Library iris.base_logic.lib.fancy_updates

From stdpp Require Export coPset.
From iris.algebra Require Import gmap auth agree gset coPset.
From iris.proofmode Require Import proofmode.
From iris.base_logic.lib Require Export own.
From iris.base_logic.lib Require Import wsat.
From iris.base_logic Require Export later_credits.
From iris.prelude Require Import options.
Export wsatGS.
Import uPred.
Import le_upd_if.

The definition of fancy updates (and in turn the logic built on top of it) is parameterized by whether it supports elimination of laters via later credits or not. This choice is necessary as the fancy update *with* later credits does *not* support the interaction laws with the plainly modality in BiFUpdPlainly. While these laws are seldomly used, support for them is required for backwards compatibility.
Thus, the invGS_gen typeclass ("gen" for "generalized") is parameterized by a parameter of type has_lc that determines whether later credits are available or not. invGS is provided as a convenient notation for the default HasLc. We don't use that notation in this file to avoid confusion.
Inductive has_lc := HasLc | HasNoLc.

Class invGpreS (Σ : gFunctors) : Set := InvGpreS {
  invGpreS_wsat : wsatGpreS Σ;
  invGpreS_lc : lcGpreS Σ;
}.

Class invGS_gen (hlc : has_lc) (Σ : gFunctors) : Set := InvG {
  invGS_wsat : wsatGS Σ;
  invGS_lc : lcGS Σ;
}.
Global Hint Mode invGS_gen - - : typeclass_instances.
Global Hint Mode invGpreS - : typeclass_instances.
Local Existing Instances invGpreS_wsat invGpreS_lc.
Global Existing Instances invGS_lc invGS_wsat.

Notation invGS := (invGS_gen HasLc).

Definition invΣ : gFunctors :=
  #[wsatΣ; lcΣ].
Global Instance subG_invΣ {Σ} : subG invΣ Σ invGpreS Σ.
Proof. solve_inG. Qed.

Local Definition uPred_fupd_def `{!invGS_gen hlc Σ} (E1 E2 : coPset) (P : iProp Σ) : iProp Σ :=
  wsat ownE E1 -∗ le_upd_if (if hlc is HasLc then true else false) ( (wsat ownE E2 P)).
Local Definition uPred_fupd_aux : seal (@uPred_fupd_def). Proof. by eexists. Qed.
Definition uPred_fupd := uPred_fupd_aux.(unseal).
Global Arguments uPred_fupd {hlc Σ _}.
Local Lemma uPred_fupd_unseal `{!invGS_gen hlc Σ} : @fupd _ uPred_fupd = uPred_fupd_def.
Proof. rewrite -uPred_fupd_aux.(seal_eq) //. Qed.

Lemma uPred_fupd_mixin `{!invGS_gen hlc Σ} : BiFUpdMixin (uPredI (iResUR Σ)) uPred_fupd.
Proof.
  split.
  - rewrite uPred_fupd_unseal. solve_proper.
  - intros E1 E2 (E1''&->&?)%subseteq_disjoint_union_L.
    rewrite uPred_fupd_unseal /uPred_fupd_def ownE_op //.
    by iIntros "($ & $ & HE) !> !> [$ $] !> !>".
  - rewrite uPred_fupd_unseal.
    iIntros (E1 E2 P) ">H [Hw HE]". iApply "H"; by iFrame.
  - rewrite uPred_fupd_unseal.
    iIntros (E1 E2 P Q HPQ) "HP HwE". rewrite -HPQ. by iApply "HP".
  - rewrite uPred_fupd_unseal. iIntros (E1 E2 E3 P) "HP HwE".
    iMod ("HP" with "HwE") as ">(Hw & HE & HP)". iApply "HP"; by iFrame.
  - intros E1 E2 Ef P HE1Ef. rewrite uPred_fupd_unseal /uPred_fupd_def ownE_op //.
    iIntros "Hvs (Hw & HE1 &HEf)".
    iMod ("Hvs" with "[Hw HE1]") as ">($ & HE2 & HP)"; first by iFrame.
    iDestruct (ownE_op' with "[HE2 HEf]") as "[? $]"; first by iFrame.
    iIntros "!> !>". by iApply "HP".
  - rewrite uPred_fupd_unseal /uPred_fupd_def. by iIntros (????) "[HwP $]".
Qed.
Global Instance uPred_bi_fupd `{!invGS_gen hlc Σ} : BiFUpd (uPredI (iResUR Σ)) :=
  {| bi_fupd_mixin := uPred_fupd_mixin |}.

Global Instance uPred_bi_bupd_fupd `{!invGS_gen hlc Σ} : BiBUpdFUpd (uPredI (iResUR Σ)).
Proof. rewrite /BiBUpdFUpd uPred_fupd_unseal. by iIntros (E P) ">? [$ $] !> !>". Qed.

The interaction laws with the plainly modality are only supported when we opt out of the support for later credits.
Global Instance uPred_bi_fupd_plainly_no_lc `{!invGS_gen HasNoLc Σ} :
  BiFUpdPlainly (uPredI (iResUR Σ)).
Proof.
  split; rewrite uPred_fupd_unseal /uPred_fupd_def.
  - iIntros (E P) "H [Hw HE]".
    iAssert ( P)%I as "#>HP".
    { by iMod ("H" with "[$]") as "(_ & _ & HP)". }
    by iFrame.
  - iIntros (E P Q) "[H HQ] [Hw HE]".
    iAssert ( P)%I as "#>HP".
    { by iMod ("H" with "HQ [$]") as "(_ & _ & HP)". }
    by iFrame.
  - iIntros (E P) "H [Hw HE]".
    iAssert ( P)%I as "#HP".
    { iNext. by iMod ("H" with "[$]") as "(_ & _ & HP)". }
    iFrame. iIntros "!> !> !>". by iMod "HP".
  - iIntros (E A Φ) "HΦ [Hw HE]".
    iAssert ( x : A, Φ x)%I as "#>HP".
    { iIntros (x). by iMod ("HΦ" with "[$Hw $HE]") as "(_&_&?)". }
    by iFrame.
Qed.

Later credits: the laws are only available when we opt into later credit support.
lc_fupd_elim_later allows to eliminate a later from a hypothesis at an update. This is typically used as iMod (lc_fupd_elim_later with "Hcredit HP") as "HP"., where "Hcredit" is a credit available in the context and "HP" is the assumption from which a later should be stripped.
Lemma lc_fupd_elim_later `{!invGS_gen HasLc Σ} E P :
   £ 1 -∗ ( P) -∗ |={E}=> P.
Proof.
  iIntros "Hf Hupd".
  rewrite uPred_fupd_unseal /uPred_fupd_def.
  iIntros "[$ $]". iApply (le_upd_later with "Hf").
  iNext. by iModIntro.
Qed.

If the goal is a fancy update, this lemma can be used to make a later appear in front of it in exchange for a later credit. This is typically used as iApply (lc_fupd_add_later with "Hcredit"), where "Hcredit" is a credit available in the context.
Lemma lc_fupd_add_later `{!invGS_gen HasLc Σ} E1 E2 P :
  £ 1 -∗ ( |={E1, E2}=> P) -∗ |={E1, E2}=> P.
Proof.
  iIntros "Hf Hupd". iApply (fupd_trans E1 E1).
  iApply (lc_fupd_elim_later with "Hf Hupd").
Qed.

fupd soundness lemmas

"Unfolding" soundness stamement for no-LC fupd: This exposes that when initializing the invGS_gen, we can provide a general lemma that lets one unfold a |={E1, E2}=> P into a basic update while also carrying around some frame ω E that tracks the current mask. We also provide a bunch of later credits for consistency, but there is no way to use them since this is a HasNoLc lemma.
Lemma fupd_soundness_no_lc_unfold `{!invGpreS Σ} m E :
   |==> `(Hws: invGS_gen HasNoLc Σ) (ω : coPset iProp Σ),
    £ m ω E ( E1 E2 P, (|={E1, E2}=> P) -∗ ω E1 ==∗ (ω E2 P)).
Proof.
  iMod wsat_alloc as (Hw) "[Hw HE]".
  iMod (later_credits.le_upd.lc_alloc m) as (Hc) "[_ Hlc]".
  set (Hi := InvG HasNoLc _ Hw Hc).
  iExists Hi, (λ E, wsat ownE E)%I.
  rewrite (union_difference_L E ); [|set_solver].
  rewrite ownE_op; [|set_solver].
  iDestruct "HE" as "[HE _]". iFrame.
  iIntros "!>!>" (E1 E2 P) "HP HwE".
  rewrite fancy_updates.uPred_fupd_unseal
          /fancy_updates.uPred_fupd_def -assoc /=.
  by iApply ("HP" with "HwE").
Qed.

Note: the _no_lc soundness lemmas also allow generating later credits, but these cannot be used for anything. They are merely provided to enable making the adequacy proof generic in whether later credits are used.
Lemma fupd_soundness_no_lc `{!invGpreS Σ} E1 E2 (P : iProp Σ) `{!Plain P} m :
  ( `{Hinv: !invGS_gen HasNoLc Σ}, £ m ={E1,E2}=∗ P) P.
Proof.
  intros Hfupd. apply later_soundness, bupd_soundness; [by apply later_plain|].
  iMod fupd_soundness_no_lc_unfold as (hws ω) "(Hlc & Hω & #H)".
  iMod ("H" with "[Hlc] Hω") as "H'".
  { iMod (Hfupd with "Hlc") as "H'". iModIntro. iApply "H'". }
  iDestruct "H'" as "[>H1 >H2]". by iFrame.
Qed.

Lemma fupd_soundness_lc `{!invGpreS Σ} n E1 E2 (P : iProp Σ) `{!Plain P} :
  ( `{Hinv: !invGS_gen HasLc Σ}, £ n ={E1,E2}=∗ P) P.
Proof.
  intros Hfupd. eapply (lc_soundness (S n)); first done.
  intros Hc. rewrite lc_succ.
  iIntros "[Hone Hn]". rewrite -le_upd_trans. iApply bupd_le_upd.
  iMod wsat_alloc as (Hw) "[Hw HE]".
  set (Hi := InvG HasLc _ Hw Hc).
  iAssert (|={,E2}=> P)%I with "[Hn]" as "H".
  { iMod (fupd_mask_subseteq E1) as "_"; first done. by iApply (Hfupd Hi). }
  rewrite uPred_fupd_unseal /uPred_fupd_def.
  iModIntro. iMod ("H" with "[$Hw $HE]") as "H".
  iPoseProof (except_0_into_later with "H") as "H".
  iApply (le_upd_later with "Hone"). iNext.
  iDestruct "H" as "(_ & _ & $)".
Qed.

Generic soundness lemma for the fancy update, parameterized by use_credits on whether to use credits or not.
Lemma fupd_soundness_gen `{!invGpreS Σ} (P : iProp Σ) `{!Plain P}
  (hlc : has_lc) n E1 E2 :
  ( `{Hinv : invGS_gen hlc Σ},
    £ n ={E1,E2}=∗ P)
   P.
Proof.
  destruct hlc.
  - apply fupd_soundness_lc. done.
  - apply fupd_soundness_no_lc. done.
Qed.

step_fupdN soundness lemmas

Lemma step_fupdN_soundness_no_lc `{!invGpreS Σ} (P : iProp Σ) `{!Plain P} n m :
  ( `{Hinv: !invGS_gen HasNoLc Σ}, £ m ={,}=∗ |={}▷=>^n P)
   P.
Proof.
  intros Hiter.
  apply (laterN_soundness _ (S n)); simpl.
  apply (fupd_soundness_no_lc _ m)=> Hinv. iIntros "Hc".
  iPoseProof (Hiter Hinv) as "H". clear Hiter.
  iApply fupd_plainly_mask_empty. iSpecialize ("H" with "Hc").
  iMod (step_fupdN_plain with "H") as "H". iMod "H". iModIntro.
  rewrite -later_plainly -laterN_plainly -later_laterN laterN_later.
  iNext. iMod "H" as "#H". auto.
Qed.

Lemma step_fupdN_soundness_no_lc' `{!invGpreS Σ} (P : iProp Σ) `{!Plain P} n m :
  ( `{Hinv: !invGS_gen HasNoLc Σ}, £ m ={}[]▷=∗^n P)
   P.
Proof.
  intros Hiter. eapply (step_fupdN_soundness_no_lc _ n m)=>Hinv.
  iIntros "Hcred". destruct n as [|n].
  { by iApply fupd_mask_intro_discard; [|iApply (Hiter Hinv)]. }
   simpl in Hiter |- ×. iMod (Hiter with "Hcred") as "H". iIntros "!>!>!>".
  iMod "H". clear. iInduction n as [|n] "IH"; [by iApply fupd_mask_intro_discard|].
  simpl. iMod "H". iIntros "!>!>!>". iMod "H". by iApply "IH".
Qed.

Lemma step_fupdN_soundness_lc `{!invGpreS Σ} (P : iProp Σ) `{!Plain P} n m :
  ( `{Hinv: !invGS_gen HasLc Σ}, £ m ={,}=∗ |={}▷=>^n P)
   P.
Proof.
  intros Hiter.
  eapply (fupd_soundness_lc (m + n)); [apply _..|].
  iIntros (Hinv) "Hlc". rewrite lc_split.
  iDestruct "Hlc" as "[Hm Hn]". iMod (Hiter with "Hm") as "Hupd".
  clear Hiter.
  iInduction n as [|n] "IH"; simpl.
  - by iModIntro.
  - rewrite lc_succ. iDestruct "Hn" as "[Hone Hn]".
    iMod "Hupd". iMod (lc_fupd_elim_later with "Hone Hupd") as "> Hupd".
    by iApply ("IH" with "Hn Hupd").
Qed.

Lemma step_fupdN_soundness_lc' `{!invGpreS Σ} (P : iProp Σ) `{!Plain P} n m :
  ( `{Hinv: !invGS_gen hlc Σ}, £ m ={}[]▷=∗^n P)
   P.
Proof.
  intros Hiter.
  eapply (fupd_soundness_lc (m + n) ); [apply _..|].
  iIntros (Hinv) "Hlc". rewrite lc_split.
  iDestruct "Hlc" as "[Hm Hn]". iPoseProof (Hiter with "Hm") as "Hupd".
  clear Hiter.
  iInduction n as [|n] "IH"; simpl.
  - by iModIntro.
  - rewrite lc_succ. iDestruct "Hn" as "[Hone Hn]".
    iMod "Hupd". iMod (lc_fupd_elim_later with "Hone Hupd") as "> Hupd".
    by iApply ("IH" with "Hn Hupd").
Qed.

Generic soundness lemma for the fancy update, parameterized by use_credits on whether to use credits or not.
Lemma step_fupdN_soundness_gen `{!invGpreS Σ} (P : iProp Σ) `{!Plain P}
  (hlc : has_lc) (n m : nat) :
  ( `{Hinv : invGS_gen hlc Σ},
    £ m ={,}=∗ |={}▷=>^n P)
   P.
Proof.
  destruct hlc.
  - apply step_fupdN_soundness_lc. done.
  - apply step_fupdN_soundness_no_lc. done.
Qed.