Library iris.base_logic.lib.mono_nat

Ghost state for a monotonically increasing nat, wrapping the mono_natR RA. Provides an authoritative proposition mono_nat_auth_own γ q n for the underlying number n and a persistent proposition mono_nat_lb_own γ m witnessing that the authoritative nat is at least m.
The key rules are mono_nat_lb_own_valid, which asserts that an auth at n and a lower-bound at m imply that m n, and mono_nat_update, which allows to increase the auth element. At any time the auth nat can be "snapshotted" with mono_nat_get_lb to produce a persistent lower-bound proposition.
From iris.proofmode Require Import proofmode.
From iris.algebra.lib Require Import mono_nat.
From Require Import fractional.
From iris.base_logic.lib Require Export own.
From iris.prelude Require Import options.

Class mono_natG Σ :=
  MonoNatG { mono_natG_inG : inG Σ mono_natR; }.
Local Existing Instance mono_natG_inG.

Definition mono_natΣ : gFunctors := #[ GFunctor mono_natR ].
Global Instance subG_mono_natΣ Σ : subG mono_natΣ Σ mono_natG Σ.
Proof. solve_inG. Qed.

Local Definition mono_nat_auth_own_def `{!mono_natG Σ}
    (γ : gname) (q : Qp) (n : nat) : iProp Σ :=
  own γ (MN{#q} n).
Local Definition mono_nat_auth_own_aux : seal (@mono_nat_auth_own_def).
Proof. by eexists. Qed.
Definition mono_nat_auth_own := mono_nat_auth_own_aux.(unseal).
Local Definition mono_nat_auth_own_unseal :
  @mono_nat_auth_own = @mono_nat_auth_own_def := mono_nat_auth_own_aux.(seal_eq).
Global Arguments mono_nat_auth_own {Σ _} γ q n.

Local Definition mono_nat_lb_own_def `{!mono_natG Σ} (γ : gname) (n : nat): iProp Σ :=
  own γ (MN n).
Local Definition mono_nat_lb_own_aux : seal (@mono_nat_lb_own_def). Proof. by eexists. Qed.
Definition mono_nat_lb_own := mono_nat_lb_own_aux.(unseal).
Local Definition mono_nat_lb_own_unseal :
  @mono_nat_lb_own = @mono_nat_lb_own_def := mono_nat_lb_own_aux.(seal_eq).
Global Arguments mono_nat_lb_own {Σ _} γ n.

Local Ltac unseal := rewrite
  ?mono_nat_auth_own_unseal /mono_nat_auth_own_def
  ?mono_nat_lb_own_unseal /mono_nat_lb_own_def.

Section mono_nat.
  Context `{!mono_natG Σ}.
  Implicit Types (n m : nat).

  Global Instance mono_nat_auth_own_timeless γ q n : Timeless (mono_nat_auth_own γ q n).
  Proof. unseal. apply _. Qed.
  Global Instance mono_nat_lb_own_timeless γ n : Timeless (mono_nat_lb_own γ n).
  Proof. unseal. apply _. Qed.
  Global Instance mono_nat_lb_own_persistent γ n : Persistent (mono_nat_lb_own γ n).
  Proof. unseal. apply _. Qed.

  Global Instance mono_nat_auth_own_fractional γ n :
    Fractional (λ q, mono_nat_auth_own γ q n).
  Proof. unseal. intros p q. rewrite -own_op -mono_nat_auth_dfrac_op //. Qed.
  Global Instance mono_nat_auth_own_as_fractional γ q n :
    AsFractional (mono_nat_auth_own γ q n) (λ q, mono_nat_auth_own γ q n) q.
  Proof. split; [auto|apply _]. Qed.

  Lemma mono_nat_auth_own_agree γ q1 q2 n1 n2 :
    mono_nat_auth_own γ q1 n1 -∗
    mono_nat_auth_own γ q2 n2 -∗
    (q1 + q2 1)%Qp n1 = n2.
    unseal. iIntros "H1 H2".
    iCombine "H1 H2" gives %?%mono_nat_auth_dfrac_op_valid; done.
  Lemma mono_nat_auth_own_exclusive γ n1 n2 :
    mono_nat_auth_own γ 1 n1 -∗ mono_nat_auth_own γ 1 n2 -∗ False.
    iIntros "H1 H2".
    by iDestruct (mono_nat_auth_own_agree with "H1 H2") as %[[] _].

  Lemma mono_nat_lb_own_valid γ q n m :
    mono_nat_auth_own γ q n -∗ mono_nat_lb_own γ m -∗ (q 1)%Qp m n.
    unseal. iIntros "Hauth Hlb".
    iCombine "Hauth Hlb" gives %Hvalid%mono_nat_both_dfrac_valid.

The conclusion of this lemma is persistent; the proofmode will preserve the mono_nat_auth_own in the premise as long as the conclusion is introduced to the persistent context, for example with iDestruct (mono_nat_lb_own_get with "Hauth") as "#Hfrag".
  Lemma mono_nat_lb_own_get γ q n :
    mono_nat_auth_own γ q n -∗ mono_nat_lb_own γ n.
  Proof. unseal. iApply own_mono. apply mono_nat_included. Qed.

  Lemma mono_nat_lb_own_le {γ n} n' :
    n' n
    mono_nat_lb_own γ n -∗ mono_nat_lb_own γ n'.
  Proof. unseal. intros. iApply own_mono. by apply mono_nat_lb_mono. Qed.

  Lemma mono_nat_lb_own_0 γ :
     |==> mono_nat_lb_own γ 0.
  Proof. unseal. iApply own_unit. Qed.

  Lemma mono_nat_own_alloc_strong P n :
    pred_infinite P
     |==> γ, P γ mono_nat_auth_own γ 1 n mono_nat_lb_own γ n.
    unseal. intros.
    iMod (own_alloc_strong (MN n MN n) P) as (γ) "[% [??]]"; first done.
    { apply mono_nat_both_valid; auto. }
    auto with iFrame.
  Lemma mono_nat_own_alloc n :
     |==> γ, mono_nat_auth_own γ 1 n mono_nat_lb_own γ n.
    iMod (mono_nat_own_alloc_strong (λ _, True) n) as (γ) "[_ ?]".
    - by apply pred_infinite_True.
    - eauto.

  Lemma mono_nat_own_update {γ n} n' :
    n n'
    mono_nat_auth_own γ 1 n ==∗ mono_nat_auth_own γ 1 n' mono_nat_lb_own γ n'.
    iIntros (?) "Hauth".
    iAssert (mono_nat_auth_own γ 1 n') with "[> Hauth]" as "Hauth".
    { unseal. iApply (own_update with "Hauth"). by apply mono_nat_update. }
    iModIntro. iSplit; [done|]. by iApply mono_nat_lb_own_get.
End mono_nat.