Commit 28c2e8e3 authored by Tej Chajed's avatar Tej Chajed Committed by Robbert Krebbers
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Add mnat library for monotonic nat ghost state

Implemented as an algebra for rules about `auth max_natUR` and a
logic-level wrapper for the auth element (a nat) and fragment (a
persistent lower-bound).

Fixes #327.
parent cfbdb375
......@@ -51,6 +51,8 @@ With this release, we dropped support for Coq 8.9.
available everywhere without further changes.
* The authoritative fragment `✓ (◯ b : auth A)` is no longer definitionally
equal to `✓ b`.
* Add `mnat_auth`, a wrapper for `auth max_nat`. The result is an authoritative
`nat` where a fragment is a lower bound whose ownership is persistent.
**Changes in `proofmode`:**
......@@ -105,6 +107,9 @@ With this release, we dropped support for Coq 8.9.
`uPred.discrete_fun_validI` to the new `base_logic.algebra` module. That
module is exported by `base_logic.base_logic` so these names are now usually
available everywhere, and no longer inside the `uPred` module.
* Add an `mnat` library on top of `mnat_auth` that supports ghost state which is
an authoritative, monotonically-increasing `nat` with a proposition giving a
persistent lower bound. See `base_logic.lib.mnat` for further details.
**Changes in `program_logic`:**
......
......@@ -40,6 +40,7 @@ theories/algebra/lib/frac_auth.v
theories/algebra/lib/ufrac_auth.v
theories/algebra/lib/frac_agree.v
theories/algebra/lib/gmap_view.v
theories/algebra/lib/mnat_auth.v
theories/si_logic/siprop.v
theories/si_logic/bi.v
theories/bi/notation.v
......@@ -89,6 +90,7 @@ theories/base_logic/lib/gen_inv_heap.v
theories/base_logic/lib/fancy_updates_from_vs.v
theories/base_logic/lib/proph_map.v
theories/base_logic/lib/ghost_var.v
theories/base_logic/lib/mnat.v
theories/program_logic/adequacy.v
theories/program_logic/lifting.v
theories/program_logic/weakestpre.v
......
From iris.algebra Require Export auth.
From iris.algebra Require Import numbers updates.
From iris Require Import options.
(** Authoritative CMRA for [max_nat]. The authoritative element is a
monotonically increasing [nat], while a fragment is a lower bound. *)
Definition mnat_auth := auth max_natUR.
Definition mnat_authR := authR max_natUR.
Definition mnat_authUR := authUR max_natUR.
Section mnat_auth.
Implicit Types (n : nat).
Definition mnat_auth_auth (q : Qp) (n : nat) : mnat_auth := {q} MaxNat n.
Definition mnat_auth_frag (n : nat) : mnat_auth := MaxNat n.
Global Instance mnat_auth_frag_core_id n : CoreId (mnat_auth_frag n).
Proof. apply _. Qed.
Lemma mnat_auth_auth_frac_op q1 q2 n :
mnat_auth_auth q1 n mnat_auth_auth q2 n mnat_auth_auth (q1 + q2) n.
Proof. rewrite /mnat_auth_auth auth_auth_frac_op //. Qed.
Lemma mnat_auth_auth_op_valid n1 n2 :
(mnat_auth_auth 1 n1 mnat_auth_auth 1 n2) False.
Proof. rewrite auth_auth_op_valid //. Qed.
Lemma mnat_auth_frac_op_valid q1 q2 n1 n2 :
(mnat_auth_auth q1 n1 mnat_auth_auth q2 n2) (q1 + q2)%Qp n1 = n2.
Proof. rewrite auth_auth_frac_op_valid. naive_solver. Qed.
Lemma mnat_auth_frag_op n1 n2 :
mnat_auth_frag n1 mnat_auth_frag n2 = mnat_auth_frag (n1 `max` n2).
Proof. rewrite -auth_frag_op max_nat_op_max //. Qed.
(** rephrasing of [mnat_auth_frag_op] useful for weakening a fragment to a
smaller lower-bound *)
Lemma mnat_auth_frag_op_le_l n n' :
n' n
mnat_auth_frag n = mnat_auth_frag n' mnat_auth_frag n.
Proof.
intros Hle.
rewrite mnat_auth_frag_op.
rewrite Nat.max_r //.
Qed.
Lemma mnat_auth_both_frac_valid q n m :
(mnat_auth_auth q n mnat_auth_frag m) q m n.
Proof.
rewrite auth_both_frac_valid_discrete.
rewrite max_nat_included /=.
naive_solver.
Qed.
Lemma mnat_auth_both_valid n m :
(mnat_auth_auth 1 n mnat_auth_frag m) m n.
Proof. rewrite mnat_auth_both_frac_valid frac_valid'. naive_solver. Qed.
Lemma mnat_auth_auth_valid n : mnat_auth_auth 1 n.
Proof. apply auth_auth_valid; done. Qed.
Lemma mnat_auth_update n n' :
n n'
mnat_auth_auth 1 n ~~> mnat_auth_auth 1 n' mnat_auth_frag n'.
Proof.
intros Hle.
apply auth_update_alloc, max_nat_local_update; done.
Qed.
Lemma mnat_auth_alloc_frag q n :
mnat_auth_auth q n ~~> mnat_auth_auth q n mnat_auth_frag n.
Proof. by apply (auth_update_frac_alloc _ _ _). Qed.
End mnat_auth.
Typeclasses Opaque mnat_auth_auth mnat_auth_frag.
(** Ghost state for a monotonically increasing nat, wrapping the [mnat_authR]
RA. Provides an authoritative proposition [mnat_own_auth γ q n] for the
underlying number [n] and a persistent proposition [mnat_own_lb γ m] witnessing that
the authoritative nat is at least m.
The key rules are [mnat_own_lb_valid], which asserts that an auth at [n] and a
lower-bound at [m] imply that [m ≤ n], and [mnat_update], which allows to
increase the auth element. At any time the auth nat can be "snapshotted" with
[mnat_get_lb] to produce a persistent lower-bound proposition. *)
From iris.proofmode Require Import tactics.
From iris.algebra.lib Require Import mnat_auth.
From iris.bi.lib Require Import fractional.
From iris.base_logic.lib Require Export own.
From iris Require Import options.
Class mnatG Σ :=
MnatG { mnatG_inG :> inG Σ mnat_authR; }.
Definition mnatΣ : gFunctors := #[ GFunctor mnat_authR ].
Instance subG_mnatΣ Σ : subG mnatΣ Σ mnatG Σ.
Proof. solve_inG. Qed.
(** [mnat_own_auth] is the authoritative nat. The definition includes the
fragment at the same value so that [mnat_get_lb] does not require an
update modality. *)
Definition mnat_own_auth `{!mnatG Σ} (γ : gname) (q : Qp) (n : nat) : iProp Σ :=
own γ (mnat_auth_auth q n mnat_auth_frag n).
Definition mnat_own_lb `{!mnatG Σ} (γ : gname) (n : nat): iProp Σ :=
own γ (mnat_auth_frag n).
Section mnat.
Context `{!mnatG Σ}.
Implicit Types (n m : nat).
Global Instance mnat_own_auth_timeless γ q n : Timeless (mnat_own_auth γ q n).
Proof. apply _. Qed.
Global Instance mnat_own_lb_timeless γ n : Timeless (mnat_own_lb γ n).
Proof. apply _. Qed.
Global Instance mnat_own_lb_persistent γ n : Persistent (mnat_own_lb γ n).
Proof. apply _. Qed.
Global Instance mnat_own_auth_fractional γ n : Fractional (λ q, mnat_own_auth γ q n).
Proof.
rewrite /mnat_own_auth. setoid_rewrite own_op.
apply fractional_sep; [|apply _].
intros p q. rewrite -own_op mnat_auth_auth_frac_op //.
Qed.
Global Instance mnat_own_auth_as_fractional γ q n :
AsFractional (mnat_own_auth γ q n) (λ q, mnat_own_auth γ q n) q.
Proof. split; [auto|apply _]. Qed.
Lemma mnat_own_auth_agree γ q1 q2 n1 n2 :
mnat_own_auth γ q1 n1 - mnat_own_auth γ q2 n2 - ⌜✓ (q1 + q2)%Qp n1 = n2.
Proof.
iIntros "[H1 _] [H2 _]".
iDestruct (own_valid_2 with "H1 H2") as %?%mnat_auth_frac_op_valid; done.
Qed.
Lemma mnat_own_auth_exclusive γ n1 n2 :
mnat_own_auth γ 1 n1 - mnat_own_auth γ 1 n2 - False.
Proof.
iIntros "[H1 _] [H2 _]".
iDestruct (own_valid_2 with "H1 H2") as %[]%mnat_auth_auth_op_valid.
Qed.
(** The conclusion of this lemma is persistent; the proofmode will preserve the
[mnat_own_auth] in the premise as long as the conclusion is introduced to the
persistent context, for example with [iDestruct (mnat_get_lb with
"Hauth") as "#Hfrag"]. *)
Lemma mnat_get_lb γ q n :
mnat_own_auth γ q n - mnat_own_lb γ n.
Proof. iIntros "[_ $]". Qed.
Lemma mnat_alloc n :
|==> γ, mnat_own_auth γ 1 n mnat_own_lb γ n.
Proof.
iMod (own_alloc (mnat_auth_auth 1 n mnat_auth_frag n)) as (γ) "Hauth".
{ apply mnat_auth_both_valid; auto. }
iModIntro. iExists γ.
iDestruct (mnat_get_lb with "Hauth") as "#Hlb".
auto.
Qed.
Lemma mnat_own_lb_valid γ q n m :
mnat_own_auth γ q n - mnat_own_lb γ m - ⌜✓ q m n.
Proof.
iIntros "[Hauth _] Hlb".
iDestruct (own_valid_2 with "Hauth Hlb") as %Hvalid%mnat_auth_both_frac_valid.
auto.
Qed.
Lemma mnat_own_lb_le γ n n' :
n' n
mnat_own_lb γ n - mnat_own_lb γ n'.
Proof.
intros Hle.
rewrite /mnat_own_lb.
rewrite (mnat_auth_frag_op_le_l n n') //.
iIntros "[$ _]".
Qed.
Lemma mnat_update n' γ n :
n n'
mnat_own_auth γ 1 n == mnat_own_auth γ 1 n'.
Proof.
iIntros (Hlb) "[Hauth _]".
iMod (own_update with "Hauth") as "$"; [|done].
apply mnat_auth_update; auto.
Qed.
Lemma mnat_update_with_lb γ n n' :
n n'
mnat_own_auth γ 1 n == mnat_own_auth γ 1 n' mnat_own_lb γ n'.
Proof.
iIntros (Hlb) "Hauth".
iMod (mnat_update n' with "Hauth") as "Hauth"; auto.
iDestruct (mnat_get_lb with "Hauth") as "#Hlb".
auto.
Qed.
End mnat.
Typeclasses Opaque mnat_own_auth mnat_own_lb.
......@@ -25,6 +25,14 @@ Section fractional.
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.
Lemma fractional_split P P1 P2 Φ q1 q2 :
AsFractional P Φ (q1 + q2) AsFractional P1 Φ q1 AsFractional P2 Φ q2
P P1 P2.
......
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