Library iris.base_logic.bupd_alt

From iris.proofmode Require Import tactics.
From iris.base_logic Require Export base_logic.

This file contains an alternative version of basic updates, that is expression in terms of just the plain modality .
Definition bupd_alt `{BiPlainly PROP} (P : PROP) : PROP :=
  ( R, (P -∗ R) -∗ R)%I.

This definition is stated for any BI with a plain modality. The above definition is akin to the continuation monad, where one should think of R being the final result that one wants to get out of the basic update in the end of the day (via bupd_alt (■ P) P).
We show that:
1. bupd_alt enjoys the usual rules of the basic update modality. 2. bupd_alt entails any other modality that enjoys the laws of a basic update modality (see bupd_bupd_alt). 3. The ordinary basic update modality |==> on uPred entails bupd_alt (see bupd_alt_bupd). This result is proven in the model of uPred.
The first two points are shown for any BI with a plain modality.

Section bupd_alt.
  Context `{BiPlainly PROP}.
  Implicit Types P Q R : PROP.
  Notation bupd_alt := (@bupd_alt PROP _).

  Global Instance bupd_alt_ne : NonExpansive bupd_alt.
  Proof. solve_proper. Qed.
  Global Instance bupd_alt_proper : Proper ((≡) ==> (≡)) bupd_alt.
  Proof. solve_proper. Qed.
  Global Instance bupd_alt_mono' : Proper ((⊢) ==> (⊢)) bupd_alt.
  Proof. solve_proper. Qed.
  Global Instance bupd_alt_flip_mono' : Proper (flip (⊢) ==> flip (⊢)) bupd_alt.
  Proof. solve_proper. Qed.

The laws of the basic update modality hold
  Lemma bupd_alt_intro P : P bupd_alt P.
  Proof. iIntros "HP" (R) "H". by iApply "H". Qed.
  Lemma bupd_alt_mono P Q : (P Q) bupd_alt P bupd_alt Q.
  Proof. by intros →. Qed.
  Lemma bupd_alt_trans P : bupd_alt (bupd_alt P) bupd_alt P.
  Proof. iIntros "HP" (R) "H". iApply "HP". iIntros "HP". by iApply "HP". Qed.
  Lemma bupd_alt_frame_r P Q : bupd_alt P Q bupd_alt (P Q).
  Proof.
    iIntros "[HP HQ]" (R) "H". iApply "HP". iIntros "HP". iApply ("H" with "[$]").
  Qed.
  Lemma bupd_alt_plainly P : bupd_alt ( P) P.
  Proof. iIntros "H". iApply ("H" $! P with "[]"); auto. Qed.

Any modality conforming with BiBUpdPlainly entails the alternative definition
  Lemma bupd_bupd_alt `{!BiBUpd PROP, BiBUpdPlainly PROP} P : (|==> P) bupd_alt P.
  Proof. iIntros "HP" (R) "H". by iMod ("H" with "HP") as "?". Qed.

We get the usual rule for frame preserving updates if we have an own connective satisfying the following rule w.r.t. interaction with plainly.
  Context {M : ucmraT} (own : M PROP).
  Context (own_updateP_plainly : x Φ R,
    x ~~>: Φ
    own x ( y, Φ y -∗ own y -∗ R) R).

  Lemma own_updateP x (Φ : M Prop) :
    x ~~>: Φ own x bupd_alt ( y, Φ y own y).
  Proof.
    iIntros (Hup) "Hx"; iIntros (R) "H".
    iApply (own_updateP_plainly with "[$Hx H]"); first done.
    iIntros (y ?) "Hy". iApply "H"; auto.
  Qed.
End bupd_alt.

The alternative definition entails the ordinary basic update
Lemma bupd_alt_bupd {M} (P : uPred M) : bupd_alt P |==> P.
Proof.
  rewrite /bupd_alt. uPred.unseal; splitn x Hx H k y ? Hxy.
  unshelve refine (H {| uPred_holds k _ :=
     x' : M, ✓{k} (x' y) P k x' |} k y _ _ _).
  - intros n1 n2 x1 x2 (z&?&?) _ ?.
    eauto using cmra_validN_le, uPred_mono.
  - done.
  - done.
  - intros k' z ?? HP. z. by rewrite (comm op).
Qed.

Lemma bupd_alt_bupd_iff {M} (P : uPred M) : bupd_alt P ⊣⊢ |==> P.
Proof. apply (anti_symm _). apply bupd_alt_bupd. apply bupd_bupd_alt. Qed.

The law about the interaction between uPred_ownM and plainly holds.
Lemma ownM_updateP {M : ucmraT} x (Φ : M Prop) (R : uPred M) :
  x ~~>: Φ
  uPred_ownM x ( y, Φ y -∗ uPred_ownM y -∗ R) R.
Proof.
  uPred.unsealHup; split; intros n z Hv (?&z2&?&[z1 ?]&HR); ofe_subst.
  destruct (Hup n (Some (z1 z2))) as (y&?&?); simpl in ×.
  { by rewrite assoc. }
  refine (HR y n z1 _ _ _ n y _ _ _); auto.
  - rewrite comm. by eapply cmra_validN_op_r.
  - by rewrite (comm _ _ y) (comm _ z2).
  - apply (reflexivity (R:=includedN _)).
Qed.