atomic.v 5.81 KB
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From stdpp Require Import namespaces.
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From iris.program_logic Require Export weakestpre.
From iris.proofmode Require Import tactics classes.
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From iris.bi.lib Require Export atomic.
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From iris.bi Require Import telescopes.
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Set Default Proof Using "Type".

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(* This hard-codes the inner mask to be empty, because we have yet to find an
example where we want it to be anything else. *)
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Definition atomic_wp `{!irisG Λ Σ} {TA TB : tele}
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  (e: expr Λ) (* expression *)
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  (Eo : coPset) (* (outer) mask *)
  (α: TA  iProp Σ) (* atomic pre-condition *)
  (β: TA  TB  iProp Σ) (* atomic post-condition *)
  (f: TA  TB  val Λ) (* Turn the return data into the return value *)
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  : iProp Σ :=
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    ( (Φ : val Λ  iProp Σ),
             atomic_update Eo  α β (λ.. x y, Φ (f x y)) -
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             WP e {{ Φ }})%I.
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(* Note: To add a private postcondition, use
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   atomic_update α β Eo Ei (λ x y, POST x y -∗ Φ (f x y)) *)

Notation "'<<<' ∀ x1 .. xn , α '>>>' e @ Eo '<<<' ∃ y1 .. yn , β , 'RET' v '>>>'" :=
  (atomic_wp (TA:=TeleS (λ x1, .. (TeleS (λ xn, TeleO)) .. ))
             (TB:=TeleS (λ y1, .. (TeleS (λ yn, TeleO)) .. ))
             e%E
             Eo
             (tele_app (TT:=TeleS (λ x1, .. (TeleS (λ xn, TeleO)) .. )) $
                       λ x1, .. (λ xn, α%I) ..)
             (tele_app (TT:=TeleS (λ x1, .. (TeleS (λ xn, TeleO)) .. )) $
                       λ x1, .. (λ xn,
                         tele_app (TT:=TeleS (λ y1, .. (TeleS (λ yn, TeleO)) .. ))
                         (λ y1, .. (λ yn, β%I) .. )
                        ) .. )
             (tele_app (TT:=TeleS (λ x1, .. (TeleS (λ xn, TeleO)) .. )) $
                       λ x1, .. (λ xn,
                         tele_app (TT:=TeleS (λ y1, .. (TeleS (λ yn, TeleO)) .. ))
                         (λ y1, .. (λ yn, v%V) .. )
                        ) .. )
  )
  (at level 20, Eo, α, β, v at level 200, x1 binder, xn binder, y1 binder, yn binder,
   format "'[hv' '<<<'  ∀  x1  ..  xn ,  α  '>>>'  '/  ' e  @  Eo  '/' '[    ' '<<<'  ∃  y1  ..  yn ,  β ,  '/' 'RET'  v  '>>>' ']' ']'")
  : stdpp_scope.

Notation "'<<<' ∀ x1 .. xn , α '>>>' e @ Eo '<<<' β , 'RET' v '>>>'" :=
  (atomic_wp (TA:=TeleS (λ x1, .. (TeleS (λ xn, TeleO)) .. ))
             (TB:=TeleO)
             e%E
             Eo
             (tele_app (TT:=TeleS (λ x1, .. (TeleS (λ xn, TeleO)) .. )) $
                       λ x1, .. (λ xn, α%I) ..)
             (tele_app (TT:=TeleS (λ x1, .. (TeleS (λ xn, TeleO)) .. )) $
                       λ x1, .. (λ xn,
                         tele_app (TT:=TeleO) β%I
                        ) .. )
             (tele_app (TT:=TeleS (λ x1, .. (TeleS (λ xn, TeleO)) .. )) $
                       λ x1, .. (λ xn,
                         tele_app (TT:=TeleO) v%V
                        ) .. )
  )
  (at level 20, Eo, α, β, v at level 200, x1 binder, xn binder,
   format "'[hv' '<<<'  ∀  x1  ..  xn ,  α  '>>>'  '/  ' e  @  Eo  '/' '[    ' '<<<'  β ,  '/' 'RET'  v  '>>>' ']' ']'")
  : stdpp_scope.

Notation "'<<<' α '>>>' e @ Eo '<<<' ∃ y1 .. yn , β , 'RET' v '>>>'" :=
  (atomic_wp (TA:=TeleO)
             (TB:=TeleS (λ y1, .. (TeleS (λ yn, TeleO)) .. ))
             e%E
             Eo
             (tele_app (TT:=TeleO) α%I)
             (tele_app (TT:=TeleO) $
                       tele_app (TT:=TeleS (λ y1, .. (TeleS (λ yn, TeleO)) .. ))
                         (λ y1, .. (λ yn, β%I) .. ))
             (tele_app (TT:=TeleO) $
                       tele_app (TT:=TeleS (λ y1, .. (TeleS (λ yn, TeleO)) .. ))
                         (λ y1, .. (λ yn, v%V) .. ))
  )
  (at level 20, Eo, α, β, v at level 200, y1 binder, yn binder,
   format "'[hv' '<<<'  α  '>>>'  '/  ' e  @  Eo  '/' '[    ' '<<<'  ∃  y1  ..  yn ,  β ,  '/' 'RET'  v  '>>>' ']' ']'")
  : stdpp_scope.

Notation "'<<<' α '>>>' e @ Eo '<<<' β , 'RET' v '>>>'" :=
  (atomic_wp (TA:=TeleO)
             (TB:=TeleO)
             e%E
             Eo
             (tele_app (TT:=TeleO) α%I)
             (tele_app (TT:=TeleO) $ tele_app (TT:=TeleO) β%I)
             (tele_app (TT:=TeleO) $ tele_app (TT:=TeleO) v%V)
  )
  (at level 20, Eo, α, β, v at level 200,
   format "'[hv' '<<<'  α  '>>>'  '/  ' e  @  Eo  '/' '[    ' '<<<'  β ,  '/' 'RET'  v  '>>>' ']' ']'")
  : stdpp_scope.
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(** Theory *)
Section lemmas.
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  Context `{!irisG Λ Σ} {TA TB : tele}.
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  Notation iProp := (iProp Σ).
  Implicit Types (α : TA  iProp) (β : TA  TB  iProp) (f : TA  TB  val Λ).

  Lemma atomic_wp_seq e Eo α β f {HL : .. x, Laterable (α x)} :
    atomic_wp e Eo α β f -
     Φ, .. x, α x - (.. y, β x y - Φ (f x y)) - WP e {{ Φ }}.
  Proof.
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    rewrite ->tforall_forall in HL. iIntros "Hwp" (Φ x) "Hα HΦ".
    iApply wp_frame_wand_l. iSplitL "HΦ"; first iAccu. iApply "Hwp".
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    iAuIntro. iAaccIntro with "Hα"; first by eauto. iIntros (y) "Hβ !>".
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    (* FIXME: Using ssreflect rewrite does not work, see Coq bug #7773. *)
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    rewrite ->!tele_app_bind. iIntros "HΦ". iApply "HΦ". done.
  Qed.
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  (* Sequential triples with a persistent precondition and no initial quantifier
  are atomic. *)
  Lemma seq_wp_atomic e Eo (α : [tele]  iProp) (β : [tele]  TB  iProp)
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        (f : [tele]  TB  val Λ) {HP : Persistent (α [tele_arg])} :
    ( Φ, α [tele_arg] - (.. y, β [tele_arg] y - Φ (f [tele_arg] y)) - WP e {{ Φ }}) -
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    atomic_wp e Eo α β f.
  Proof.
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    simpl in HP. iIntros "Hwp" (Φ) "HΦ". iApply fupd_wp.
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    iMod ("HΦ") as "[#Hα [Hclose _]]". iMod ("Hclose" with "Hα") as "HΦ".
    iApply wp_fupd. iApply ("Hwp" with "Hα"). iIntros "!>" (y) "Hβ".
    iMod ("HΦ") as "[_ [_ Hclose]]". iMod ("Hclose" with "Hβ") as "HΦ".
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    (* FIXME: Using ssreflect rewrite does not work, see Coq bug #7773. *)
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    rewrite ->!tele_app_bind. done.
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  Qed.
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End lemmas.