{dq} v ∗ (l +ₗ 1) ↦∗{dq} vs) Q →
Frame false R (l ↦∗{dq} (v :: vs)) Q | 2.
Proof. by rewrite /Frame array_cons. Qed.
Lemma update_array l dq vs off v :
vs !! off = Some v →
⊢ l ↦∗{dq} vs -∗ ((l +ₗ off) ↦{dq} v ∗ ∀ v', (l +ₗ off) ↦{dq} v' -∗ l ↦∗{dq} <[off:=v']>vs).
Proof.
iIntros (Hlookup) "Hl".
rewrite -[X in (l ↦∗{_} X)%I](take_drop_middle _ off v); last done.
iDestruct (array_app with "Hl") as "[Hl1 Hl]".
iDestruct (array_cons with "Hl") as "[Hl2 Hl3]".
assert (off < length vs) as H by (apply lookup_lt_is_Some; by eexists).
rewrite length_take min_l; last by lia. iFrame "Hl2".
iIntros (w) "Hl2".
clear Hlookup. assert (<[off:=w]> vs !! off = Some w) as Hlookup.
{ apply list_lookup_insert_eq. lia. }
rewrite -[in (l ↦∗{_} <[off:=w]> vs)%I](take_drop_middle (<[off:=w]> vs) off w Hlookup).
iApply array_app. rewrite take_insert_ge; last by lia. iFrame.
iApply array_cons. rewrite length_take min_l; last by lia. iFrame.
rewrite drop_insert_lt; last by lia. done.
Qed.
Lemma pointsto_seq_array l dq v n :
([∗ list] i ∈ seq 0 n, (l +ₗ (i : nat)) ↦{dq} v) -∗
l ↦∗{dq} replicate n v.
Proof.
rewrite /array. iInduction n as [|n'] "IH" forall (l); simpl.
{ done. }
iIntros "[$ Hl]". rewrite -fmap_S_seq big_sepL_fmap.
setoid_rewrite Nat2Z.inj_succ. setoid_rewrite <-Z.add_1_l.
setoid_rewrite <-loc_add_assoc. iApply "IH". done.
Qed.
End array_lemmas.
([∗ list] i ∈ seq 0 n, (l +ₗ (i : nat)) ↦{dq} v) -∗
l ↦∗{dq} replicate n v.
Proof.
rewrite /array. iInduction n as [|n'] "IH" forall (l); simpl.
{ done. }
iIntros "[$ Hl]". rewrite -fmap_S_seq big_sepL_fmap.
setoid_rewrite Nat2Z.inj_succ. setoid_rewrite <-Z.add_1_l.
setoid_rewrite <-loc_add_assoc. iApply "IH". done.
Qed.
End array_lemmas.